Loughborough University Institutional Repository
The calculation of thermodynamic properties and phase equilibria using a new cubic equation of state
This item was submitted to Loughborough University's Institutional Repository by the/an author.
Additional Information:
• A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.
Metadata Record: https://dspace.lboro.ac.uk/2134/11044
(https://dspace.lboro.ac.uk/) under the following Creative Commons Licence conditions.
For the full text of this licence, please go to: http://creativecommons.org/licenses/by-nc-nd/2.5/
i ('
--- -- - ----- - ---------------- - - -- --- - --- -. - - -_._-- - .... - : • ,ACCESSION/COPY NO.
\ 8 NUV lUUU ---:--- or.' ;e.' .
. "" . .--c--
,-
.. :
,:
j j j j j j j j j j j j j j j j j j
j
j j
AND PHASE EQUILIBRIA USING A NEW CUBIC
EQUATION OF STATE
Navin Chhotubhai Patel, B.Sc. (D.I.S)
A Doctoral Thesis Submitted in Partial Fulfilment of the Requirements for the Award of
Doctor of Philosophy of the Loughborough University of Technology
October 1980
' .. '; , . -..
l.eu!lhborough University
~ 'fq.,.,. 8 I CI.n
ABSTRACT
A new three constant equation of state for fluids and fluid mixtures has been proposed. The equation contains three constants and is a more general form of the Peng-Robinson equation. In addition to the critical temperature and the critical pressure, two parameters are required to characterize a particular fluid. These parameters have been evaluated by minimizing the deviation
in the saturated liquid densities while simultaneously satisfying the equilibrium criteria (equality of fugacities) along the satu­ ration curve. Thus, prediction of volumetric properties in the
saturation region and other regions of the phase diagram is improved, while accuracy in the prediction of vapour-liquid equilibrium is maintained. Parameters for hydrocarbons and non­
hydrocarbons of importance to the synthetic fuel industry are presented. The new equation is extended to mixtures by using mixing rules similar to those used by Peng and Robinson. Only one binary interaction coefficient is sufficient for the accu­ rate prediction of vapour-liquid equilibria. Optimum values of the binary interaction coefficients for hydrocarbon-hydrocarbon and hydrocarbon-non-hydrocarbon systems have been obtained using the new equation, the Peng-Robinson equation and the Soave modi­
fication of the Redlich-Kwong equation. The criterion used for selecting the optimum interaction coefficient is the minimization of deviations in bubble point pressures. The new equation has been
tested for the prediction of volumetric behaviour of pure fluids and the phase and volumetric behaviour of binary, ternary and multicomponent systems. Comparisons with conventional equations (P-R, S-R-K, R-K and B-W-R-S) are shown.
The applicability of van-der-Waals orie fluid model to _ generalised equations of state is demonstrated. The equations
of Han and Starling, Simonet and-Behar and Chaudron et al have been compared for volumetric predictions. The Han and Starling equation has also been used with two sets of mixing rules to predict
vapour-liquid equilibrium. The van-der-Waals one fluid model has been
i i
shown to be a simple and effective method of applying the compli­ cated equations of state to the prediction of thermodynamic properties of mixtures.
A generalised form of the corresponding states principle using two non-spherical reference fluids is presented. This represents an alternative method of extending the equation of state approach to fluids and fluid mixtures when experimental data are not available.
i i i
ACKNOWLEDGEMENTS
I should like to express my sincere appreciation to the following:
OR A S TEJA for his unfailing friendship, encouragement
and patient supervision of all the work presented in this thesis and for his invaluable guidance and help.
MR R H BERESFORD, MR J GLOVER and OR P RICE for their
helpful discussions.
Staff of the Advisory Office of the University Computer Centre, for their constant advice, guidance and encouragement.
Mrs Janet Smith for typing this thesis.
All my friends who, in one way or another, contributed towards
the making of this thesis, and above all my brother and parents for their years of patience and encouragement.
iv
TITLE PAGE ABSTRACT ACKNOWLEDGEMENTS CONTENTS LisT OF PRINCIPAL SYMBOLS AND ABBREVIATIONS
CONTENTS
1. INTRODUCTION 2. LITERATURE REVIEW: EQUATIONS OF STATE AND THE
CORRESPONDING STATES PRINCIPLE 2.1 Equations of State
2.1.1 The Ideal Gas Equation of State 2.1. 2 The van der Waals Equation of State
(1873) · .... 2.1.3 The Virial Equation of State 2.1 .4 The Redlich-Kwong Equation of State
(1949) · .... 2.1 .5 Modifications of the Redlich-Kwong
Equa ti on 2.1.6 A Modified Redlich-Kwong Equation by
Chaudron, Asselineau and Renon (1973) 2.1. 7 A Modified Redlich-Kwong Equation by
Simonet and Behar (1976) 2.1.8 The Soave-Redlich-Kwong Equation (1972) 2.1.9 The Peng and Robinson Equation of State
(1976 ) · .... 2.1.10 A Modified Peng and Robinson Equation of
State by Heyen (1980)
34 2.1.11 A Modified van der Waa1s Type Equation of
38 State by Schmidt and Wenzel (1980) 2.1.12 The Han-Star1ing Equation of State (1972) 41
2.2 The Corresponding States Principle 46
2.2.1 CSP and Two-Parameter Equations of State 48 2.2.2 Extension of the CSP 49 2.2.3 Extension to ~li xtures 51
3. CHOICE OF AN EQUATION OF STATE 56
4. A NEW CUBIC EQUATION OF STATE 61
v
4.1 Evaluation of the Parameters 4.2 Pure Component Calculations
4.2.1 Density 4.2.2 Vapour Pressure •.•.• 4.2.3 The Enthalpy Departure 4.2.4 The Entropy Departure
4.3 Extension to New Substances 4.4 Extension to New Substances via the Genera­
lized Corresponding States Principle 4.5 Comparison of the Methods Used for Genera­
. 1 i zi ng the Equati on of State 4.6 Use of the Exponential Temperature Function
5. PHASE EQUILlBRIUH IN MIXTURES 5.1 Thermodynamic Considerations
6. CALCULATION OF MIXTURE PROPERTIES 6.1 Extension of the New Equation of State to
Mixtures 6.2 Evaluation of the Binary Interaction Coeffi-
Page 62 69
cient 125 6.3 Vapour-Liquid Equilibrium (VLE) Calculations 130
6.3.1 Calculation of the Bubble Point Pressure 130
'6.3.2 Calculation of the Bubble Point Temperature 134 6.3.3 Calculation of the Equilibrium Ratios
(K values) 6.3.4 Effect of Temperature and Pressure on
Sijopt
6.4 Calculation of the Mixture Density 6.5 Prediction of Saturated Liquid Volumes
.7. THE VAN DER WAALS ONE FLUID MODEL AND GENERALIZED EQUATIONS OF STATE ..... 7.1 Mixture Density 7.2 VLE Calculations
7.2.1 Predictions of Bubble Point Pressure
vi
134
135
135
137
156
157
157 158
7.2.2 Predictions of the Equilibrium Ratios 7.2.3 Volumes and Fugacities of Co-existing
Phases ..... 7.3 Comparison of Computer Time
8. SUMMARY OF RESULTS AND CONCLUSIONS 8.1 Conclusions 8.2 Suggestions for Future Work
GRAPHS (Figures 6.1 to 6.113) REFERENCES REFERENCES OF SATURATION DATA REFERENCES OF VOLUMETRIC DATA REFERENCES OF VAPOUR-LIQUID EQUILIBRIUM DATA
APPENDICES:
A. Parameters for the New Equation of State B. Critical Properties of Pure Components.
Conversion Factors.
C. Constants Used in the Equations of State. Generalized Coefficients d.(K) 'and g.(K) in Chaudron Equation 1 1
Values of Parameters A. and B J . in the Generalized
Han and 'Starling Equatlon Optimized Coefficients for the Chaudron Equation Individually Fitted Constants for the Han and Star- 1 ing Equation
Page 159
161 161
329 331 332
333
334
335
336
D. . Analytic Solution of A Cubic Equation. 337 E. Expressions for the Enthalpy and the Entropy Depar- 340
tures. F. Calculation of Pure Component Fugacity. 347
Fugacity of Component i in a Fluid Mixture. 349 ~ Fugacity of the ith component in a Fluid Mixture Using
Han and Starling Equation. 359 Differentials of Pseudo Critical Constants.
G. Algorithms for Vapour-Liquid Equilibrium Calculations. Prediction of the Vapour Pressure. Prediction of the Bubble Point Pressure. Prediction of the Bubble Point Temperature. Prediction of tbe Equilibrium Ratios.
H. Listings of Computer Programs.
367
371
372
Roman Capitals
A Molar Helmholtz free energy. Constant in Equation of State; may be Function of Temperature.
B Second Virial Coefficient Constant in Equation of State; may be Function of Temperature.
C Third Virial Coefficient. B i·nar:y-I·nteraet-ion-Parameter. Constant in Equation of State; may be Function of Temperature. S pe&-i-f ic-Heat-Gapa c.i.ty .. -
D Constant in Equation of State.
E Constant in Equation of State; may be Function of Temperature.
F Constant in Equation of State; may be Substance Dependent.
~ Molar Gibbs Free Energy.
H Molar Enthalpy. Henry~s_ ~0.I1?tant.
K Constant in Equation of State; may be Substance Dependent. [qui.] ibr·ium-Ra ti o.
M Constant
N Constant
S Molar Entropy. Sum to be Minimized.
T Absolute Temperature.
<R) Mole Fraction in Liquid Phase. Variable.
LV) Mole Fraction in Vapour Phase. Variable.
Z Compressibility Factor. Parameter in the New Equation of State. Feed Composition.
Roman Lower Case
a Constant in Equation of State; may be Function of Temperature.
b Constant in Equation of State.
c Constant in Equation of State.
d Optimized- Coefficient in Chaudron Equation. Constant in Equation of State
e Constant in Equation of State.
ix
t8 Binary Interaction Parameter.
m Constant; may be Substance Dependent..
n No. of moles; No. of Components; No. of Carbon Atoms;
No. of Data Points; Constant.
q Constant. A Thermodynamic Function.
(:D Cons tant.
&' Constant in Equation of State.
x Mole Fraction in Liquid Phase. Vari ab le.
y ·Mole Fraction in Vapour Phase.
Variable.
Gree k Symbo 1 s
a Riedel Parameter. Constant in Equation of State; may be Function of Temperature.
S Constant in Equation of State; may be Function of Temperature .
. y Constant in Equation of State.
Activity Coefficient.
8 Substance Dependent Constant.
p Molar Density.
w Pitzer's Acentric Factor.
ab Ratio of the values corresponding to a and b.
c Critical property.
CRK Critical value from Redlich-Kwong Equation.
. CvdW Critical value from van der Waals Equation .
cm Pseudo Critical Property.
xi
n Constant Composition.
o Constant Value.
opt Optimum Value.
P Constant Pressure.
r Reduced Property.
s . Saturation Property.
S Constant Entropy.
T Constant Temperature.
Roots of Cubic EquationjUsed to Identify Different Constants.
Tota 1 Quantity.
CH Refers to Hydrocarbon.
i ,j, k Optimi zed Coeffi ci ents in Chaudron Equa tion.
L Li qui d Phase.
o Reference State; Simple Fluid; Standard State; Ideal Gas State.
r Reference Fl ui d.
rl, r2 Reference Fluid 1, Reference Fluid 2.
S Solid Phase.
Partial Molar Quantity.
Used to Differentiate between two Quantities Represented by the Same Symbol. Differential of Given Quantity.
Reference Fluids in Generalized Corresponding States Principle. Generalized Coefficients in Simonet-Behar Equation.
Abbreviations
P-V-T
~R-K
vdW
VLE
Modified Peng and Robinson Equation (the New Equation presented in this work).
Peng and Robinson Equation.
INTRODUCTION
Knowledge of accurate thermodynamic properties is an essenti a 1 requi rement of desi gn work in the chemi ca'l i ndus try. Failure of a particular design can often be directly attributed to a lack of accurate data.
The design of experiments to provide thermodynamic data
is a cumbersome and often inaccurate process. It is also difficult to reproduce in a small scale laboratory experiment the conditions that will prevail in an actual industrial set~ up. These difficulties have contributed to the formulation of different methods for the prediction of unmeasured properties.
Currently available methods of correlation and prediction can be divided into three main categories:
Acti vity coeffi ci ent methods are_ at- best, empil:.i.ca t cpr:re- _ -lations of data and -are -nofconsidered further in this work. -- '
Of the first two methods, th_eCorresp-onding States Principle is ~ --- - ----. - .... -.- -. -- - --------------- """-
the oldest_and within cer:tain limits, has the strongest theore-, -. - - -- - - , - ... - - - - . - . - - tical basis. Historically, the CSP was introduced by van der Waals (1873), and it can be said that the principle is the most useful by product of the van der Waals equation of state. ,Qne _. of- the limita ti ons of the_ ~ta ti sti ca 1 mechani ca 1 CSP is its
----- -- restriction -to simple pure substances. Many proposals have been made for overcoming these difficulties (Rowlinson, 1954; Pitzer,
1939; Riedel, 1954; Viswanath and Su, 1965). The most widely used method of extending the sets of substances over which the
(extended) principle can be used is the introduction of the
1
\ ,
acentric factor by Pitzer (1939). The CSP can also be extended
to mixtures by means of a one-fluid model whereby the properties of a given mixture can ·be represented by those of a single
~hypothetical pure fluid.
\
Equations of state can either be theoretical or semi­ theoretical. Theoretical equations are based on either kinetic theory or statistical mechanics involving intermolecular forces while semitheoretical equations combine theory with a limited amount of experimental data. Semi theoretical equations ·of state have had the greatest success in representing data and in the calculation of derived properties.
When combined with appropriate thermodynamic relations, a well behaved equation of state can.give accurate predictions of
isothermal changes in heat capacity, enthalpy, entropy, fugacity, vapour pressure, latent heat of vaporization, activity coefficients
and vapour-liquid equilibrium'(VLE) in mixtures. The main advan­ tage of using an equation of state procedure compared to the method using activity coefficients for VLE calculations, is that it avoids defining different reference states for the activity coefficients when one or more components are supercritical and, therefore, non existent as liquids at the given pressure an~ temperature. Equations of state on the other hand, may be applied to gases, liquids and solids, whether pure or mixed. For the
\ above reasons, the search for simple and accurate equations of
~state continues to be of importance.
\
Since the time of van der Waals (1873), a large number of equations of state have appeared in the literature. These have ranged from simple expressions with one or two constants to compli­ cated forms with more than 50 constants. The longer equations have been uti 1 i zed for hi gh preci si on work and one can fi nd in the 1 it­
erature many interesting and useful forms such as the. Bene~i~~-Webb~ RubMin (1940), the Strobr'idge (1962), the Virial form of Onnes (1901), ~--,.. ""--- -------~ ~ - ~- .
the Martin-Stanford (1974) and the Han-Starling (1972) equations. '~-..- ......... - - ---'-
Although these long and complex equations are desirable for the
.2
dynamic properties. they are not generally preferred for involved thermodynamic calculations such as those for vapour pressures. latent heat of vaporization and multiCQmponent vapour-liquid equilibrium ratios because they require tedious manipulation and excessive computer storage in lengthy iterative calculations. The application of these long equations is frequently limited by
the inaccessibility of the constants for the PVT conditions of interest. Establishing these constants is usually a lengthy process. requiring curve fitting. careful weighting of the experimental data and trial and error procedures. What appears
to be limiting is the fact that extensive PVT data of the sub­ stance must be available in order to begin establishing these constants. In addition. some thermodynamic properties. such as the' vapour pressure. must also be known.
The attractiveness of the shorter equations of state lies in their simplicity of calculation. Many of these short equa­ tions are cubic in volume and these include such well known forms
as the v~deLWaals_(187}1.Clausius (1880). Berthelot (1899). Redlich-Kwong (1949). Wilson (1964). Soave (1972). Chaudron et ----- -"""'- _. - -.- - ...... --_.- -- "
a 1 (1973). STiiiOnet-Behar (1976). Peng-Robi nsoQ (1976) and ... -- .
Fuller. (1976). To increase the accuracy of prediction. the cubic equations are often designed for the calculation of a specific thermodynamic property. No cubic equation has yet been found which can represent both low and high pressure volumetric beha­ viour and give accurate prediction of VLE. One of the most successful of the cubic equations has been the modified Redlich­
Kwong equation proposed by Peng and Robi nS9n -<1976). This equa- -"---"-- --- - - --
tion has been used with considerable success for the prediction of vapour-liquid equilibrium (Peng and Robinson. 1976; API. 1977). three phase equil ibria (Peng and Robinson. 1976) and critical states of multi component mixtures (Peng and Robiris,on. 1977). Although the Peng and Robinson equation gives very good repre­ sentation of VLE. the liquid phase density predicted by this , equation is not reliable-j' One of the objectives of this research -
3
was therefore to modify the Peng and Robinson equation in order
to improve the prediction of liquid phase density while main­ taining the accuracy in the prediction of VLE.
A second objective was to examine alternative ways of extending the resulting equation to the calculation of phase
equilibria and phase densities of mixtures._)jmp~~~n..s __ of state may be extended to mixtures either by means of the van
r-d~r Waals one fluid model and the extended corresponding states P?lnciple (Teja, 1980; Teja and Sandler, 1980; Teja, Sandler
/and Patel, 1980) or by means of arbitrary mixing rules whereby their constants are repl~ced by composition dependent terms. Both approaches have been followed in this work. Wherever possible, comparisons with other equations and with alternative approaches will be shown and the relative advantages of each method will be outlined.
4
LITERATURE REVIEW: EQUATIONS OF STATE AND THE CORRESPONDING STATES PRINCIPLE
2.1 Equations of State
An equation of state attempts to express in an analytical
form the relationship existing between pressure (P) - Volume (V) - Temperature (T) and composition (x or y) in the absence of special field forces. Via this relationship for pure components and mixtures with fixed composition, it becomes possible to express an otherwise complicated three dimensional relationship or surface in terms of two dimensions by keeping one variable constant and varying the other two. The general form of this relationship is given by:
f [P, V , T ,xl = 0 (2. 1 )
Two approaches to the development of an equation of state have been followed in the past. One is the theoretical approach based on either kinetic theory or statistical mechanics involving intermolecular forces. This is exemplified by such equations as those of van der Waals (1873), Lennard-Jones and Devonshire (1937),
Flory, Orwall and Vrij (1964), Carnahan and Starling (1972), Beret and Prausnitz (1975) and De Santis, Gironi and Marrelli (1976). The other approach is empirical or at best only semi­ theoretical. This is exemplified by equations such as those of
Clausius (1880), Beattie and Bridgeman (1928), Benedict, Webb and Rubin (1940), Redlich and Kwong (1949), Bender (1973), Han and Starling (1972) and Peng and Robinson (1976).
The widespread activity in the field of equations of state over an extended period of time since the introduction of the van der Waals equation can be attributed to the exceptional power
5
and utility of an equation of state. In addition to P-V-T data smoothing and interpolation, an equation of state can be used
to calculate derived thermodynamic properties through various thermodynamic relationships. When combined with appropriate thermodynamic relations, a well behaved equation of state can predict with high precision thermodynamic properties commonly needed in process engineering, such as:
1) Isothermal entropy and entha1py departures.
2) Heats of mixing. 3) Isothermal changes in heat capacities. 4) Fugacities of liquid and vapour phases. 5) Vapour pressures. 6) Densities. 7) Latent heats of vaporization.
But the utility of an equation of state extends far beyond the
the prediction of the above-mentioned thermodynamic properties. An equation of state even offers assistance in transport property calculations. Perhaps the most important industrial applications of equations of state lie in phase-equilibrium predictions and in calculations listed below.
a) Vapour-liquid equilibrium - bubble point, dew point and flash calculations.
b) Liquid-liquid equilibrium calculations.
d) Simultaneous phase and chemical reaction equi1ibria.
Even the simple operations of compression and expansion of gases and. liquids often involve vapour-liquid equilibrium.
Equations of state have been used for the prediction of
critical points and…