ISBN: 3540673113
TITLE: Multiparameter Equations of State
AUTHOR: Span, R.
TOC:

Nomenclature XI
1 Introduction 1
2 History and Potentials - an Overview 5
2.1 A Brief History 5
2.2 Publications on Thermodynamic Reference Data 11
2.3 Current Work and Future Challenges 12
3 Using Multiparameter Equations of State for Pure Substances 15
3.1 Different Kinds of Multiparameter Equations of State 15
3.1.l Equations in Terms of the Helmholtz Energy 16
3.1.1.1 Correlations for the Helmholtz Energy of the Ideal Gas 18
3.1.1.2 Common Formulations for the Residual Helmholtz Energy 23
3.1.2 Equations in Terms of Pressure 25
3.1.2.1 Integration of Pressure Explicit Equations 26
3.1.3 Special Functional Forms 28
3.1.3.1 Hard Sphere Terms 29
3.1.3.2 Critical Region Terms 31
3.1.4 Simple Equations for Gas Phase Calibrations 35
3.2 Calculating Thermodynamic Properties from the Hehnholtz Energy 35
3.2.1 Properties in the Homogeneous Region 36
3.2.2 Properties of the Vapour-Liquid Equilibrium Phases 42
3.2.3 Properties in the Vapour-Liquid Two Phase Region 45
3.3. Iterative Procedures 46
3.3.1 Calculations Based on Temperature and Pressure 48
3.3.2 Calculations Based on Pressure and Density 51
3.3.3 Calculations Based on Pressure and Enthalpy 52
3.3.4 Calculations Based on Pressure and Entropy 54
3.3.5 Calculation of Phase Equilibria 54
3.3.5.1 Multiple Maxwell Loops 56
4. Setting Up Multiparameter Equations of State for Pure Substances 61
4.1 Linear and Nonlinear Fitting 62
4.2 Describing the Helmholtz Energy of the Ideal Gas 66
4.3 Describing the Residual Helmholtz Energy 74
4.3.1 The Multiproperty Approach 74
4.3.2 Defining Residua for Linear Algorithms 77
4.3.3 Defining Residua for Nonlinear Algorithms 80
4.3.4 Assigning Weights to Experimental Data 85
4.4 Optimising the Functional Form 90
4.4.1 Defining a Bank of Terms 91
4.4.1.1 Hard Sphere Terms 93
4.4.2 Setting Up a Regression Matrix 97
4.4.3 The Stepwise Regression Analysis (SEEQ) 98
4.4.3.1 Introduction of a Pairwise Exchange 103
4.4.4 The Evolutionary Optimisation Method (EOM) 106
4.4.5 The Optimisation Algorithm by Setzmann and Wagner (OPTIM) 107
4.4.5.1 Adapting OPTIM to Equations of State 114
4.4.5.2 Introduction of a Pair-wise Exchange 116
4.4.6 The Nonlinear Optimisation Algorithms by Tegeler et al. 117
4.4.6.1 The Nonlinear Quality Criterion 118
4.4.6.2 The Nonlinear Stepwise Regression Analysis (NLREG) 121
4.4.6.3 The Nonlinear Optimisation Algorithm (NLOPT) 125
4.4.6.4 Speeding Up Nonlinear Optimisation Algorithms 126
4.4.7 Automated Optimisation Algorithms 128
4.4.7.1 Optimisation of Simplified Equations for Mixtures with Constant Composition 129
4.4.8 Independent Developments and Future Perspectives 131
4.5 Describing Properties in the Critical Region a 133
4.5.1 Predictions from Theory 134
4.5.1.1 Theoretically Founded Equations of State 138
4.5.2 Capabilities of Empirical Multiparameter Equations of State 141
4.5.3 Setting Up Equations with Modified Gaussian Bell Shaped Terms 151
4.5.4 Setting Up Equations with Nonanalytic Terms 152
4.5.5 Semiempirical Approaches 156
4.5.5.1 The Use of Switching Functions 157
4.5.5.2 The Transformation Approach 158
4.5.5.3 The Approach of Kiselev and Friend 159
4.6 Consideration of the Extrapolation Behaviour 161
4.6.1 Comparisons with Data Beyond the Range of Primary Data 163
4.6.2 The Influence of the Functional Form 166
4.6.3 The Representation of Ideal Curves 168
5 The Performance of Multiparameter Equations of State 173
5.1 Comparisons with Thermal Properties 177
5.2 Comparisons with Caloric Properties 191
5.3 Properties at Vapour-Liquid Phase Equilibrium 206
5.4 Some General Assessments 214
6 Generalised Functional Forms 219
6.1 Simultaneous Optimisation of Functional Forms 223
6.2 Simultaneously Optimised Equations for Technical Applications 227
6.2.1 Accuracy Versus Numerical Stability - A Compromise 235
6.2.1.1 An Investigation of Numerical Stability 237
6.2.1.2 The Influence of Uncertain Critical Parameters 242
6.2.1.3 Conclusions Regarding Required Data Sets 245
6.2.2 Results for Non- and Weakly Polar Fluids 248
6.2.3 Results for Polar Fluids 262
6.3 Simultaneously Optimised Reference Equations of State 275
7 Generalised Equations of State 277
7.1 BACKONE Equations of State 279
7.1.l Fitting BACKONE Equations of State to Data 281
7.1.2 Some Results 285
7.2 Generalised Empirical Equations of State 291
7.2.1 The Approach by Platzer and Maurer 292
7.2.1.l Some Results 293
7.2.2 The Approach by Span and Wagner 300
7.2.2.1 Fitting the Substance Specific Parameters 303
7.2.2.2 Results for Non- and Weakly Polar Fluids 306
7.2.2.3 Numerical Stability 314
8 Describing Mixtures with Multiparameter Equations of State 319
8.1 Using Composition Dependent Sets of Coefficients 320
8.1.1 The AGA8-DC92 Equation of State for Natural Gases 324
8.2 Extended Corresponding States Approaches 327
8.2.1 Interpolation Between Reference Fluids 328
8.2.2 The Shape Factor Concept 329
8.3 Helmholtz Models with Departure Functions 332
8.3.1 Mixture Specific Departure Functions 335
8.3.2 Generalised Departure Functions 336
References 341
Index 363
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