Multiparameter Equations of State: An Accurate Source of Thermodynamic Property Data

384 Pages·2000·9.4 MB·English

by  Dr.-Ing. Roland Span (auth.)

Span, Multiparameter Equations of State
Springer-Verlag Berlin Heidelberg GmbH
ONLINE LIBRARY
Engineering
http://www.springer.de/engine/
R. Span
Multiparameter
Equations of State
An Accurate Source
of Thermodynamic Property Data
With 151 Figures and 36 Tables
Springer
Dr.-Ing. Roland Span
Ruhr-Universitat-Bochum
Lehrstuhl fiir Thennodynamik
44780 Bochum / Gennany
e-mail: [email protected]
ISBN 978-3-642-08671-7
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ISBN 978-3-642-08671-7 ISBN 978-3-662-04092-8 (eBook)
DOI 10.1007/978-3-662-04092-8
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© Springer-Verlag Berlin Heidelberg 2000
Originally published by Springer-Verlag Berlin Heidelberg New York in 2000
Softcover reprint of the hardcover 1 st edition 2000
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Preface
As a basis for printed property charts and tables, empirical multiparameter equa
tions of state have been the most important source of accurate thermodynamic
property data for more than 30 years now. However, due to increasing demands on
the accuracy of thermodynamic property data in computerised calculations as well
as the availability of appropriate software tools, and the ever increasing computer
power, such formulations are nowadays becoming a valuable tool for everyday
work. This development has substantially increased the number of scientists, engi
neers, and students who are working with empirical multiparameter equations of
state, and it continues to do so.
Nevertheless, common knowledge on this kind of thermodynamic property
models and on the ongoing progress in this scientific discipline is still very limited.
Multiparameter equations of state do not belong to the topics which are taught
intensively in thermodynamic courses in engineering and natural sciences and the
books and articles where they are published mainly deal with the thermodynamic
properties of certain substances rather than with the theoretical background of the
used equations of state. In contrast to this, my concern mainly was to give a survey
of the theoretical background of multiparameter equations of state both with regard
to their application and their development. This book is written for users of multi
parameter equations of state who need some background information for their own
application developments as well as for scientists who are active in the develop
ment of such equations or who want to become involved in it. As the only current
and comprehensive work of this kind, it is intended to improve common knowl
edge on multiparameter equations, to fuel their application, and to attract more
scientists to the work on such thermodynamic property models.
For my own part, I have been involved in the development of multiparameter
equations of state since I joined the research group of Wolfgang Wagner at the
Ruhr-Universitat Bochum as a student almost 15 years ago. After all this time, I
am firstly and above all grateful for all the support I have received from Prof.
Wagner. He is not only an outstanding expert in this field, but also an extraordi
nary personality who has over the years slowly turned from a professor into a col
league - it has always been fun to work in his group. Many of the assessments
given in this book result from intensive discussions with him and the main reason
why this book is authored only by me and not by Span and Wagner is that it is
based on the habilitation thesis which I submitted last year. Secondly, I am grateful
to V. Marx who introduced me to the work on multiparameter equations of state
years ago. I wish I had as much time for my students as he had for me. Besides
this, I would like to thank all those who later worked with me on this topic, namely
VI Preface
A. PruB, Ch. Tegeler, A. Kruse, J. Smukala, M. Kubessa, R. Klimeck, M. Klinne,
H.-J. Collmann, C. Bonsen, D. Blicker, and M. Waschke. In very different ways,
they have all contributed to the scientific foundation of this book. However, work
on multiparameter equations of state is always closely related to experimental
work as well. In place of all the experimentalists who have helped me to under
stand the possibilities and the limitations of experimental work, I would like to
thank R. Kleinrahm, N. Kurzeja, R. Gilgen, and M. Trusler for many fruitful dis
cussions. And finally, I am grateful to E. W. Lemmon not only for the inspiring
discussions which we have had over the course of several co-operative projects but
most of all for turning the manuscript of this book into readable English.
Bochum, February 2000 Roland Span
Contents
Nomenclature ........................................................................................... XI
1 Introduction ........................................................................................... 1
2 History and Potentials - an Overview ................................................. 5
2.1 A Brief History ......................................................................................... 5
2.2 Publications on Thenuodynamic 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
3.1.1 Equations in Tenus of the Helmholtz Energy .............................. 16
3.1.1.1 Correlations for the Helmholtz Energy of the
Ideal Gas ........................................................................ 18
3.1.1.2 Common Fonuulations for the Residual Helmholtz
Energy ............................................................................ 23
3.1.2 Equations in Tenus of Pressure ................................................... 25
3.1.2.1 Integration of Pressure Explicit Equations ..................... 26
3.1.3 Special Functional Fonus ............................................................. 28
3.1.3.1 Hard Sphere Tenus ........................................................ 29
3.1.3.2 Critical Region Tenus .................................................... 31
3.1.4 Simple Equations for Gas Phase Calibrations .............................. 35
3.2 Calculating Thenuodynamic Properties from the Helmholtz 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
VIII Contents
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 Pairwise 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 ......................................... 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
Contents IX
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
7.1.1 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
7.2.1.1 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