Thermodynamic Properties from Cubic Equations of State
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. 103
vn Figure 4.33 Region III : Reduced Gibbs energy departure functions of n-octane (Tr=1.76) and ethanol (Tr=1.95) versus reduced pressure 103
Figure 4.34 Region III : Reduced internal energy departure functions of n-octane (Pr = 4.03) and ethanol (Pr=1.63) versus reduced temperature. 104
Figure 4.35 Region III : Reduced internal energy departure functions of n-octane (Tr=1.76) and ethanol (Tr=1.95) versus reduced pressure 104
Figure 4.36 Region III : Fugacity coefficients of n-octane (Pr = 4.03) and ethanol (Pr=1.63) versus reduced temperature 105
Figure 4.37 Region III : Fugacity coefficients of n-octane (Tr=1.76) and ethanol (Tr=1.95) versus reduced pressure 105
Figure 4.38 Region IV : Compressibility factors of n-heptane (Pr = 0.73) and methanol (Pr = 0.63) versus reduced temperature 116
Figure 4.39 Region IV : Compressibility factors of n-heptane (Tr = 0.67) and methanol (Tr = 0.70) versus reduced pressure 116
Figure 4.40 Region IV : Reduced enthalpy departure functions of n-heptane (Pr=0.73) and methanol (Pr = 0.63) versus reduced temperature 117
Figure 4.41 Region IV : Reduced enthalpy departure functions of n-heptane (Tr = 0.67) and methanol (Tr = 0.70) versus reduced pressure 117
Figure 4.42 Region IV : Reduced entropy departure functions of n-heptane (Pr=0.73) and methanol (Pr = 0.63) versus reduced temperature 118
Figure 4.43 Region IV : Reduced entropy departure functions of n-heptane (Tr=0.67) and methanol (Tr = 0.70) versus reduced pressure 118
Figure 4.44 Region IV : Reduced Helmholtz departure functions of n-heptane (Pr = 0.73) and methanol (Pr = 0.63) versus reduced temperature 119
Figure 4.45 Region IV : Reduced Helmholtz departure functions of n-heptane (Tr=0.67) and methanol (Tr=0.70) versus reduced pressure 119
Figure 4.46 Region IV : Reduced Gibbs energy departure functions of n-heptane (Pr=0.73) and methanol (Pr = 0.63) versus reduced temperature. 120
Figure 4.47 Region IV : Reduced Gibbs energy departure functions of n-heptane (Tr = 0.67) and methanol (Tr = 0.70) versus reduced pressure. 120
Figure 4.48 Region IV : Reduced internal energy departure functions of n-heptane (Pr = 0.73) and methanol (Pr = 0.63) versus reduced temperature. 121
Vlll Figure 4.49 Region IV : Reduced internal energy departure functions of n-heptane (Tr = 0.67) and methanol (Tr = 0.70) versus reduced pressure. 121
Figure 4.50 Region IV : Fugacity coefficients of n-heptane (Pr = 0.73) and methanol (Pr = 0.63) versus reduced temperature 122
Figure 4.51 Region IV : Fugacity coefficients of n-heptane (Tr = 0.67) and methanol (Tr = 0.70) versus reduced pressure 122
Figure 4.52 Inversion curve of methane calculated from the SRK, PR, LM, and GCP equations 128
Figure 4.53 Inversion curve of propane calculated from the SRK, PR, and LM equations 129
Figure 4.54 Inversion curve of n-butane calculated from the SRK, PR, and LM equations 130
Figure 4.55 Inversion curve of carbon monoxide calculated from the SRK, PR, LM, and GCP equations 131
Figure 4.56 Inversion curve of carbon dioxide calculated from the SRK, PR, and LM equations 132
Figure 4.57 Inversion curve of ethylene calculated from the SRK, PR, LM, and GCP equations 133
Figure 4.58 Inversion curve of p-hydrogen calculated from the SRK, PR, and LM equations 134
Figure 4.59 Inversion curve of ammonia calculated from the SRK, PR, and LM equations 135
Figure 4.60 Inversion curve of argon calculated from the SRK, PR, LM, and GCP equations 136
Figure 5.1 Second virial coefficients of n-butane and benzene versus reduced temperature 148
Figure 5.2 Second virial coefficients of n-octane and argon versus reduced temperature 149
ix LIST OF TABLES
Table 3.1 Calculated and Fitted p and q Constants 39
Table 3.2 Vapor Pressure RMS % Error 41
Table 3.3 Acentric Factor from the Proposed, Lee-Kesler, and Edmister Correlations 43
Table 4.1 Summary of Physical Properties 51
Table 4.2 Region I - Vapor Pressure RMS % Error 53
Table 4.3 Region I - Liquid and Vapor Volume RMS % Errors 55
Table 4.4 Region I - Enthalp3f and Entropy of Vaporization : Average Absolute Deviation 57
Table 4.5 Overall Average Errors (RMS % and AAD) for the Four Regions
(NC = number of compounds ; N = number of data points) 59
Table 4.6 Coefficients of Ideal Gas Enthalpy Polynomial 75
Table 4.7 Coefficients of Ideal Gas Entropy Polynomial 75
Table 4.8 Region II - Volume RMS % Error 76
Table 4.9 Region II - Pressure and Temperature RMS % Errors 77
Table 4.10 Region II - Enthalpy and Entropy Departure Functions : Average Absolute Deviation 78 Table 4.11 Region II - Helmholtz and Gibbs Free Energy Departure Functions : Average Absolute Deviation 79
Tbale 4.12 Region II - Internal Energy Departure Functions and Fugacity
Coefficient : Average Absolute Deviation 80
Table 4.13 Region II - Isobaric Heat Capacitj' : Average Absolute Deviation ...81
Table 4.14 Region II - Joule-Thomson Coefficient RMS % Error 81
Table 4.15 Region III - Volume RMS % Error 93
Table 4.16 Region IU - Pressure and Temperature RMS % Errors 94
Table 4.17 Region III - Enthalpy and Entropy Departure Functions : Average Absolute Deviation 95
x Table 4.18 Region III - Helmholtz and Gibbs Free Energy Departure Functions : Average Absolute Deviation 96
Tbale 4.19 Region III - Internal Energy Departure Functions and Fugacity
Coefficient : Average Absolute Deviation 97
Table 4.20 Region III - Isobaric Heat Capacity : Average Absolute Deviation .. 98
Table 4.21 Region III - Joule-Thomson Coefficient RMS % Error 98
Table 4.22 Region IV - Volume RMS % Error 110
Table 4.23 Region IV - Pressure and Temperature RMS % Errors Ill
Table 4.24 Region IV - Enthalpy and Entropy Departure Functions : Average Absolute Deviation 112 Table 4.25 Region IV - Helmholtz and Gibbs Free Energy Departure Functions : Average Absolute Deviation 113
Tbale 4.26 Region IV - Internal Energy Departure Functions and Fugacity
Coefficient : Average Absolute Deviation 114
Table 4.27 Region IV - Isobaric Heat Capacity : Average Absolute Deviation 115
Table 4.28 Inversion Pressure RMS % Error 127
Table 4.29 Maximum Reduced Inversion Pressure and Temperature 127
Table 5.1 Second Virial Coefficient : Average Absolute Deviation (cc/mole) 147
xi ACKNOWLEDGEMENT
My sincere thanks to professor Janis Lielmezs for his guidance, his patience, and his support in carrying out this project and his assistance in the preparation of this manuscript.
I also wish to thank the University of British Columbia for a Graduate
Scholarship. Thanks are also due to the financial aid provided by the Natural
Sciences and Engineering Research Council of Canada.
xn CHAPTER 1. INTRODUCTION
Physical and thermodjmamic properties calculations have always been an essential part of engineering in the process industries. Many methods have been proposed and used in past years. The older methods have usually relied on charts or graphs. For the process engineer, the task has been time consuming and tedious. Accuracy of the results were usually limited by the graph or chart used. Due to the advent of computers, routine calculations such as multicomponent vapor-liquid equilibria based on state equations can now be done in minutes rather than hours. Hence, the demand for an accurate equation of state has increased in the past decade.
An equation of state (EOS) refers to the equilibrium relation of state parameters such as pressure, volume, temperature, and composition. In functional form this relation is
f(P,V,T,x) =0 (L1)
Such an equation of state may be applied to gases, liquids, and solids.
Recent advances in computers have permitted wide spread efforts in finding an equation of state which is simple to use and yet accurate enough for most engineering calculations. Two approaches become evident in the literature: theoretical and semi-empirical. When the theoretical approach is still in its infancy stage, the semi-empirical approach is usually preferred and enjoys the
1 2 greatest success.
The so-called semi-empirical equation of state refers to an equation which has a limited theoretical framework. This form of the equation is usually based on the corresponding states principle which states that similar thermodynamic behavior is possessed by different chemical substances when compared at the same reduced conditions. The constants of the semi-empirical equation are, in many cases, obtained from experimental data. There are two types of semi-empirical equation of state of general importance : multi-parameter and cubic.
The term "multi-parameter" equation of state implies an equation with more than five constants. These constants are usually substance-dependent.
Examples of this type are the equations of Beattie-Bridgeman [5],
Benedict-Webb-Rubin [6,7], Lee-Kesler [59], and Starling [114,115]. This type of equation is usually reserved for highly accurate work. To define the constants, vast amount of input data are needed for multiproperty regression routine. The
Lee-Kesler equation is the only exception which has a set of generalized constants for hydrocarbons. However, no constants have yet been developed for polar substances for the Lee-Kesler equation.
Cubic equation of state, on the other hand, refers to an equation which, if expanded, is cubic in volume or cubic in compressibility factor. Many of the common two-constant cubic equations of state can be expressed by the general form suggested by Schmidt and Wenzel [103] : 3 RT a(T) p = (1.2) V-b V2+ubV+wb2
From here on Eq.(1.2) is known as the generalized cubic equation of state. With different u and w values, most of the well-known cubic EOS can be reproduced: for instance when u = 0 and w = 0, Eq.(1.2) becomes the van der Waals equation
[125]; with u=l and w = 0, Eq.(1.2) becomes the various versions of the
Redlich-Kwong equation (RK) [93], and assigning u = 2 and w = -l, the
Peng-Robinson (PR) [87] equation is reproduced.
One of the reasons for the lack of acceptance of multi-parameter state equations in volume calculation is the use of a time consuming trial-and-error procedure. In many instances, the initial guess must be close to the solution in order to have convergence. The cubic equation, on the other hand, can be solved analytically, even though numerical techniques are often used. The constants required are readily attainable from minimal input data. In many of the well-accepted cubic equations only critical properties and the acentric factor are necessary to define the constants. The Soave modification of the RK equation
(SRK) [110] and the PR equation are such equations.
Since the introduction of the SRK and PR equations in 1972 and 1976, respectively, their capabilities have well been recognized by various industries
[2,16,32,33,43,74,86]. One simple reason for their wide acceptance in calculating fluid thermodynamic properties is due primarily to their simplicity and generality combined with reasonable accuracy. However, both of these equations represent polar substances poorly. Lielmezs and Merriman (LM) [64] realized this limitation 4 and proposed a modified PR equation. They claimed that their modification has improved significantly vapor pressure prediction and slightly better saturated liquid compressibility than the original PR equation. In addition, the LM equation can be applied to polar compounds. Mak [70] further modified the LM equation so that it can be used for volume calculations in the subcritical region. The accuracy of the resulting modification is comparable to that of the multi-parameter Lee-Kesler equation. However, the modification needed experimental PVT data as the input data in order to determine the required constants and therefore limited its application. Although improvements have been made by Lielmezs and Merriman, their equation cannot be used in the region outside of the critical isotherm on the PVT surface.
The purpose of this work is, therefore, twofold: 1) to extend the capabilhVy of the modified PR equation by Lielmezs and Merriman to the entire PVT surface and 2) to test the modification for various property predictions. Since the
SRK and PR equations are popular in industries, it is of interest to test these two equations along with the present work in property predictions that have never been tested before. This work is limited to pure compound properties. It is important to study pure substances because an equation of state that is unable to adequately represent properties of a substance in the pure state, cannot, in general, be expected to handle accurate^ mixtures containing the same substances.
The PVT surface is divided into four regions: saturation, subcritical, supercritical, and compressed. The subcritical region refers to the area above the 5 saturated vapor curve and beneath the critical isotherm. The supercritical region is the area above the critical isotherm and the compressed liquid region is the area under the the critical isotherm and above the saturated liquid curve. These regions are shown in Figure 1.1. The properties included in the testing for the saturation region are: vapor pressure, vapor and liquid volumes, and enthalpy and entrop.y of vaporization. In the subcritical and supercritical regions, the properties included are: pressure, volume, temperature, departure functions, fugachty coefficient, isobaric heat capacity and Joule-Thomson coefficent. In the compressed liquid region, the properties are basically the same as those in the subcritical and supercritical states except Joule-Thomson coefficient that has not been included due to the lack of experimental data. In addition to the above properties, the inversion curve and the second virial coefficient are also included in this work. Tc
Region III : Supercritical
L Critical VPoint
Region IV : / \\ Region II : Compressed/* \\ Subcritical Liquid / *
\Region I :/ V\\ Saturation Curve \ \
Volume
Figure 1.1 Division of the Pressure-Volume surface. CHAPTER 2. LITERATURE REVIEW
Ever since van der Waals [125] first proposed his equation of state more than a centur\' ago, scores of modifications of his equation have been made.
Most of these equations have been empirical and arbitrary with parameters that are adjustable to fit certain kinds of experimental data such as vapor pressure, liquid or vapor volume. Literature on this subject has grown to large proportions.
A number of reviews have been published which addressed the merits and limitations of some of these modifications. Tsonopoulos and Prausnitz [124] reviewed a number of equations that are important for engineering purposes.
Shah and Thodos [104] pointed out the advantages of cubic equations over multi-parameter equations. Martin [71] discussed some of the essential features an equation of state should possess. The following review will be confined to cubic equation of state of the van der Waals type and is by no means exhaustive.
This review will focus on three areas: 1) what approaches have been adopted in past j'ears for improving the van der Waals t3'pe equation, 2) what properties and t3'pes of compounds have been tested and, 3) what are the limitations of these modifications.
The van der Waals equation may be written as the sum of two terms:
P " PR + PA (2.1)
where PR is the repulsion pressure and is the attraction pressure. Practically all cubic equations of state of the semi-empirical t3'pe uses the van der Waals' 8 hard sphere equation to represent the repulsion pressure:
RT P = (2.1a) R V-b
while the attraction pressure can be expressed by the following equation:
a(T) (2.1b) V*+ubV+wb2
Most modifications are based on modifj'ing the attraction term by 1) proposing a different temperature dependent function for a(T) and/or, 2) using different u and w values. One of the most successful modification of the van der Waals equation is the two-parameter equation proposed by Redlich and Kwong in 1949 [93] where the attraction pressure is expressed as
a(T) (2.2) V(V+b)
and the two constants a and b are defined by Eq.(2.2a) to (2.2c)
a(T) = fla a (2.2a)
a = T -0.5 (2.2b) r
b(T) = % — (2.2c) 9
The two constants Sl^ and fl^ at the critical point have numerical values of
0.42748 and 0.08664, respectively. The derivations of Eq.(2.2a) to (2.2c) can be found in Appendix A. Although predictions of vapor phase properties have been improved, liquid phase representations were still poor.
After the work of Redlich and Kwong, Wilson [127,128] proposed that the attraction constant "a" be made temperature dependent. The temperature dependence is established from vapor pressure data at reduced temperature of 0.7 and 1.0. However his proposed equation did not gain much attention due to lack of interest in cubic equations at that time.
Instead of making the constant "a" temperature dependent, Chueh and
Prausnitz [14,15] proposed a modified RK equation with two sets of
substance-dependent constants Sl& and fl^, one set for the vapor phase and another set for the liquid phase, which can be calculated from volumetric data.
The accuracy from the modified equation in estimating both vapor and liquid volumetric properties has been increased, but the equation lacks internal consistence. Joffe et al [50,130] recognized this problem and modified the
Chueh-Prausnitz equation by using only one set of constants which were fitted with saturated liquid volume data. Similar to its predecessor, the equation is not a generalized correlation. That is, experimental volumetric data are the necessary input data in order to establish the two constants.
The acceptance of a cubic instead of a multi-parameter equation of state
for industrial application was due to the efforts of Soave [110] who proposed a 10 modified RK equation. This modified equation was generalized, simple to use and its range of application exceeded all the other modifications. The necessary input data for this modification are the critical pressure and temperature and acentric factor, (j. Soave utilized Wilson's original idea by making the attraction constant
"a" temperature dependent. Vapor pressure data at T =0.7 and the critical point were used to obtain the required temperature dependance. The a-function is
generalized as a function of reduced temperature Tf and acentric factor u. For vapor-liquid equilibrium calculations, the Soave modification satisfies the equal fugacity criterion, that is the vapor phase fugacity is equal to the liquid phase fugacity. The importance of this fugacity equality condition will be discussed further in the next chapter. Instead of the a defined b}' Eq.(2.2b), Soave defined
a by Eq.(2.3) and (2.3a) as:
5 2 a = [ 1 + m(l - Tr°' ) ] (2.3)
m = 0.480 + 1.574u - 0.176w2 in o„i
Eq.(2.3) and (2.3a) can only be applied to nonpolar and slightly polar compounds
[110]. However the accuracy in vapor pressure prediction had increased more
than 100 % over the original RK [93] equation for a number of hydrocarbons.
Chaudron [12], Simonet and Behar [107] generalized the Q& and 0^
constants with the acentric factor but without incorporating the equal fugacity
criterion. Therefore, their equations do not satisfy the thermodynamic requirement
for phase equilibrium. Hence, for saturated vapor-liquid equilibrium calculation, these equations give unacceptable results even though volumetric property
representations in the single phase region are reasonable.
Peng and Robinson [87] furthered the work of Soave by fitting the
a-function with vapor pressure data from the normal boiling point to the critical
point. In other words, the a multiplier was used to optimize the vapor pressure
as predicted by the EOS in question. In addition, the u and w values in the
denominator of the attraction pressure term have also been changed to 2 and
-1, respectively. The repulsion pressure term PR remains the same as before but
the attraction term is now given by the following:
a(T) Pa V2+2bV-b2 ^2'4^
The a-function has the same form as Eq.(2.3) but the coefficients of the constant m generalized in u take on new values:
m = 0.37464 + 1.54226u - 0.26992u2 (2.4a)
The two constants Q& and have values of 0.45724 and 0.07780 [87], respectively. As the result of their modification, Peng and Robinsion claimed that
their equation is superior to the SRK equation for saturated liquid volume
predictions. It was shown by a number of studies that their claim was indeed
valid [66,67,84,129].
Graboski and Daubert [41,42] adopted the same approach as Peng and 12
Robinson by refitting the a-function of the SRK equation with the entire vapor pressure curve instead of just two points as was done by Soave [110]. For hydrocarbons and slightly polar compounds, the form of the a-function is the same as the SRK equation (Eq.(2.3)) but the coefficients of the constant m generalized in CJ has been modified [41],
2 m = 0.48508 + 1.55171w - 0.15613w (2.5)
In the case of hydrogen, the a-function is no longer written as Eq.(2.3) but is expressed as [42]:
a = 1.202exp(-0.30228Tr) (2.6)
Hamam et al [47], Leiva [60], and Paunovic et al [84] modified the RK equation by expressing both constants a and b temperature dependent. They generalized these two constants with acentric factors. Both vapor pressure and saturated liquid volume data were used to obtain the two constants, a and b.
These studies have shown that saturated liquid volume predictions have improved while vapor pressure predictions have remained approximately the same as those for the SRK and PR equations. However, saturated vapor volume predictions have drastically decreased in accuracy compared to the SRK and PR equations.
Study by Paunovic et al [84] has shown that by allowing both a and b to be functions of temperature, derivative properties along the saturated liquid-vapor equilibrium curve when estimated by this type of equation, have decreased in accuracy. 13
For polar compounds, none of the above equations of state are sufficiently suitable for representing saturation properties. This is mainly due to the fact that polar compounds do not follow the usual three-parameter corresponding states principle proposed by Pitzer [88-90]. This principle states that in addition to reduced pressure and temperature, a third parameter is necessar}' to account for the deviation of normal fluid behavior from simple fluid. Pitzer defined normal fluid as the ones with zero or small dipole moment and he proposed that the acentric factor defined as
u = -log (at Tr = 0.7) - 1.0 {2.1)
be this third parameter. In simple terms, the acentric factor accounts for the effects of the size and shape of different molecules. In order to obtain the
acentric factor value, reduced vapor pressure (Pypr) at Tr = 0.7 is required. The three-parameter corresponding states principle with the acentric factor u as the third parameter is, - therefore, inadequate to account for the dipole moment found in polar compounds. Hence, for this reason, a number of investigators have suggested a fourth parameter to characterize polar compounds. Such parameters have been proposed, for instance, by Halm and Stiel [46], O'Connell and
Prausnitz [81], and Tarakad and Danner [120]. Although each of the proposed fourth parameter has correlated, with some degree of success, the thermodynamic functions, however, none of them as yet has gained wide acceptance. It appears that in order to use an equation of state for polar compounds, generalized constants with acentric factor are unlikely to provide good representation of state behavior; indeed substance-dependent constants seem to be the most logical way 14 to achieve high degree of accuracy.
This route was followed by Soave [111] who realized that his earlier equation was inadequate for polar compounds. He suggested a different a-function characterized now by two substance-dependent constants m and n :
a = 1 + (1 - Tr)(m + n/Tr) (2.8)
These two constants can be calculated from vapor pressure data. Lielmezs et al
(LHC) [63] modified the RK equation by defining a different a-function, also with two substance dependent constants p and q:
a = 1.0 + p(T*) (2.9)
where
(2.9a)
These two constants can also be established from vapor pressure data. Similarly,
Lielmezs and Merriman (LM) [64] applied the same a-function to the PR equation. The present forms of both the LHC and LM equations are incapable of representing fluid properties above the critical isotherm because a is undefined, * that is, T becomes negative and a negative number to the power of a non-integer, q is not defined mathematically. 15
Recently Stryjek and Vera [117,118] modified the PR equation by introducing a more complicated temperature dependent a-function:
0 5 2 a = [ 1 + K<1 - Tr * ) ] (2 10)
K is a complex function of reduced temperature and the acentric factor u:
5 K- = «•„ K (K ~ + [ Ki + 2 3 Tr)(l " Tr°' ) ] •
5 (1 + Tr°' )(0.7 - Tr) (2.10a)
where »c0 is given by
2 K0 = 0.378893 + 1.4897153w - 0.17131848a)
+ 0.0196544w3 (2.10b)
and KLT K2 , and *c3 are substance-dependent constants. In addition to these three constants, acentric factor, critical temperature and pressure are required.
Stryjek and Vera provided their own set of acentric factors and they claimed their values are the most accurate and which should be used in conjuction with their modification. In' other words, in order to achieve the accuracy claimed, four constants instead of three are now necessary. Both polar and nonpolar compounds have been tested and vapor pressure predictions have indeed been improved over the original PR equation. Second virial coefficient calculation from EOS is also part of their comparison and the performance is of the typical cubic equation of 16
state. More on this subject will be said in the chapter on second virial coefficient. In the supercritical region, only the fugacity coefficient has been tested
against experimental data. The accurac}' from- the modification for this property
is about the same as the PR equation.
Peneloux et al [85] proposed a modification of the SRK equation by
utilizing the volume-translation technique; the specific volume and the volume
correction constant b are corrected by c.Th e new volume and volume correction
constant are now V and b', respectively:
V = V - c (2.11)
b" = b - c (2.12)
The equation of state after translation becomes
RT a(T) P = - (o ni V-b' (V'+c)(V'+b,+2c) v • ;
Peneloux and co-workers have shown that such translation does not alter the
vapor pressure predictions from the original SRK equation. In fact, this shift in
volume by the amount c changes the location of the isotherm and does not
affect the equilibrium condition. Peneloux et al suggested that this constant c be
established from saturated liquid volume at Tr = 0.7. They claimed that at such
temperature, the prediction gives a balanced representation of the liquid volume
along the saturation curve. They generalized this constant c with the Rackett
[92,113] compressibility factor, 17 RT
c = 0.40768 —£ (0.29441 - Z^) (2.i4)
This correction should be subtracted from the volume calculated by the SRK equation. They tested the predictions from their modification for saturated liquid volume with experimental data for 233 compounds. Both polar and nonpolar compounds were included. The results from this correction are encouraging.
However, this correction can have adverse effects on vapor volume predictions because the isotherm has shifted away from the experimental saturated vapor curve in order to provide a better representation of the saturated liquid curve.
Soave [112] used the same technique for the van der Waals equation and
obtained the constant c at the reduced normal boiling point instead of Tr = 0.7.
Up to now, the two most popular cubic EOS, SRK and PR, have been tested mostly with saturation properties. Rarely do comparisons extend to the supercritical or compressed liquid region. Recently Tannar et al [119] tested their modification of the LHC [63] equation with the SRK and Lee-Kesler [59] equations for volumetric calculations in the subcritical and compressed liquid regions. Comparison of derived or derivative properties such as enthalpy and entropy of vaporization or departure functions found in the literature are scarce.
In the original work of Peng and Robinson [87], only the enthalpy departure functions for five compounds were tested against experimental data. Mohanty et al [79] tested the PR equation for enthalpy of vaporization predictions for
straight chain hydrocarbons from Cx to C20. Mihajlov et al [76] compared the
SRK equation along with three other cubic equations for saturation property predictions for 14 substances. The properties included were vapor pressure, 18 fugacity coefficient and volumes of both vapor and liquid, enthalpy and entropy of vaporization. Tarakad et al [121] compared eight cubic equations, including the
SRK EOS, for saturated vapor and superheated gas phase density and fugacity predictions for polar and nonpolar compounds. Probably the most comprehensive review yet on EOS is the one by Yu et al [129]. Their comparison involved 14 cubic equations, both PR and SRK were included, for eight properties of straight
chain hydrocarbons from d to C10. Reasonable ranges of pressure and temperature were covered in their stud3'.
From the above discussion, it is clear that a more elaborate comparison on cubic equations of state with experimental data is necessary. Thus, the effort of this work is justified. CHAPTER 3. DEVELOPMENT OF AN EQUATION OF STATE
In the last chapter a number of different approaches have been discussed in modifying or improving the cubic equation of state. In the first part of this chapter, some of the fundamental features one must consider in designing an equation of state is discussed.
The Lielmezs-Merriman equation has briefly been mentioned earlier. The present form of the LM equation is not capable of representing compound properties for the region above the critical isotherm. In order to remove this limitation, the equation must be modified and the second part of this chapter will be devoted to the formulation of the LM equation for the supercritical region.
3.1. THEORETICAL CONSIDERATIONS
Equation of state is most frequently used in vapor-liquid equilibrium calculations and the conditions of phase equilibrium for a pure substance are given by Eq.(3.1) to (3.3)
pl ' pV (3.2)
G " G (3.3)
These conditions state that at equilibrium, temperature, pressure and Gibbs free
19 20 energy of the two phases, liquid and vapor, must be identical. Eq.(3.3) can alternatively be expressed in terms of the chemical potential, M
1 v u = U (3.4)
and since fugacit.y is related to chemical potential by Eq.(3.5),
u - u° = RTln (3.5)
condition (3.3) can now be written in terms of fugacit}', f
1 v f = f (3.6)
Frequently Eq.(3.6) is used as the third condition instead of Eq.(3.3). In order to
accurately reproduce vapor-liquid equilibrium properties from an equation of state,
these three conditions, Eq.(3.1), (3.2), and (3.6) must be incorporated into the
EOS. The following discussion will show how such a task can be accomplished.
Gibbs free energy is the sum of the Helmholtz free energy, A and the
product of pressure, P and volume, V :"
G = A + PV (3.7)
Since condition (3.3) requires that the Gibbs free energy be equal for the two phases, or
1 V G - G = 0 (3.3a)
Substituting Eq.(3.7) into (3.3a) yields
1 V 1 1 V V A - A + P V - P V = 0 (3.8)
But A^ - AV can be written as
3A A1 - AV = / dv av (3.9) and from the fundamental equations of thermodynamics,
3A = -P 3V (3.10) JT
By combining Eq.(3.2), (3.8), (3.9), and (3.10), condition (3.3) can now be expressed in terms of pressure and volume.
J , PdV + P(V1 - Vv) = 0 1 (3.11) V where P is the vapor pressure. Eq.(3.11) is shown graphically in Figure 3.1.
The first term of Eq.(3.11) gives the area underneath the isotherm or 22
Vv
J x PdV = area 1-2-3-4-5-6-7-1 (3 12)
and the second term represents the area bound by the isobar P and the isochors
1 V V (=V7) and V (=V6) or
V 1 P(V - v ) = area 1-3-5-6-7-1 (3 13)
But
area 1-2-3-4-5-6-7-1 = area 1-3-5-6-7-1 - area 1-3-2-1
+ area 3-4-5-3 (3.14)
Combination of Eq.(3.13) and (3.14) gives
area 1-3-2-1 = area 3-4-5-3 (3.15)
The volume V^- is the saturated liquid and Vv is the saturated vapor volume.
Eq.(3.11) is formally known as the Maxwell-rule of equal areas.
Since the Maxwell-rule is derived from the conditions of phase equilibrium, an alternative way of satisfying the three conditions of equilibrium is to incorporate the Maxwell-rule into the equation of state. The most commonly used technique to do so is to fit the constant "a" of the attraction pressure term with vapor pressure data. The reason is that by fitting the constant "a" with 23 experimental vapor pressure data, it is possible to adjust the two areas such that their sum equals zero. From the Gibbs phase rule:
F = m + 2 - (3.16)
where F is the number of degrees of freedom, m is the number of components and 7r is the number of phases, it is clear that there is one degree of freedom in the saturation region. Therefore an iterative routine must be used to establish the constant a. The algorithm used is given in a later section of this chapter.
In addition to the Maxwell-rule of equal areas, two other criteria must also be considered. These two criteria are 1) an equation of state should reduce to the ideal gas law when pressure approaches zero or
PV lim — = 1 (3.17) P—>0 RT and 2) the EOS should satisfy the thermodynamic stability criteria at the critical point. From Figure 1.1, it is clear that the critical point is not only an inflection point, but it is also a point which has zero slope. In other words, at the critical point, the first and second derivatives of pressure with respect to volume at constant temperature are zero:
3P = 0 (3.18) 3V 24
92P = 0 av2 (3.19) JT
Another important aspect of an equation of state that must be considered
is the functional dependence of the constants. When the generalized cubic equation of state, Eq.(1.2) is expanded in terms of density or volume, the following
results (Appendix A):
a 1 uab 2 Z = 1 + b — + b + RT V RT
(w-u2)ab2 1 3 b + (3.20) RT V3
The virial equation:
BCD Z = 1 + - + — + — + (3.21) V V2 y 3
where B, C, D etc are the second, third, and fourth virial coefficients.
Comparing the coefficients of Eq.(3.20) and (3.21), the virial coefficients in terms
of a, b, u, and w are as follows:
B = b - (3.22) RT
uab 2 C = b + (3.23) RT 25 (w-u2)ab2 D = b3 + RT (3.24)
Since virial coefficients are functions of temperature only, the constants a, b, u, and w can either be constant or temperature dependent.
The above criteria are not restricted to cubic equations of state but are applicable to equations of state in general. In order to have a theoretically sound
EOS, all of the above aspects should be considered.
3.2. LIELMEZS-MERRIMAN EQUATION OF STATE
Lielmezs and Merriman [64] modified the Peng-Robinson equation of state by defining a new a term of Eq.(2.2a) but retaining the same u and w values as the PR equation. To express the a term as a continuous, temperature dependent function along the vapor-liquid equilibrium curve, the previously * * proposed T coordinate system by Lielmezs [34,61] was introduced; T is given by Eq.(2.9a):
* _ Tc/T ' 1
where T , T^, T are the critical temperature, normal boiling point temperature, * and the state temperature, respectively. This T coordinate system have been applied to a wide variety of thermodynamic and transport properties. These include vapor pressure-temperature relation [62], heat of vaporization [34,101], self-diffusion coefficient [61], thermal conductivity [49], and surface tension [65] 26 correlations.
* The a term was written in terms of the proposed T as
o(T*) = 1 + p(T*)3 (29)
where p and q were characteristic constants of the given pure substance. One advantage of using substance-dependent constants is that the arbitrarily chosen third parameter such as the acentric factor is eliminated. Another advantage is that quantum or polar effects are absorbed by the two constants and the knowledge of the structure or type of compound is not necessary. These two constants are determined by fitting experimental vapor pressure data to the a-function with the least-squares routine. For the sake of completeness, the algorithm given by Merriman [75] is shown in Figure 3.2. The list of compounds studied in this work along with the fitted p and q values are given in Table
3.1. Some of the p and q values are different than the ones given previously by Lielmezs and Merriman [64]. This is due to the fact that different data sets are used in the curve-fitting.
When experimental data are not available, use of an independent vapor pressure equation is recommended. In this work, five vapor pressure equations have been evaluated. These included the Lee-Kesler (LK) [59], Riedel [94],
Riedel-Plank-Miller (RPM) [94], Frost-Kalkwarf-Thodos (FKT) [94], Gomez-Thodos
(GNT) [95] equations. These are all generalized equations and only critical temperature and pressure, normal boiling point temperature, acentric factor, and 27 molecular weight of the compound are needed as input data. To evaluate these five equations, they were used to predict vapor pressure of 36 compounds studied in this work. The predicted results are compared with the experimental data used to obtain the p and q constants. The comparison is based on root mean square % error (RMS %)
I (% error)2 RMS % Error = (3.25) N where
Experimental - Calculated % Error = X 100 Experimental and the results are shown in Table 3.2.
Of the five equations studied, the GNT and FKT equations seem to give the most reliable results for both polar and nonpolar compounds. However, pressure is expressed implicitly in the FKT equation and solution must be obtained by an iteration routine. On the other hand, the GNT equation is a pressure explicit equation and is simpler to solve. Therefore, the GNT equation is recommended when no experimental data are available. The conclusion here is consistent with that of Reid et al [94].
In the course of evaluating different vapor pressure equations, the parameter, s [28,77,94] : 28 T, ln(P /P") s = -£= (3.26) l-T.b r
which is the negative of the slope between the critical point and normal boiling point of the vapor pressure equation:
lnPr= s 1 - (3.27)
kept on appearing in the literature [9,28,36-38,77,94,95]. The critical pressure is expressed in atm and P° is one atmosphere. Miller [77] suggested that this parameter s be treated as a fundamental characteristic parameter similar to the
Riedel factor aQ [96], the critical compressibility Zc, and the acentric factor o.
Giacalone [35] proposed a correlation for estimating the heat of vaporization at the normal boiling point with the parameter s:
AH , = sRT vb c (3.28)
Edmister [28] correlated the acentric factor u with s b3r the following equation:
u = 1 1^ 5 ~ 1,0 (3-29)
Gomez-Nieto and Thodos [36-38] also used this characterization parameter s in their vapor pressure equations and later Campbell and Thodos [9] used it in their saturated liquid density correlation. Since the constants p and q are so closely related to vapor pressure, a relation might exist between the constants p and q and the parameter s. 29
To explore this possibility, 34 of the 36 compounds studied here were divided into three groups according to the nature of the compounds: nonpolar
(Group 1), polar and slightly polar (Group 2), and inert and quantum (Group 3):
Group 1 : Ethane, Propane, n-Butane, i-Butane, n-Pentane, i-Pentane, Neopentane, n-Hexane, n-Heptane, n-Octane, Benzene, Ethylene, Propylene, 1-Butene, Nitrogen, Oxygen.
Group 2 : Carbon Monoxide, Carbon Disulfide, Hydrogen Sulfide, Sulfur Dioxide, Methanol, Ethanol, 1-Propanol, Tertiarj' Butanol, Water, Ammonia.
Group 3 : Methane, Deuterium, n-Hydrogen, p-Hydrogen, Neon, Argon, Krypton, Xenon.
Carbon dioxide and acetylene do not have normal boiling points because their triple point pressures are above one atmosphere, therefore they are not included.
But if they were included, both of them would belong to Group 1. From Table
3.1, one can see that all the q values are relatively constant within each group.
In light of this fact, the constant q of the compound is taken to be the average value for the group. For Group 1, the average value of q is 0.83, for Group 2, q is 0.83, and Group 3 compounds take on the value of 0.78.
For the constant p, values vary from 0.01582 to 0.47769. These p values for the three groups have been plotted against both the parameter s and the acentric factor o> in Figure 3.3. Also least squares fits have been used to express p as functions of s and u. The results are as follows:
Group 1 :
2 p = -1.1977373 + 0.39942704s - 0.026211814s (3.30)
variance = 0.00004469 30
p = 0.19011333 + 0.67187907a) - 0.85099293a)2 (3.31)
variance = 0.00004475
Group 2
2 p = -1.8772891 + 0.55686202s - 0.033057663s (3 32)
variance = 0.00024869
2 p = 0.1639189 + 1.1372604a) - 1.0533208o> (333)
variance= 0.00023931
Group 3 :
p = 0.34065806 - 0.2439171s + 0.040416063s2 (3.34)
variance = 0.00003188
p = 0.18425207 .+ 1.1592480a) + 1.8186255a)2 (3.35)
variance = 0.00041294
For Groups 1 and 2, the correlations of p with s and a) produce approximately the same variance. For Group 3, the variance produced by fitting p with s is ten times smaller than for the fit of p with a). This indicates that the correlation of p with s is stronger than with a). The advantage is more revealing as is shown in Figure 3.3. Not only does p correlates better with s, the
determination of s is simpler than is o>. Only T , T^, and Pc are needed for
calculating s and vapor pressure at Tr = 0.7, Pc, and T£ are required for a>. In many instances, the vapor pressure value is not available, therefore an interpolation or extrapolation technique must be used to get the required data 31 point. This kind of treatment can lead to a wide range of results. For example, in the case of methane, five different values have been reported. Reid et al gave a value of 0.008 in 1977 [94] and 0.011 in 1987 [95]. Stryjek and Vera [117] report a value of 0.01045 and which is supposedly based on the most accurate vapor pressure data available. Passut and Danner [83] tabulate the acentric factors for 192 hydrocarbons, and for methane, the value of 0.0072 is given.
Edmister and Lee [29] report a value of 0.0115. These different values of acentric factors are mainly caused by the uncertaintj' in determining the vapor
pressure at Tr = 0.7. On the other hand, the boiling point temperature, critical pressure and temperature required in calculating the parameter s can be determined more accurately and therefore subject to less uncertainty.
Since the correlation of u with s has been proven possible by Edmister
[28], it might be possible to improve his correlation. The plots of p=f(s) look
similar to that of p = f(u) for all groups, therefore the correlation of acentric
factor with s is again divided into the same three groups. The results of the
correlations are plotted in Figure 3.4 and the correlations are as follows :
Group 1 :
2 u = -0.65652243 + 0.079558804s + 0.0080858283s (3 36)
variance = 0.00000867
Group 2 :
2 u = -0.71488058 + 0.10168299s + 0.0058534228s (3.37)
variance = 0.00001785 32
Group 3 :
2 u = -1.7653114 + 0.5377135s - 0.03883573s (3 38)
variance = 0.00000316
Neon has not been included in Group 3 for the curve-fit because the acentric factor value, 0.0 given by Reid et al [94] in their 1977 version is substantial^' different than the value, -0.029 given in their 1987 version [95]. By visual inspection of Figure 3.4, the value, -0.029 is definitely the more reasonable one.
For the sake of comparison, the predictions from the proposed correlations,
Eq.(3.36) to (3.38), were compared with that of the Lee-Kesler equation [59], and the Edmister correlation, Eq.(3.29). The results from the correlations are shown in Table 3.3. The proposed correlations are the best in estimating the acentric factors for all three types of compounds. This shows that the characterization parameter s can definitely be treated as a fundamental constant similar to the
acentric factor, Riedel factor, or the critical compressibility.
3.3. MODIFIED LIELMEZS-MERRIMAN EQUATION OF STATE
Before the work of Lielmezs and Merriman [64], the proposed a-function,
Eq.(2.9) had also been applied to two other equations of state: van der Waals
and Redlich-Kwong, by Law and Lielmezs (LL) [58], and Lielmezs, Howell and
Campbell (LHC) [63]. Recently this a-function has also been applied to the
Martin equation (CLI) [11]. All four modifications, LL, LHC, LM, and CLI give excellent results in the saturation region. However all suffer the same
shortcoming, that is, they cannot be applied, without modification, to the region 33 above the critical isotherm or the supercritical region. For temperatures above the * critical, T is negative and this leads to an undefined a. Therefore, in order to extend the applicability of the LM equation to the entire PVT surface, the T coordinate system must be modified. One criteria in modifying the LM equation is that no more constants should be added; the equation should be as simple as the present form. Additional constants will only complicate the equation and defeat the purpose of using a cubic equation of state. Furthermore, in order to determine the additional constants, more experimental data will be required.
The proposed modification is to let the numerator of the T coordinate be positive in all regions. To do so, two ways have been considered. One is to let
the numerator become T/T -1 and the other is 1-TC/T. In an exploratory study of two compounds, methane and water, the RMS % error in predicting volume by the first modification is slighty lower than the second modification. For methane, the RMS % errors from the first and second modifications are 4.95 and 6.53, respectively, and for water, 3.21 and 3.06. Therefore, the first *
modification is adopted and the new T coordinate is now defined as:
T/Tc - 1
and
q a = 1 - p(T*) (3.40)
The two constants, p and q, are the same in all regions. 34
The proposed modification of the a-function has kept the simplicity of the original LM equation and yet it is applicable to the entire PVT surface for volumetric prediction. For thermodynamic property calculations where the derivative of the a-function with respect to temperature is involved, the proposed modification is applicable to all regions except along the critical isotherm. The reason is that such derivative function is undefined at T , that is, at the critical c point the derivative of a with respect to temperature is infinite. From the practical point of view, this is not a problem because most processes do not operate near the critical point. Process designers trj' to avoid such conditions because near the critical, it is extremely difficult to have stable processes; fluids that are near their critical points exhibit exceptional physical and thermodynamic behavior. Therefore the inadequacy of the present proposed equation does not pose a major limitation on its application. 35
Figure 3.1 Maxwell-rule of equal areas. 36
INPUT 7
\ INPUT / \ T t P/
CALCULATE T
ASSUME a
CALCULATE A & B ADJUST a SOLVE ZJ -(1-B)Z1+(A-3B1-2B)Z-(AB-B1-B')=0 FOR ZV & Z
CALCULATE fV & f1
NO
YES
NO
CURVEFIT ^ a = 1 + p(T )q
OUTPUT"\ p & q 7
Figure 3.2 Algorithm for deterniining the substance-dependent constants p and q. Figure 3.3 Correlations of p with parameter s and with acentric factor 3 0.2
8.0
10.0
3 -0.1
e.o
Figure 3.4 Correlations of acentric factor a with parameter Table 3.1 Calculated and Fitted p and q Constants
p q Compound Fitted p = f(s) p = f(u) Fitted Avg. s u Group
Methane 0 19584 0 2041 1 0 19364 0 78426 0 78 5 41072 0 008 3 E thane 0 25183 0 24740 0 24778 0 83742 0 83 5 91060 0 098 1 Propane 0 27413 0 27269 0 27258 0 85176 0 83 6 221 12 0 152 1 n-Butane 0 28984 0 28986 0 28809 0 87067 0 83 6 47932 0 193 1 1-Butane 0 279G8 0 28358 0 28200 0 87124 0 83 6 37862 0 176 1 n-Pentane 0 30395 0 30421 0 30514 0 86468 0 83 6 75195 0 251 1 1-Pentane 0 29387 0 29805 0 29878 0 85979 0 83 6 62562 0 227 1 Neopentane 0 27709 0 28839 0 28945 0 87028 0 83 6 45483 0 197 1 n-Hexane 0 30876 0 31384 0 31443 0 81677 0 83 6 99910 0 296 1 n-Heptane 0 32020 0 32062 0 321 10 0 82035 0 83 7 26413 0 351 1 n-Octane 0 32632 0 32357 0 32273 0 81321 0 83 7 50371 0 394 1 Benzene 0 30668 0 29424 0 29430 0 82281 0 83 6 55501 0 212 1 Carbon Monoxide 0 20444 0 21358 0 21712 0 80737 0 83 5 64933 0 049 2 Carbon Dioxide 0 31364 0 89550 -• ... 0 225 - Carbon Disulfide 0 28184 0 28700 0 28077 0 72832 0 83 6 08380 0 1 15 2 Hydrogen Sulfide 0 28655 0 26616 0 2671 1 0 84269 0 83 5 95272 0 100 2 Sulfur Dioxide 0 36256 0 38402 0 38301 0 83570 0 83 6 83047 0 251 2 Methanol 0 47075 0 46780 0 47050 0 80070 0 83 8 39992 0 559 2 Ethanol 0 47769 0 46101 0 45955 0 84658 0 83 8 87656 0 6436 2 1-Propanol 0 45585 0 46438 0 46343 0 91571 0 83 8 74510 0 624 2 Tertiary Butanol 0 45101 0 46469 0 46446 0 90728 0 83 8 73054 0 618 2 Acetylene 0 30632 0 80464 -- -- 0 184 - Ethylene 0 24542 0. 24149 0 24107 0 81586 0 83 5 84587 0 085 1
CO Table 3.1 Cont'l nued
p q Compound Fitted p = f(s) p = f(u) Fitted Avg. s u Group
Propylene 0., 27311 0 .26929 0 .27091 0 .83694 0 .83 6 . 17555 0. 148 1 1-Butene 0..2908 5 0 .28708 0,, 28600 0 .85773 0 .83 6 .43366 0. 187 1 Water 0,.4422 1 0..4284 2 0..4304 9 0 .73237 0 .83 7 . 33080 0. 344 2 n-Deuterlum 0..0639 4 0..0624 5 0.,0642 8 0 .61602 0,.7 8 4 .50830 -0.13 3 n-Hydrogen 0.,0187 7 0..0183 8 0.,0172 4 0,,3672 0 0 . 78 4 .08143 -0. 22 3 p-Hydrogen 0.,0158 2 0..0166 4 0..0172 4 0..3080 6 0.. 78 4 .06100 -0. 22 3 Nf trogen 0. 20477 0. 21681 0. 21563 0.,8171 3 0. 83 5 .59774 0.040 1 Ammon1 a 0. 38595 0. 38180 0. 38240 0. 85842 0. 83 6 .8094. 8 0.250 2 Oxygen 0. 20734 0. 20271 0. 20385 0. 81007 0. 83 5 .4687. 8 0.021 1 Neon 0. 1446G 0..1496 4 0. 18425 0.,7635 3 0..7 8 5 .1102. 9 0.000 3 Argon 0. 18893 0. 18789 0. 17964 0. 78649 0..7 8 5 .32535 -0.004 3 Krypton 0. 19626 0. 19068 0. 18194 0. 80560 0. 78 5,.3402 9 -0.002 3 Xenon 0. 20060 0. 19503 0. 18658 0. 79778 0. 78 5 .3633. 1 0.002 3
1^- o Table 3.2 Vapor Pressure RMS % Error
Compound LK RIEDEL RPM FKT GNT Tr Range N
Methane 0. 660 0 .787 0,.43 1 0. 749 0., 565 ' 0.,586 --o .990 29 Ethane 0. 542 0 . 569 0,.44 2 0. 202 0,. 185 0..604 --o .999 23 Propane 0. 542 0 .498 0,. 750 0. 059 0.. 148 0., 625-0 .991 30 n-Butane 0. 330 0 . 267 0..73 3 0. 194 0 . 282 0..641 --0 . 998 32 i-Butane 0. 357 0 .312 0,.73 1 0. 186 O.. 345 0.,640 --0 .998 30 n-Pentane 0. 678 0 . 565 0,,50 0 0. 184 0., 236 0,,658 --0 .999 27 i-Pentane 0. 501 0 .404 0.,52 6 0. 164 0,, 146 0,,654 --0 .999 27 Neopentane 0. 203 0 . 140 0..74 1 0. 246 0.,42 6 0.,652 --0 .999 25 n-Hexane 1 .46 4 1.. 136 0.,40 3 0. 573 0.,44 5 0.,538 --0..99 5 43 n-Heptane 1 .58 3 1.. 128 0.,35 1 0. 455 0., 241 0.,545 --0..98 7 30 n-Octane 1 .84 1 0 .894 0.,60 1 0. 543 0., 751 0.,488 --0..97 6 27 Benzene 1 .00 8 0,.89 6 0.,74 5 0. 495 0.,63 6 0.,553 --0..99 7 47 Carbon Monoxide 0. 730 0..69 9 1. 254 0. 925 1 .. 188 0. 513--0 .981 24 Carbon Dioxide 16.99 17.28 18.79 17.83 16.84 0. 712--0.,99 5 32 Carbon Disulfide 3. 836 3..84 1 2. 482 3. 310 3 .76 1 0. 500--0..97 6 26 Hydrogen Sulfide 1 .58 0 1 ,59. 3 1 .76 7 1 .37 8 1 .81 4 0. 565--o.,99 7 30 Sulfur Dioxide 1 .92 1 1 ,72, 3 0. 806 1 .1 16 4 .40 4 0. 593--0..99 2 35 Methanol 5. 409 4 ,61. 8 3. 211 4 .65 4 1 .47 7 0. 532--0,,99 6 44 Ethanol 3. 595 2 .76. 1 2. 940 3. 378 0. 684 0. 543--0.,97 9 29 1-Propano1 5. 667 6..87 6 7 .56 5 6. 800 2 .48 8 0. 555--0. 993 13 Tertiary Butanol 1 .12 1 1 .29. 5 1 .83 2 1 .70 9 2. 945 0. 701--0. 993 12 Acetylene 2. 391 2.,42 3 3. 067 2 .66 8 2 .47 9 0. 623--0.,98 9 20 Ethy 1ene 0. 924 0. 951 0. 914 0. 923 0. 815 0. 599--0. 981 19 Table 3.2 Continued
Compound LK RIEDEL RPM FKT GNT Tr Range N
Propylene 0. 688 0..65 1 0,,70 7 0.. 139 0,, 168 0,.618 --o. .989 26 1-Butene 0..97 2 0.,88 6 0,,87 6 0.,45 0 0,.53 1 0 .651--o .979 26 Water 7 .389 6..57 4 2,, 1 16 7 . 154 1 .71, 5 0,.422 --0,.99 5 43 n-Deuter i urn 0,.89 7 1 .64. 7 2 ,87, 8 2., 134 2 .82. 4 0,,488 --o. ,991 21 n-Hydrogen 0,.25 1 0.,66 5 1 ,, 135 1 .46 9 0,.61 3 0,.615 --0,.96 4 13 p-Hydrogen 0,.09 7 0.,53 3 1 ,01, 0 1 ,. 384 3 .02, 3 0,.615 --0,.97 0 13 Ni trogen 0..69 5 0,,78 7 0,.58 6 0, 578 0,.44 9 0,.614 --0,.99 1 18 Ammon i a 2..05 3 1..85 3 1 ,66, 4 1 ,33. 5 0,.81 1 0,.602 --0,.98 6 29 Oxygen 0..24 8 0..40 8 0,.49 3 0., 194 0,, 303 0,.583 --0,.98 6 24 Neon 0..53 9 0.,79 7 0,,54 5 0. 753 0,,29 4 0,. 563-0- ,.96 8 19 Argon 0,.22 4 0..44 2 0,,24 4 0. 293 0,, 155 0,,555 --o. .994 15 Krypton 0,. 183 0. 245 0..46 9 0. 243 0,.49 0 0.. 573-0- ,.98 4 20 Xenon 0,.22 5 0.,41 7 0,,49 2 0. 138 0,,33 8 0,.569 --0..93 2 12 Table 3.3 Acentric Factor from the Proposed, Lee-Kesler, and Edmister Correlations
Compound s o Proposed LK Edmister
Methane 5 .41072 0 .0080 0 .0072 0 .0097 0 .0071 Ethane 5 .91060 0 .0980 0 .0962 0 .0939 0 . 1001 Propane 6 .221 12 0 . 1520 0 . 1514 0 . 1493 0 . 1579 n-Butane 6 .47932 0 . 1930 0 . 1984 0 . 1969 0 .2060 1-Butane 6 .37862 0,. 1760 0,. 1799 0 .1791 0 . 1872 n-Pentane 6 .75195 0..251 0 0,, 2493 0,.248 6 0,. 2567 1-Pentane 6 .62562 0,. 2270 0,.225 6 0 .2251 0,. 2332 Neopentane 6 .45483 0.. 1970 0,. 1939 0,. 1944 0,.201 4 n-Hexane 6 .99910 0..296 0 0., 2964 0,. 2965 0,. 3027 n-Heptane 7., 26413 0.,351 0 0.,348 1 0.,348 8 0.. 3520 n-Octane 7 .5037, 1 0. 3940 0. 3957 0. 3970 0., 3966 Benzene 6,.5550 1 0. 2120 0. 2124 0.,207 3 0., 2201 Carbon Monoxide 5 .6493 3 0. 0490 0. 0464 0.,050 6 0.,051 5 Carbon Dioxide 0. 2250 Carbon Disulfide 6. 08380 0. 1 150 0. 1204 0. 1201 0. 1324 Hydrogen Sulfide 5. 95272 0. 1000 0. 0978 0. 0967 0. 1080 Sulfur Dioxide 6 .8304 7 0. 2510 0. 2528 0. 2501 0. 2713 Methanol 8 .3999 2 0. 5590 0. 5523 0. 5424 0. 5634 Ethanol 8 .8765 6 0. 6436 0. 6489 0. 6430 0. 6522 1-Propanol 8. 74510 0. 6240 0. 6220 0. 6227 0. 6277 Tertiary Butanol 8 .7305 4 0. 6180 0. 6190 0. 6253 0. 6250 Acetylene 0. 1840 Ethylene 5 .8458 7 0. 0850 0. 0849 0. 0825 0. 0881 Table 3.3 Continued
Compound s u Proposed LK Edmister
Propylene 6 . 17555 0. 1480 0.1432 0.1404 0.1494 1-Butene 6 .43366 0. 1870 0.1900 0. 1880 0.1975 Water 7 . 33080 0.3440 0.3451 0. 3213 0.3645 n-Deuter1um 4 .50830 -0.1300 -0.1305 -0.1437 -0.1609 n-Hydrogen 4 .08143 -0.2200 -0.2176 -0.2157 -0.2403 p-Hydrogen 4 .06100 -0.2200 -0.2221 -0.2193 -0.2441 N1trogen 5 . 59774 0.0400 0.0422 0.0418 0.0419 Ammon1 a 6 .80948 0.2500 0.2489 0.2409 0.2674 Oxygen 5 .46878 0.0210 0.0204 0.0191 0.0179 Neon 5 .11029 0.0000 -0.0316 -0.0408 -0.0488 Argon 5 .32535 -0.0040 -0.0032 -0.0044 -0.0088 Krypton 5 .34029 -0.0020 -0.0013 -0.0020 -0.0060 Xenon 5 . 36331 0.0020 0.0015 0.0016 -0.0017 CHAPTER 4. APPLICATIONS OF THE CUBIC EQUATION OF STATE
This chapter is divided into five parts. The first four parts are discussions of various property predictions from the three cubic equations of state: SRK, PR, and LM in the four regions on the Pressure-Volume surface shown in Figure
1.1: saturation, subcritical, supercritical, and compressed. The last part is devoted to the inversion curve estimation. All the equations derived from the generalized cubic equation of state for calculating different properties are shown in the
Appendix A. The physical constants required as input data have been tabulated
in Table 4.1. These include molecular weight (MW), critical pressure (Pc), critical
temperature (Tc), normal boiling point (T^), acentric factor ( The normal boiling points of carbon dioxide and acetylene shown in Table 4.1 are temperatures of the compounds at one atmosphere. These physical constants have been used in all of the calculations found in this work. Although small variations of these constants appear in the literature, the assumption here is that these differences cause negligible errors. Due to the large number of properties being studied, there was a need to use various sources of experimental data; even though the ideal is to use a single source. Often, the so called experimental data found in the literature are not data measured from experiments but data calculated from an empirical equation with its constants fitted with experimental data. Since different data sources use different sets of units, to be consistent, all data are converted to the 45 46 same set of units and these units are as follows: Pressure - atm Volume - cm3/g-mole Temperature - K Energy - calorie The following conversion constants are used and are taken from [30,102,126]: 1 atm = 14.696 psi = 1.01325 bar 1 ft3/lb = 0.062428 cm3/g 0°C = 273.15 K = 32°F = 460.0 R atm cm3 cal Gas constants, R = 82.05606 = 1.9872 g-mole K g-mole K 4.1. REGION I : SATURATION Properties tested in this region included vapor pressure, liquid and vapor volumes, and enthalpy and entropy of vaporization. The experimental data sources are given in Table 4.2. In order to ensure internal consistency, all the properties of a compound were taken from the same source. The algorithm for calculating these properties from an EOS is given in Figure 4.1. The results from this work (LM(1), LM(2)) together with those of the SRK and PR are reported in Tables 4.2-4.4. The LM(1) equation refers to the LM equation with the two constants, p and q, fitted with experimental vapor pressure data. The LM(2) equation refers to the LM equation with the constants p correlated with the parameter s (Eq.(3.30), (3.32) and (3.34)) and the constant q calculated as the 47 average value of the group. In this region, a total of 36 compounds are tested and classified into three groups as before : 1. Nonpolar 2. Polar and slightly polar 3. Inert and quantum In the vapor pressure calculation, both LM(1) and PR give approximately the same predictions for the Group 1 compounds. The PR equation gives slightty lower RMS % errors for hydrocarbons. The LM(2) equation does not give satisfactory predictions for all compounds. This is due to the fact that LM(2) only gives approximate p and q values and vapor pressure is extremely sensitive to these two constants, especially p. For Groups 2 and 3, the LM(1) equation is the best among the three equations. This is the strength of any correlation which uses individual constants instead of generalized ones such as the SRK or PR equation. The SRK and PR equations were designed not for Groups 2 and 3 types of compounds. Therefore it is not surprising to find that they both give such poor results for these two groups. The % errors of vapor pressure, liquid, and vapor volumes are plotted against the reduced temperature, Tr in Figures 4.2 to 4.5 for four compounds selected from the three groups (group 1: n-Pentane; group 2: 1-Propanol; group 3: Methane and Argon). These figures show that the LM (LM refers to the LM(1) equation) equation gives vapor pressure errors that are evenly distributed along the temperature range studied whereas the SRK and PR either over predict in one compound and under predict in another. Near the lower end of the temperature range, the errors from all three equations seem to increase 48 rapidly. This behavior suggests that all the equations considered here should not be used for extrapolation purposes; they should be used within the temperature range given in Table 4.2. For the saturated liquid and vapor volume predictions, the PR and LM(1) equations give almost identical results. The LM(1) equation gives slightly lower errors near the critical point. In terms of RMS % error, the SRK equation gives almost double that of the PR or LM(1) equation. From Figures 4.2-4.5, it can be seen that the SRK equation tends to over predict liquid volumes along the entire saturation curve for all three groups of compounds. For the PR and LM(1) equations, the errors are distributed more evenly. It is interesting to note that the shape of the error distribution curves for the three equations are almost identical. The only difference between them is how much they have shifted away from the others. There is evidence in the literature [129] that this shift may be directly connected to the two constants u and w in the generalized cubic equation, Eq.(1.2). For the PR and LM equations, they have the same u and w values, and their error distribution curves are indeed on top of each other and are shown in Figures 4.2 to 4.5. From the evidence presented up to this point, one may conclude that altering the a-function will not dramatically improve volume predictions. By using different u and w values, it is possible to shift the low-error portion of the distribution curve to the desired temperature range. Table 4.4 contains the results of the enthalpy and entropy of vaporization 49 from the four equations. Usually entropy of vaporization is not tabulated in the experimental data source, but it can be calculated from the enthalpy of vaporization at the given temperature: AH AS„ = —* (4.1) v T Enthalpj' of vaporization is just the difference between the vapor and liquid phase enthalpy: v 0 1 AHv = (H - H°) - (H - H ) (4.2) where H° is the enthalpy at zero pressure. Some data sources did not specify their reference state or the value of the reference state. Therefore, instead of comparing the departure function with that predicted from an EOS, the difference between the departure functions, which is the enthalpy of vaporization, is used as the basis of comparison. It should be pointed out that Table 4.4 gives average absolute deviation (AAD) which is defined as I |Deviation| = ~ ' (4.3) where Deviation = Experimental - Calculated and not RMS % error. Figures 4.6 to 4.9 have the plots of reduced enthalpy and entropy of vaporization against reduced temperature. Even though in terms of AAD both PR and LM(1) give similar results, Figures 4.6-4.9 reveal the strength of each. For both enthalpy and entropy of vaporization calculations, the PR equation gives better predictions for the lower temperature region and the LM is better for the higher end. The separation is in the vicinity of 50 Tr = 0.7. The SRK equation gives similar results as the LM(1) equation at the lower end and similar to the PR equation at the higher end. The limitation of the LM(1) equation is that it is not applicable at the critical point. The LM(2) equation gives reasonable predictions for all compounds and in some cases, it gives the lowest AAD. When no p and q values are available, LM(2) is recommended. Table 4.5 has the overall average errors of the five properties tested in this region. Among the three equations, the LM(1) equation is the most accurate in estimating vapor pressure and saturated liquid and vapor volumes. The PR equation gives the lowest deviations for the two derivative properties, namely, enthalpy and entropy of vaporization. Table 4.1 Summary of Physical Properties * Compound MW Pc (atm) Tc (K) Tb (K) u s t Methane 16.042 45 .8016 190 .65 1 1 .71 0 0..00 8 5 .41072 Ethane 30.068 48 .2000 305 .42 184 .47 0,,09 8 5 .91060 Propane 44.094 42 .0162 369 .96 231 . 10 0.. 152 6 .221 12 n-Butane 58.123 37 .4700 425 . 16 272 .67 0., 193 6 .47932 i-Butane 58.123 36 .0000 408 . 13 261 . 32 0.. 176 6 .37862 n-Pentane 72.1498 33 .2500 469 .65 309 . 19 0..25 1 6 .75195 i-Pentane 72.1498 33 . 3700 460 .39 301 .025 0.,22 7 . 6 .62562 Neopentane 72.1498 31 .5450 433 .75 282 .628 0., 197 6 .45483 n-Hexane , 86 . 170 29 .9197 507 .87 341 . 87 0.. 296 6 .99910 n-Heptane 100.205 27 .0000 540 . 20 371 .60 0. 351 7 .26413 n-Octane 1 14.232 24 ,5000 568 .80 398 .80 0. 394 7 .50371 Benzene 78.108 48..700 3 562 .65 353 . 25 0. 212 6 .55501 Carbon Monoxide 28.010 34 ,526. 4 132 .92 81 .7 '0 0. 049 5 .64933 Carbon Dioxide 44.011 72..806 2 304 . 19 194 .70 0. 225 Carbon Disulfide 76.131 75., 1900 546 . 15 319 . 37 O. 115 6 .08380 Hydrogen Sulfide 34.0758 88 .200. 0 373 .07 212 . 875 0. 100 5,.9527 2 Sulfur Dioxide 64.066 77 .796 7 430 .65 263 .00 0. 251 6 ,8304, 7 Methanol 32.042 78. 5928 513 . 15 337 .696 0. 559 8 . 39992 Ethanol 46.069 60..567 5 513 .92 351 .443 0. 6436 8 .8765, 6 1-Propanol 60.090 50..210 9 537 .04 370 .93 0. 624 8 ,7451, 0 Tertiary Butanol 74.1224 41 .770. 0 508 .87 356 .48 0. 618 8 .7305, 4 Acetylene 26.036 61 .649. 4 308 .69 189 .20 0. 184 Ethylene 28.054 50. 5001 283 .05 169 .40 0. 085 5 .8458. 7 Tab1e 4.1 Cont i nued Compound MW Pc (atm) Tc (K) Tb (K) u s t Propylene 42.078 45.6090 364.91 225.45 0. 148 6 .17555 1-Butene 5G.104 39.6707 419.59 266.90 0. 187 6 .43366 Water 18.0152 218.300 647.30 373. 15 0. 344 7 .33080 n-Deuterium 4.032 16.4300 38.35 23.66 -0.130 4 .50830 n-Hydrogen 2.016 12.9800 33. 18 20. 38 -0.220 4 .08143 p-Hydrogen 2.016 12.7590 32.976 20.268 -0.220 4 .06100 Ni trogen 28.016 33.4900 125.95 77 .40 0.040 5 .59774 Ammoni a 17.032 111.348 405.59 239.70 0. 250 6 .80948 Oxygen 32.000 50.0817 154.76 90.2056 0.021 5 .46878 Neon 20.179 26.1900 44.40 27 .09 O.OOO 5 .11029 Argon 39.944 48.3395 150.86 87 . 29 -0.004 5 , 32535 Krypton 83.80 54.2512 209.39 1 19.80 -0.002 5..3402 9 Xenon 131.30 57.6000 289.74 165.02 0.002 5..3633 1 + Physical properties that are not given in the original data source are taken from reference 94. t Calculated from Tb, tc, and Pc. Table 4.2 Region I - Vapor Pressure RMS % Error Compound SRK PR LM(1) LM(2) Tr Range N Ref. t Methane 1 . S05 1 . 592 0. .423 2 .323 0 .586- -O..99 0 29 10 Ethane 0 .920 0 .420 0, .812 1 .079 0 .604- -0..99 9 23 30 Propane 1 .027 0 .421 0, .714 0 .839 0 .625- -0, .991 30 10 n-Butane 1 .115 0 . 769 0. . 702 1 . 294 0 .641- -o..99 8 32 18 i -Butane 1 .03, 6 0,.70 0 0, , 742 1 .803 0,.640 - -o, 998 30 19 n-Pentane 0,.99 1 0,.33 6 0. ,803 1 . 299 0,.658 - -o.,99 9 27 20 i-Pentane 0 .987 0,.49 3 0. ,502 1 . 602 0 .654- -0. 999 27 21 Neopentane 0 .687 0 . 358 0. . 555 3 .046 0, .652- -o. 999 25 22 n-Hexane 1..72 5 1 .82, 2 1 ,32. 5 3 . 135 0. . 538--0. 995 43 10 n-Heptane 1..86 5 1 ,03, 4 1 .41. 2 1 .75, 4 0, , 545--o. 987 30 115 n-Octane 1,.62 0 4 , 395 2. 711 3 .37, 4 0, .488- -0. 976 27 1 15 Benzene 1.. 297 1 ,0. 1 1 1 .07 7 4, ,551 0. ,553- -o. 997 47 10 Carbon Monoxide 0..89 7 2. .455 0. 972 4 .09, 6 0. ,513- -o. 981 24 10 Carbon Dioxide 0. .540 0. ,704 0. 385 0. .712- -0. 995 32 10 Carbon Disulfide 4 ,49, 0 4 .85, 0 2 . 317 5 .99, 6 0. .500- -0. 976 26 80 Hydrogen Su1f i de 1 ,. 138 1 .80 8 1 .63 0 5 ., 524 0. .565- -0. 997 30 115 Sulfur Dioxide 2. ,244 2 .02 2 1 .26 3 5. ,817 0. 593- -O. 992 35 10 Methanol 8. 043 4 . 442 4 . 504 5 . 564 0. 532- -0. 996 44 108 Ethanol 3. 877 2. 315 4. 480 10.21 0. 543- -o. 979 29 109 1-Propanol 4 . 743 10.01 3. 413 6 . 669 0. 555- -o. 993 13 17 Tertiary Butanol 1 .769 2. 121 1 .36 5 6. 410 0. 701- -o. 993 12 56 Acetylene 2 .58 3 1 .92 9 0. 415 0. 623- -0. 989 20 10 Ethylene 1 .53 3 1 .22 6 0. 772 1 .62 7 0. 599- -0. 981 19 10 CO Table 4.2 Continued Compound SRK PR LM(1) LM(2) Tr Range N Ref. t Propylene 1 .25 4 0 .666 0 .699 1 .05 1 0 .618--o .989 26 10 1-Butene 1 .53 6 0 .923 0 .874 1 .03 4 0 .651- -0 .979 26 10 Water 13.34 7 .905 5,.82 3 12.19 0,, 422--0, .995 43 10 n-Deuter1 urn 9. 206 5 .065 1 .. 359 2 .80 1 0. 488--0 .991 21 97 n-Hydrogen 2. 096 3..67 1 0.,69 0 1 .75 3 0.,615 - -o. .964 13 69 p-Hydrogen 2. 157 2,,91 0 0..70 5 1 .65 5 0. 615- -o, .970 13 97 Ni trogen 1 .50 3 1 .. 560 0.,63 3 2 .89 2 0.,614 --0, .991 18 10 Ammon i a 0. 910 0 .746 1..46 5 1 .59 9 0,,602 --o. .986 29 10 Oxygen 1 .14 0 0 . 795 0,.64 3 1 .79 3 0., 583--0, .986 24 10 Neon 9 .07 4 7 .71. 6 0.. 77 1 1 .78 2 0. 563- -O,.96 8 19 126 Argon 1 .37 1 1 .45. 0 0. 677 0. 683 0. 555- -0, .994 15 126 Krypton 1 .10 4 1 ,04. 3 0. 599 1 .27 7 0. 573- -0. ,984 20 126 Xenon 1 .17 4 1 ,31. 3 0. 694 1 .53 9 0. 569- -0. 932 12 1 16 t Vapor pressure, liquid and vapor volumes, enthalpy and entropy of vaporization data are taken from the same source. Table 4.3 Region I - Liquid and Vapor Volume RMS % Errors Liquid Volume Vapor Volume Compound SRK PR LM(1) LM(2) SRK PR LM( 1) LM(2) Methane 9. 722 8 .52 3 8. 372 8. 549 2 . 765 4 .554 2 ,71, 0 3.767 Ethane 18 .03 9. 290 8 .15 7 8 .08 6 3 . 140 2 .385 0.,84 2 0.946 Propane 15 .35 6 .77 5 6. 109 5 .89 6 0 .771 2 .863 1 ,74. 1 1 .410 n-Butane 22 .61 1 1.9 1 10. 73 9. 883 4 .93. 2 2 . 280 2 .32 3 2.998 1-Butane 20 .71 1CI . 53 9. 456 8. 623 6,, 244 1 .544 3. 156 4.975 n-Pentane 24 .27 13 .41 121. 05 1 1. 30 4 .84. 3 0 .981 2 .37 1 3.780 i-Pentane 22 .91 12 .56 1 1. 19 10. 53 5 ,88. 4 1 . 122 2 .91 7 4 . 543 Neopentane 20 .57 10i . 85 9. 842 9. 035 3 .71 8 2 , 195 1 .01 5 4 . 220 n-Hexane 17 .71 7 .17 9 5. 912 6 .01 8 1 .93 6 1 ., 763 2 .83 8 4 . 376 n-Heptane 18 .20 6 .89 1 5. 790 5 .90 6 2 ,66. 0 1 ,87, 1 2 .31 8 2.578 n-Octane 20 .08 7. 086 6. 779 6 .92 9 1 .32 5 3 ,, 799 3. 413 3 . 724 Benzene 14 .60 6. 009 5. 144 5. 236 2 .36 6 3.,67 9 2. 308 5.892 Carbon Monoxide 5.819 9. 227 9. 286 9 .45 8 4 .79 7 3,,61 8 4.07 1 7.277 Carbon Dioxide 15 .78 6. 191 5.80 4 2 .29 3 1 .77 2 1 .86 7 Carbon Disulfide 9. 156 6. 861 6. 888 6 .86 6 5 .52 8 5. 023 8. 480 8.411 Hydrogen Sulfide 12 . 19 7 .73 8 7. 378 7. 129 4 .28 3 6 .41 0 5. 522 8 .038 Sulfur Dioxide 18 .41 7 .47 5 6 .13 3 5. 734 2. 456 1 .51 0 2 .32 5 7.878 Methanol 41 .69 26 .02 24 .40 25i.0 1 13 . 77 8 .51 9 1 1. 20 9.961 Ethanol 26 .45 12 .92 1 1.8 6 12 .05 5 .04 1 1 .64 1 5 .28 6 8.747 1-Propanol 23 . 18 1 1.0 5 10i .77 9. 147 8 .31 1 10.03 5. 927 11.51 Tertiary Butanol 9.883 4 .60 5 4. 700 5. 806 5. 584 3. 926 5.05 1 12.99 Acetylene 14 .68 6. 344 5. 314 1 .28 5 1 .23 9 2 .45 9 E thy1ene 9.648 7 .24 6 7 .05 4 6 .97 1 2 .51 7 2 .06 8 2. 483 2.006 Table 4.3 Continued L1qu1d Volume Vapor Volume Compound SRK PR LM( 1 ) LM(2) SRK PR LM( 1 ) LM(2) Propylene 11 .85 6.004 5.539 5 .473 2 .887 1 .58 2 1 .83. 5 1 .876 1 -Butene 13.36 5. 147 4.727 4.555 1.615 2. 627 1 .73 2 0.983 Water 41.19 25.82 23.44 25.63 19.28 10. 16 10.97 13.31 n-Deuterium 7.544 15.98 15.92 15 .66 14 . 10 7 .69 4 6 ,28. 6 4.311 n-Hydrogen 6.990 15.46 16 . 20 15 .84 2 .982 3. 090 2 .31 6 2.352 p-Hydrogen 6.932 14. 10 14.66 14.39 2 .689 5. 333 1 .98 9 4 .080 Nitrogen 9.506 8.853 8.750 9.000 3.477 5 .19 8 3 ..62 6 4 .016 Ammonia 30.75 15.90 15.48 15.13 7 .656 6. 036 7 .28 6 8.378 Oxygen 6.661 8.828 8 .873 8.688 3.860 4. 888 3 .69 5 3 .999 Neon 5.449 13.23 12.62 12.72 1 1 . 77 9. 438 0. 779 1 .969 Argon 7.579 9.876 9.864 9.853 2.248 2. 887 1 .31 5 1 .202 Krypton 7.946 8.528 8.488 8.412 1 .849 3. 044 1 .81 8 1 .772 Xenon 6.481 7.900 7 .995 7.912 1 . 335 2 .10 9 0. 915 1 .437 Table 4.4 Region I - Enthalpy and Entropy of Vaporization : Average Absolute Deviation Enthalpy (cal/mole) Entropy (cal/mole-K) Compound SRK PR LM( 1) LM(2) SRK PR LM( 1 ) LM(2) Methane 48.9 30.8 30.6 36.3 0..33 3 0.. 183 0,.21 0 0.278 Ethane 97. 1 82. 1 56.8 51.9 0.. 356 0., 283 0,.22 8 0.202 Propane 87.3 66 .0 71 .9 48.3 0., 283 0. 205 0.. 252 0. 176 n-Butane 129.7 131.7 94 .0 43.7 0.. 321 0. 330 0.. 260 0. 127 1-Butane 69.0 63.7 40. 1 83.8 0. 181 0. 167 0.. 130 0.233 n-Pentane 167.3 139.8 95. 1 60.7 0..40 6 0. 327 0..25 7 O. 168 1-Pentane 141.6 124.3 52.6 75.6 0.. 337 0. 289 0.. 142 0. 196 Neopentane 164.0 142.4 112.9 78 .9 0. 415 0. 348 0.,30 2 0. 239 n-Hexane 1 16.6 102.2 84.4 144.6 0. 287 0. 262 0. 250 0.442 n-Heptane 127 .0 93.0 100. 3 123 .0 0. 309 0. 224 0. 277 0.344 n-Octane 113.2 141.5 159 .5 194 .7 0. 236 0. 369 0, 457 0.548 Benzene 130.4 121.2 74 . 3 105 . 1 0. 296 0. 285 0. 200 0. 232 Carbon Monoxide 22.4 26.9 16.0 35. 1 0. 226 0. 312 O. 193 0.432 Carbon Dioxide 57.9 41.3 59.8 0. 231 0. 162 0. 236 Carbon Disulfide 224.4 146.7 241 .3 325. 3 0. 587 0. 362 0. 606 0.949 Hydrogen Su1f i de 107 .5 136.5 145 .8 183 .0 0. 301 0. 435 0. 468 0.587 Sulfur Dioxide 99.2 77.7 118.8 262 . 1 0. 187 0. 141 0. 359 0.793 Methano1 563.3 433.8 402 . 1 406 .0 1.64 3 1 .25 0 1 .16 6 1 .235 Ethanol 236.2 87 . 2 281 .6 283.5 0. 703 0. 250 0. 856 0. 781 1-Propanol 275. 1 365.9 309.7 363. 1 0. 620 0. 942 0. 844 0.915 Tertiary Butanol 246.8 195.0 298 .7 473.9 0. 574 0. 437 0. 705 1 .056 Acetylene 1 10.2 97.5 49.9 0. 441 0. 397 0. 213 E thy1ene 52.9 27.4 36.9 25.8 0. 239 0. 145 0. 201 0. 153 Table 4.4 Continued Enthalpy (cal/mole) Entropy (cal/mole-K) Compound SRK PR LM( 1) LM(2) SRK PR LM( 1 ) LM(2) Propylene 82.7 48.8 47 .0 41.1 0.. 268 0. 174 0.. 158 0. 138 1-Butene 84.6 67.5 98 . 7 72.3 0. 241 0. 184 0,.30 1 0.217 Water 426.8 284.5 349.5 462.8 1 ,, 144 0. 729 0,.89 4 1 .378 n-Deuterlum 27.0 18.9 13.4 13.9 1 ,05. 7 0. 719 0.,46 7 0.539 n-Hydrogen 9. 10 5.20 5.20 6.70 0..38 6 0. 203 0..22 0 0.281 p-Hydrogen 9.20 6 . 70 4.40 7 .00 0,, 380 0. 252 0.. 180 0. 281 Nltrogen 28.3 28 .8 26.4 31.5 0..25 0 0. 254 0,, 247 0.326 Ammonia 209.8 174.5 220.5 213.2 0,,43 4 0. 326 0..48 9 0.443 Oxygen 39.4 23.8 25. 1 25.6 0., 330 0. 181 0.,22 1 0.215 Neon 26.5 20. 4 3.50 6.80 0. 851 0. 648 0. 1 18 0. 229 Argon 32.4 17 . 1 24.9 23.2 0., 291 0. 122 0. 239 0.221 Krypton 42.0 30.4 38.3 30.0 0. 260 0. 171 0. 241 0. 175 Xenon 39 .9 25 .6 38 .0 34 .6 0., 192 0. 118 0. 186 0. 156 Table 4.5 Overall Average Errors (RMS % and AAD) for the Four Regions (NC = number of compounds ; N = number of data points) Region I : Saturation Property SRK PR LM(1) LM(2)t NC N P (RMS %) 2.57 2.31 1.36 3.24 36 933 V1 (RMS %) 15.94 10.34 9.77 9.92 36 933 Vv (RMS %) 4.78 3.86 3.53 4.99 36 933 123.5 100.7 103.6 128.6 36 933 AHv (AAD:cal/mole) AS„ (AAD:cal/mole-K) 0.433 0.339 0.355 0.432 36 933 Region II : Subcritical Property SRK PR LM NC N P (RMS %) 1.00 0.92 0.96 22 947 V (RMS %) 1.81 1.45 1.53 22 947 T (RMS %) 0.61 0.60 0.63 22 947 H-H° (AAD:cal/mole) 49.5 45.3 41.7 22 947 S-S° (AAD:caymole-K) 0.111 0.112 0.097 22 947 A-A° (AAD:cal/mole) 17.5 17.7 17.5 22 947 G-G° (AAD:cal/mole) 19.7 21.3 21.6 22 947 U-U° (AAD:caymole) 44.2 44.3 40.8 22 947 f/P (AAD) 0.0064 0.0081 0.0084 22 947 Cp (AAD:cal/mole-K) 0.4 0.4 0.7 10 370 u (RMS %) 13.7 14.2 18.4 6 233 Region III : Supercritical Property SRK PR LM NC N P (RMS %) 24.98 11.80 12.26 22 3417 V (RMS %) 7.45 4.57 5.10 22 3417 T (RMS %) 3.40 3.00 3.87 22 3417 H-H° (AAD:caymole) 77.6 86.1 106.1 22 3417 S-S° (AAD:cal/mole-K) 0.152 0.155 0.202 22 3417 A-A° (AAD:cal/mole) 37.9 35.3 35.4 22 3417 G-G° (AAD:cal/mole) 55.1 42.9 45.5 22 3417 U-U° (AAD:cal/mole) 81.5 85.9 106.9 22 3417 f/P (AAD) 0.0217 0.0156 0.0174 22 3417 Cp (AAD:cal/mole-K) 3.5 3.5 4.0 10 1486 M (RMS %) 21.2 35.5 34.1 5 432 Table 4.5 Continued Region IV : Compressed Liquid Property SRK PR LM NC N P (RMS %) >100. >100. >100. 18 733 V (RMS %) 14.92 7.97 7.79 20 878 T (RMS %) 17.55 8.42 8.36 18 733 H-H° (AAD:cal/mole) 108.2 120.7 132.4 20 878 S-S° (AAD:cal/mole-K) 0.553 0.765 0.621 7 261 A-A° (AAD:cal/mole) 61.4 56.7 59.5 7 261 G-G° (AAD:cal/mole) 55.5 55.2 58.2 7 261 U-U° (AAD:cal/mole) 108.8 119.6 130.3 20 878 f/P (AAD) 0.0361 0.0218 0.0228 7 261 CD (AAD:cal/mole-K) 3.4 2.5 4.6 8 417 t Only 34 compounds (N = 881) were tested. 61 ASSUME P CALCULATE a, A, B ADJUST P SOLVE FOR Z1 & ZV CALCULATE fV & f1 NO CALCULATE AH & AS CALCULATE ERRORS CALCULATE AAE & RMS % ERROR NO YES /OUTPUT Z1, ZV, P\ yAH, AS, ERRORSy Figure 4.1 Algorithm for calculating saturation properties. LM SRK PR Figure 4.2 Region I : Error distribution curves of methane. LM SRK PR Figure 4.3 Region I : Error distribution curves of n-pentane. 64 Figure 4.4 Region I : Error distribution curves of 1-propanol. LM SRK PR Figure 4.5 Region I : Error distribution curves of argon. 66 o EXPT LM SRK PR o i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 10 8 6 2 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 Reduced Temperature. Tr Figure 4.6 Region I : Enthalpy and entropy of vaporization of methane. 67 Figure 4.7 Region I : Enthalpy and entropy of vaporization of n-pentane. 68 Figure 4.8 Region I : Enthalpy and entropy of vaporization of 1-propanol. Figure 4.9 Region I : Enthalpy and entropy of vaporization of argon. 70 4.2. REGION II : SUBCRITICAL The subcritical region is considered to be the area between the saturated vapor curve and the critical isotherm (Figure 1.1). In this region, predictions of PVT, five departure functions which include enthalpy (H-H°), entropy (S-S°), Helmholtz free energy (A-A°), Gibbs free energy (G-G°), and internal energy (U-U°); fugacity coefficient (f/P), isobaric heat capacity (Cp), and Joule-Thomson coefficient (u) from the SRK, PR, and LM equations have been tested against experimental data. The data sources of the 22 compounds are given in Table 4.8. Since the LM(2) equation does not show any real advantages over the LM(1) equation in various property predictions in Region I, the LM(2) equation is not considered any further. The Lielmezs-Merriman equation is now simply LM instead of LM(1) used in Region I. Data taken from reference [122] are in departure function form and some of the data are calculated from a modified Benedict-Webb-Rubin [18-22] equation, and some are calculated from the Boublik-Alder-Chen-Kreglewski [3,8,13,52-55,106] equation. Only PVT, enthalpy and entropy departure functions data were taken because other departure functions can be calculated as follows : (G G°) = (H -- H°) - T(S - S°) (4.4) (A - A°) = (G -- G°) - RT(Z •- 1) (4.5) (U - U°) = (H -- H°) - RT(Z •- 1) (4.6) f H - H° s - S° po In - = — — + In — (4.7) P RT R P 71 where P°, 1 atm, is the reference state pressure. Canjar et al [10] have tabulated H°, S°, H, and S in their data tables. In order to convert these data to departure functions, H° and S° were fitted by the method of least-squares to a potynominal expression of the form 2 3 H° = A + BT + CT + DT (4>8) and 2 3 S° = E + FT + GT + HT (A q-i where H° is in kcal/mole, S° in cal/mole-K, and T in K. The coefficients of the polynomials are tabulated in Tables 4.6 and 4.7. The data, H and S, are converted to departure functions by the followings: (H - H°) = H (from source) - H° (from polynomial) (4.10) (S - S°) = S (from source) - S° (from polynomial) (4.11) The enthalpy departure function and fugacity coefficient equations derived from the generalized cubic EOS, Eq.(1.2), are shown in the Appendix A. The results of the PVT predictions from the three EOS are tabulated in Tables 4.8 and 4.9. All three equations are very similar in accuracy for pressure and temperature predictions. The compressibility factors of i-butane and water are plotted against reduced temperature in Figure 4.10 and against reduced pressure in Figure 4.11. These two figures show that the predictions of 72 the PR and LM equations are almost identical for both nonpolar (i-butane) and polar (water) compounds. From Figure 4.10, it is evident that the SRK equation is more accurate near the critical isotherm whereas the PR and LM equations are better in the region below Tr = 0.95. The SRK equation tends to overpredict the compressibility factors for the entire temperature range considered. The PR and LM equations underpredict in the region above Tr = 0.95 and overpredict for the rest. Although only one isobar has been plotted for each of the two compounds, the above observation is true in general for other isobars and compounds. Figure 4.11 shows that as the pressure increases, the prediction from the SRK equation deviates more and more from the experimental data. The PR and LM equations are practically identical along the entire pressure range. Table 4.5 shows that for this region, the PR equation is the most accurate in volume prediction and the SRK is the least accurate. The results of the departure functions: enthalpy, entropy, Helmholtz, Gibbs energy, and internal energy, and fugacity coefficient are tabulated in Tables 4.10 to 4.12. Note that the errors are not in RMS % but AAD. The reason is that departure functions can take on both positive and negative values or even zero. Therefore deviation from experimental data is chosen over % error as the basis of comparison. These tables show that there are very small differences among the three equations in accuracy. Figures 4.12 to 4.23 have the plots of the five departure functions and fugacity coefficient of i-butane and water versus reduced temperature and pressure. For the enthalpy, entropy, and internal energy departure functions, the LM equation behaves rather radically in the region above Tr = 0.95 and below this temperature, the predictions are very similar to the 73 SRK and PR equations as can be seen in Figures 4.12, 4.14, and 4.20. This radical behavior of the LM equation is due to the fact that the derivative of the a-function with respect to temperature becomes infinite as it approaches the critical temperature. This causes the three departure functions rise rapidly. This behavior, however, does not appear in the Helmholtz free energy and Gibbs free energy departure functions, and fugacity coefficient because these three functions involve both the enthalpy and entropy and the subtraction of these two functions cancels out the effect near the critical region. Thus, these three functions are well-behaved even near the critical temperature. Although this behavior of the LM equation is so different than the PR and SRK equations, the overall average deviations from the LM equation for the enthalpy, entrop37, and internal energy departure function predictions are the lowest among the three equations. For Helmholtz and Gibbs free energy departure functions, the predictions from the three equations are very similar. Figures 4.16, 4.17, 4.18, and 4.19 show that the three curves are practically on top of each other. Figures 4.22 and 4.23 have the plots of fugacity coefficients versus reduced temperature and pressure. Again, these two figures show that almost no differences are observed between the PR and LM equations. In terms of overall average deviations, the SRK equation provides the most accurate prediction for fugacity coefficient. The isobaric heat capacity data taken from reference [122] are given in the departure function form, (C -C °). To convert these data to heat capacity Cp, the ideal gas heat capacity is calculated from correlations given by Reid et al [94]. Table 4.13 has the results from the three equations. The results show that the PR and SRK equations are practically identical in estimating this property. 74 The higher errors found in the LM equation are again due to the points near the critical isotherm. For the region away from Tr = 0.95, the predictions of the LM equation are similar to the other two equations. Only six compounds are tested for Joule-Thomson coefficient, LL. It is difficult to make a general statement about the predictive accuracy of the three equations at this stage. However, for the six compounds studied here, the RMS % errors from the SRK and PR equations are almost the same. The LM equation gives the highest % errors among the three equations. The results are summarized in Table 4.14. From the above discussion, it is clear that very small differences are observed among the three equations. No one equation exhibits clear advantage over the others in all property estimations. For the region below Tr = 0.95, the PR and LM equations are identical in accuracy in almost all properties tested here. Both of these equations give lower errors than the SRK equations. One disadvantage of the LM equation is that near the critical temperature, the a-funtion becomes infinite and therefore the equation gives unreasonable predictions. Table 4.6 Coefficients of Ideal Gas Enthalpy Polynomial Compound A B(10') C(10s) D(10*) Temp. Range Variance n-Hexane -30.8531 7 . 19076 : 52.6134 -11.044 298-1500 0 .0030616 Carbon Dioxide -93.9989 6.. 16733 5.33631 -1.25588 200-1500 0 .00020066 Sulfur Dioxide -70.3422 7..1513 6 4.83196 -1.23718 200-1500 0 .00016076 Acetylene 53.8343 8 .07757 5.58979 - 1.08690 298-1500 0 .00013930 Propylene 8.74913 3 .61053 22.3577 -4.58043 298-1500 0,.0004815 0 Water -56.9968 7.. 14874 1 .36210 -0.0135023 200-1500 0..0000345 2 Ammon1 a -9.23274 6. 82980 2.33560 0.704254 200-800 0..0000132 5 Table 4.7 Coefficients of Ideal Gas Entropy Polynomial Compound E F(10!) G(10') H(10") Temp. Range Variance n-Hexane 55.8375 13.3747 -3.36104 4.19050 298-1500 00020731 Carbon Dioxide 40.7596 4.04572 -2.25648 5.70186 200-1500 01 134 Sulfur Dioxide 48.0444 4.44643 -2.61622 6.73076 200-1500 01998 Acetylene 36.4355 4.53894 -2.32769 5.58302 298-1500 003852 Propylene 47.4536 5.94086 -1 .58523 2.24722 298-1500 00016728 Water 35.5415 3.84262 -2.54623 7.13967 200-1500 03261 Ammonia 33.7072 5.54379 -5.65984 26.9856 200-800 003709 Table 4.8 Region II - Volume RMS % Error Compound SRK PR LM Tr Range Pr Range N Ref. t Methane 0 . 565 2 . 759 3,. 258 0,.629 - -0 .986 0 .022- -0 .895 51 122 E thane 1 .227 1 . 535 2,,09 5 0,.655 - -0 .982 0 .021- -0,.89 2 26 31 Propane 1 .298 1 . 1 18 1 ,, 369 0,.649 - -0 .995 0 .024- -0 .952 40 122 n-Butane 0 .802 1 .034 1 ,, 138 0,.659 - -0 .988 0 .027- -0,. 747 35 18 i-Butane 2 . 779 0 .998 1 ., 1 15 0,.662 - -0,.99 9 0 .028- -0,.97 2 59 19 n-Pentane 1 .740 0 .589 0,,70 8 0,.660 - -0,.98 0 0 .030- -0,.84 2 31 20 i-Pentane 3 .279 1 .058 0.,65 9 0.,673 - -0..98 8 0 .030- -0,, 899 37 21 Neopentane 1 . 197 1 .211 1 .45 5 0,,669 - -o,,99 1 0 .032- -o,.91 9 28 22 n-Hexane 3 . 224 1 . 154 0.,84 0 0.. 722-o- .,99 5 0 .033- -0,,91 0 40 10 n-Heptane 0 .432 1 .425 1.,72 9 0..703 - -0..96 3 0 .037- -0,. 731 35 122 n-Octane 0 .739 1 .778 2, 065 0..703 - -0..98 5 0,.041 - -0..80 6 37 122 Benzene 0 . 335 0..81 6 0. 990 i'O..658 - -0,.97 8 0,.021 - -0. 811 51 122 Carbon Dioxide 3 .904 2..38 1 2. 388 0. 730- -0. 986 0,.014 - -0. 888 33 10 Sulfur Dioxide 1 .45. 3 1 .. 140 1 .50 0 0. 645- -0. 993 0..013 - -o. 918 56 10 Methanol 1 .64. 4 2..50 1 2. 795 0. 702- -o. 974 0.,013 - -o. 628 51 122 Ethanol 0..78 4 1 ,65. 0 1 .87 1 0. 701- -o. 973 0.,017 - -o. 652 43 122 1-Propanol 0.,60 2 1 ,31. 8 1 .32 1 0. 708- O. 968 0. 020- -o. 590 40 122 Acetylene 6 . 196 2 ,96, 1 2 .32 6 o. 97 1--o. 998 0..016 - -0. 971 43 10 Propylene 2 . 101 1 ,, 256 1 .26 6 0. 822- -0. 989 0, 022- -0. 895 37 10 1-Butene 0.. 283 0. 678 0. 737 0. 667- -0. 953 0.,025 - -o. 498 33 122 Water 2 , 157 1 .19 9 o. 840 0. 652- -0. 995 0. O05--0. 935 93 10 Ammon i a 3 .08. 1 1 .35 4 1. 196 0. 630- -0. 986 0. 009- -0. 856 48 10 t PVT, departure function, and fugacity coefficient data are taken from the same source. Table 4.9 Region II - Pressure and Temperature RMS % Errors Pressure Temperature Compound SRK PR LM SRK PR LM Methane 0 .309 1 .458 1 .644 0 . 207 0 .878 0 .947 E thane 0 .581 0 . 797 0 .929 0 .371 0 .487 0 .531 Propane 0 .650 0 .711 0 . 784 0 .456 0 .476 0 .520 n-Butane 0..60 3 0 . 737 0 .800 0 .428 0 . 494 0 .523 1-Butane 1 .09, 6 0 .635 0 .684 0 . 546 0 .412 0 .506 n-Pentane 1 .03, 3 0 .462 0,.52 0 0 .647 0 . 346 0 .372 i-Pentane 1 ,. 324 0,.38 8 0,, 336 0,.66 3 0.. 223 0..23 0 Neopentane 0.,69 3 0,.71 4 0., 783 0,.48 0 0.. 462 0,.48 2 n-Hexane 2.,03 0 0. 828 0,,67 0 1 .22. 3 0..58 2 0..50 2 n-Heptane 0.,34 9 1, 010 1, 135 0..26 7 0.. 738 0.,80 3 n-Octane 0. 563 1.,26 3 1.,40 1 0..41 6 0. 911 0.,98 0 Benzene 0. 170 0. 621 0. 686 0. 1 14 0. 478 0.,51 1 Carbon Dioxide 2. 687 1. 732 . 1. 729 1 .66 2 1. 125 1 ,. 127 Sulfur Dioxide 0. 851 0. 631 0. 769 0. 536 0. 395 0. 441 Methanol 1 .21 6 1. 836 2. 022 0. 765 1. 188 1. 297 Ethanol 0. 579 1. 193 1 .32 8 0. 368 0. 761 0. 826 1 -Propanol 0. 474 1.01 5 1 .01 0 0. 327 0. 699 0. 693 Acetylene 2. 523 1. 141 0. 908 1 .20 5 0. 511 0. 390 Propylene 1 .07 4 0. 816 0. 864 0. 658 0. 530 0. 558 1 -Butene 0. 209 0. 543 0. 581 0. 154 0. 422 0. 445 Water 1 .20 6 0. 756 0. 702 0. 801 0. 560 0. 554 Ammon) a 1 .85 9 0. 850 0. 795 1 .09 9 0. 563 0. 546 Table 4.10 Region II - Enthalpy and Entropy Departure Functions : Average Absolute Deviation Enthalpy (cal/mole) Entropy (cal/mole-K) Compound SRK PR LM SRK PR LM Methane 8 . 1 5.0 10.6 0.053 0.058 0.051 Ethane 35.9 29 . 2 19.9 0. 120 0. 123 0.093 Propane 35.5 29 . 1 24.2 0. 101 0. 102 0.076 n-Butane 40.5 33.2 29.2 0.098 0.099 0.090 i-Butane 86.8 73.3 61.2 0.205 0. 206 0. 162 n-Pentane 59.8 51 .5 47.3 0. 126 0. 127 0.119 1-Pentane 54.6 44 .0 28.9 0. 107 0. 108 0.076 Neopentane 49.4 40.7 31.2 0.116 0.117 0.094 n-Hexane 183.0 171.2 146.5 0. 173 0. 172 0. 1 10 n-Heptane 13.2 16.8 20.0 0.024 • 0.025 0.034 n-Octane 22.2 27.2 37. 1 0.034 0.035 0.054 Benzene 7.8 9.2 12.5 0.017 0.018 0.025 Carbon Dioxide 32. 3 35 . 1 38 . 2 0. 147 0. 148 0. 149 Sulfur Dioxide 48. 1 40. 7 33.5 0.118 0. 118 0.092 Methanol 24. 1 27 .0 33.8 0.038 0.038 0.053 Ethanol 16.8 19. 1 24.2 0.030 0.030 0.042 1-Propanol 15.0 16.9 15.5 0.028 0.029 0.029 Acetylene 65.9 56.0 33 .8 0. 154 0. 154 0.099 Propylene 61.1 54 .8 48.2 0. 137 0. 138 0. 123 1-Butene 9.5 8.5 9.0 0.029 0.031 0.032 Water 72.7 68 . 1 80.8 0.181 0. 182 0. 160 Amnion 1 a 147 . 1 139.0 131.6 0.398 0.398 0.381 oo Table 4.11 Region II - Helmholtz and Gibbs Free Energy Departure Functions : Average Absolute Deviation Helmholtz (cal/mole) Gibbs (cal/mole) Compound SRK PR LM SRK PR LM Methane 0.5 0.7 0.6 1 . 3 5.7 6. 1 Ethane 1 . 1 1 .6 1 .0 2.0 5.9 6.3 Propane 1 .8 2.2 1 .9 2.0 6. 1 6 . 4 n-Butane 0.7 1 .0 0.8 2.9 5.7 6.0 1-Butane 3.5 4.6 4 . 1 5.9 8.7 9 . 1 n-Pentane 1 .4 1 . 7 1 .4 6.3 4 . 2 4.7 1-Pentane 3.0 3.6 3.0 8 . 2 3.3 3.9 Neopentane 1 . 1 1 .6 1 . 3 3.7 6.5 6 . 9 n-Hexane 94.5 94 . 4 94 . 5 109.0 101 .0 100. 2 n-Heptane 1 .6 1 .7 1 .4 4.5 10. 1 10.7 n-Octane 1 .0 1 . 1 1 .0 6. 1 12.5 13.4 Benzene 1 . 5 1 .6 1 .5 2. 1 6.5 6.8 Carbon Dioxide 65 .8 66 . 1 65.9 53.3 59.6 59.7 Sulfur Dioxide 28.0 27.8 28 . 1 31.7 26.6 26.2 Methanol 0.9 0.9 0.8 8.3 12.5 13.3 Ethanol 1 .4 1 . 5 1 . 3 5. 1 9.3 9.9 1-Propanol 1 .0 1 .0 1 .0 4.2 8. 1 8.0 Acety1ene 17.6 17.7 17.5 24.3 19.9 19.5 Propylene 30.3 30.0 30.2 34 .8 27 .9 27 .6 4.4 4.5 1-Butene 1 .3 1 . 3 1 .3 1 . 2 Water 58.4 58.6 58 .0 52 .2 58.8 60.3 Ammon1 a 67.9 67 . 9 68.0 64 .6 66 . 3 66 .4 Table 4.12 Region II - Internal Energy Departure Function and Fugacity Coefficient Average Absolute Deviation Internal Energy (cal/mole) Fugacity Coefficient Compound SRK PR LM SRK PR LM Methane 8.6 9.4 8.6 0.0029 0.0123 0.0132 Ethane 32.8 33 . 1 24 .8 0.0030 0.0078 0.0083 Propane 31.9 31.9 24.3 0.0029 0.0071 0.0074 n-Butane 36.9 37 . 1 33.5 0.0035 0.0060 0.0063 1-Butane 77 .5 76.7 62.3 0.0059 0.0084 0.0087 n-Pentane 52 . 1 52 . 2 48 .8 0.0064 0.004 1 0.0045 1-Pentane 43.4 43.0 29. 1 0.0077 0.0031 0.0036 Neopentane 44 .6 44 . 5 34 .5 0.0042 0.0064 0.0067 n-Hexane 164 . 3 163.7 140.8 0.0216 0.0119 0.0109 n-Hep tane 12.0 12.4 16.3 0.0041 0.0093 0.0098 n-Octane 18.4 18 . 7 28.9 0.0053 0.0108 0.0115 Benzene 7.4 7.9 11.8 0.0020 0.0058 0.0060 Carbon Dioxide 38.2 38 .4 41.3 0.0117 0.0204 0.0205 Sulfur Dioxide 43.5 43 . 1 33.6 0.0039 0.0078 0.0084 Methanol 18.2 18.0 24.8 0.0079 0.0118 0.0126 Ethanol 14.4 14.3 19.4 0.OO49 0.0088 0.0093 1-Propano1 13.3 13.5 13.1 0.0040 0.0076 0.0074 Acetylene 49.8 49 . 1 34.5 0.0158 0.0044 0.0037 Propylene 56 . 3 56 . 5 50.5 0.0044 0.0109 0.0113 1-Butene 8.9 9 . 4 9.9 0.0015 0.0053 0.0055 Water 64.2 64.5 77.5 0.0057 0.0056 0.0058 Amnion 1 a 136.5 136 .2 129.4 0.0114 0.0030 0.0028 00 o Table 4.13 Region II - Isobar1c Heat Capacity : Average Absolute Deviation (cal/mole-K) Compound SRK PR LM Tr Range Pr Range N Ref . E thane 1 .8 1 ..7 2 .4 0..655- 0 .982 0 .021-0 .892 26 122 Propane 1 ..4 1 .3 3 .8 0..649-0 ..99 5 0 .024-0,,95 2 40 122 n-Butane 1 .. 1 1 .1 1 ..9 0.,659-0 ,.98 8 0,.027-0 ..74 7 35 122 i-Butane 2 .6 2 .6 3 .3 0..662-0 ..98 0 0..028-0 ..86 1 30 122 n-Heptane 0,, 3 0.. 3 0 .4 0..703-0 ..96 3 0..037-0 .. 731 35 122 n-Octane 0., 2 0.. 3 0 .8 0..703-0 ..98 5 0..041-0 ..80 6 37 122 Methanol 0.. 3 0. 3 0 , 6 0..702-0 .. 974 0..013-0 .. 628 51 122 Ethanol 0.. 2 0. 2 0.. 6 0,,701-0 ,.97 3 0..017-0 ..65 2 43 122 1-Propanol 0. 2 0. 2 0.. 3 0., 708-0.96, 8 0..020-0 ., 590 40 122 1-Butene 0.. 3 0. 4 0,, 4 0. 667-0,.95 3 0..025-0 .,49 8 33 122 Table 4.14 Region II - Joule-Thomson Coefficient RMS % Error Compound SRK PR LM Tr Range Pr Range N Ref . Methane 10.88 13.13 15.92 0.551-0.970 0.011-0.754 132 4 Ethane 9.04 11.55 15.80 0.964 0.021-0.706 6 100 Propane 11.47 10.04 10.80 0.795-0.976 0.041-0.810 32 99 n-Butane 13.97 13.30 13.36 0.692-0.888 0.027-0.409 36 51 n-Pentane 18.51 17.71 17.94 0.698-0.804 0.030-0.164 20 51 Argon 18.18 19.32 36.54 0.684-0.982 0.021-0.414 7 98 82 0.60 0.90 0.88 0.88 - N 0.84 0.80 0.66 0.64 Figure 4.10 Region II : Compressibilitj' factors of i-butane (Pr = 0.56) and water (Pr = 0.31) versus reduced temperature. 1.0 0.9 - o.e tSJ IS 0.7 0.6 0 85 0.4 Figure 4.11 Region II : Compressibility factors of i-butane (Tr = 0.98) and water (Tr=0.96) versus reduced pressure. 83 Figure 4.12 Region II : Reduced enthalpy departure functions of i-butane (Pr = 0.56) and water (Pr=0.31) versus reduced temperature. Figure 4.13 Region II : Reduced enthalpy departure functions of i-butane (Tr = 0.98) and water (Tr = 0.96) versus reduced pressure. Figure 4.15 Region II : Reduced entropy departure functions of i-butane (Tr = 0.98) and water (Tr = 0.96) versus reduced pressure. 85 3.10 4.4 i-Butane Data 3 05 — SRK — PR LM 3.00 2 95 < i < Z.B5 2 80 i.i.i 0.90 0.92 0.94 0.96 0.98 1.00 1.02 Tr Figure 4.16 Region II : Reduced Helmholtz departure functions of i-butane (Pr = 0.56) and water (Pr = 0.31) versus reduced temperature. 86 87 0.85 0.55 i-Butane 0.80 • Data SRK PR 0.75 LM CH 0.70 I 0.65 0.60 0.55 0.25 0.90 0.92 0.94 0.96 0.98 1.00 1.02 Tr Figure 4.20 Region II : Reduced internal energy departure functions of i-butane (Pr = 0.56) and water (Pr=0.31) versus reduced temperature. 0.7 Figure 4.21 Region II : Reduced internal energy departure functions of i-butane (Tr = 0.98) and water (Tr = 0.96) versus reduced pressure. 88 0.84 0.90 0.82 0.88 - 0.80 0 86 0.78 a, 0.84 0.78 - 0.82 - 0.74 0.72 0.80 Figure 4.22 Region II : Fugacity coefficients of i-butane (Pr = 0.56) and water (Pr = 0.31) versus reduced temperature. 1.05 1.05 Water 1.00 1.00 • Data SRK PR LM 0 90 0 90 OH 0.85 0.85 0.80 0.80 0.75 0.75 0.70 0.70 0.85 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Pr Figure 4.23 Region II : Fugacity coefficients of i-butane (Tr = 0.98) and water (Tr = 0.96) versus reduced pressure. 89 4.3. REGION III : SUPERCRITICAL In the supercritical region (Figure 1.1), predictions of PVT, five departure functions which include the enthalpy, entropy, Helmholtz free energy, Gibbs free energy, and internal energy; fugacit3' coefficient, isobaric heat capacity, and Joule-Thomson coefficient obtained from the SRK, PR and modified LM equations have been tested against experimental data. The data sources of the 22 compounds are summarized in Table 4.15. The results of the PVT predictions from the three EOS are tabulated in Tables 4.15 and 4.16. The overall average errors for this region are summarized in Table 4.5. For pressure prediction, the PR and LM equations are very similar in accuracy. The overall average RMS % error from the SRK equation is about double that of the PR or LM equation. For temperature prediction, all three equations are about the same in accuracy where the PR produces the smallest overall average RMS % error while the LM produces the largest. For volume prediction, The PR and LM equations are again almost identical in accuracy for most compounds. However, in terms of the overall average RMS % error, the PR equation gives the lowest. In many cases, the error from the SRK equation is almost twice that of the PR or LM equation. This observation also has been found earlier in the saturation region. Figures 4.24 and 4.25 present the plots of compressibility versus reduced temperature and versus reduced pressure for n-octane and ethanol. These two figures clearly indicate that all three equations are poor for polar compounds and reasonable for nonpolar compounds. These figures once again reinforce the fact that the PR and LM equations are indeed 90 very similar in accuracy for volume prediction and the SRK is the least accurate among the three. The results for the five departure functions and fugacity coefficient are summarized in Tables 4.17 to 4.19. In terms of overall average deviations (Table 4.5), the SRK and PR equations are very similar for enthalpy, entropy, and internal energy departure function predictions. The LM equation is poor near the critical temperature because the derivative of the a-function with respect to temperature approaches infinity as the temperature approaches the critical value. Hence, these three departure functions, enthalpy, entropy, and internal energy, increase rapidly as the critical temperature is approached. Above Tr=1.05, the predictions from the LM equation are very similar to those of SRK and PR equations. This observation can be seen in Figures 4.26, 4.27, and 4.34. For Helmholtz and Gibbs free energy departure functions, error cancellation near the critical points resulted in similar accuracy for all three equations in this region. Figures 4.30 and 4.31 represent the plots of the Helmholtz free energy versus reduced temperature and pressure for n-octane and ethanol. Figures 4.32 and 4.33 are the plots of Gibbs free energy versus reduced temperature and pressure for the same compounds. For fugacity coefficient calculation, the LM equation is inferior to the PR equation. The SRK equation exhibits the highest overall average deviation among the three equations. Fugacity coefficients of n-octane and ethanol are plotted against reduced temperature in Figure 4.36 and against reduced pressure in Figure 4.37. The average absolute deviations of the isobaric heat capacity for ten 91 compounds are tabulated in Table 4.20 along with the experimental data sources. It should be noted that, in general, the isobaric heat capacity data are not accurate near the critical region where the heat capacity changes rapidly with temperature and pressure. In many cases, all three equations give similar predictions. The exceptionally high deviations found in some of the compounds: ethane, propane, n-butane, and i-butane, may be due to the inabilitj' of the EOS to predict this property in the critical region or may be due to the low accuracy of the experimental data. The predictions of the Joule-Thomson coefficient obtained from the three EOS have been tested against experimental data for five compounds. The results are summarized in Table 4.21. The SRK equation gives the lowest RMS % error for four of the five compounds. In terms of overall average RMS % error shown in Table 4.5, The SRK is also the lowest among the three equations. Again, the number of compounds studied is not large enough to justify broad conclusions at this time. Out of the eleven properties studied in this region, the PR equation gives the lowest overall average RMS % errors for pressure, volume, and temperature predictions and also the lowest overall average deviations for Helmholtz free energy, Gibbs free energy, and fugacity coefficient predictions. It is interesting to note that the PR and LM a-functions, Eq.(2.3) and (3.39), are so different, and yet, the accuracy of the two equations is similar for a number of properties. One important point that is observed in Table 4.5 is that between Region H and III, the accuracy of the three equations decrease substantially. The departure 92 function deviations of Region HI, in many cases, are double that of Region II. For pressure, temperature, and heat capacity predictions, the accuracy reduces at least by an order of magnitude for the same number of compounds. Table 4.15 Region III - Volume RMS % Error Compound SRK PR LM Tr Range Pr Range N Ref. t Methane 2. 189 4 .692 4 ,95. 0 1 .007--7 . 343 0 .022- -21 .83 272 122 Ethane 7. 583 5 .013 5.,26 6 1 .002- -1 .768 0 .021--10i . 37 108 31 Propane 6 .42 2 4 .687 4 .93 8 1 .000- -1 .892 0 .024- -16 .66 180 122 n-Butane 9 .79 1 5 .439 6 .05 7 1 .002- -1 .411 0 .027- -18 .68 69 18 i-Bu tane 6. 377 3 . 775 3 .98 9 1 .005--1 .715 0 .028- -1 1 . 1 1 165 122 n-Pentane 8 .97 7 7 .627 7 ,94. 7 1 .001--1 . 491 O .030- -21 .05 144 122 i-Pentane 8. 896 4 . 382 4 .79 4 1 .02 1--1 . 738 0 .030- -8 .99 0 160 122 Neopentane 6. 121 4 .332 4. 694 1 .014- -1 .614 0 .032- -1 1 . 10 153 122 n-Hexane 5. 853 2 .951 3 .68 7 1 .006- -1 . 159 0 .033- -1 .36 5 174 10 n-Heptane 8. 112 3..90 3 4. 332 1 .037- -1 .851 0,.037 - -18 .28 156 122 n-Octane 10.25 5,.54 2 6 .49 1 1 .020- -1 .758 0,.041 --20 .14 143 122 Benzene 5 .15 5 2 ,51, 5 3 .08 2 1 .031--1 . 777 o,,0 2 1--2 .02 7 132 122 Carbon Dioxide 3. 104 1 .46. 8 2 .00 3 1 .022- -4 . 492 0,,014 - -2 .80 4 120 10 Sulfur Dioxide 7 .97 1 3 ,47. 6 4 .23 8 1 .019- • 1.21 2 0..013 - -3 .93 6 180 10 Methanol 1 1. 17 7. 620 8 .18 6 1 .013- -1 .94. 9 0. 013- -6 .90 7 208 122 Ethanol 9 .88 4 6 .12 2 6 .95 4 1 .012- -1 ,. 946 0. 017- -8 .14 7 182 122 1 -Propanol 9. 300 5 .48 0 6 .05 5 1 .043- • 1,86 , 2 0. 020- -9 .87 8 169 122 Acety1ene is l. 60 10.63 12.34 1 .001-• 1,03 , 4 0. 016- • 1 43. 5 63 10 Propylene 3. 565 1 .57 4 1 .90 2 1 .020- •3,.74 5 0. 022- -4 .47 6 130 10 1-Butene 4 .42 1 1 .66 4 1 .66 6 1 .049- •2 ., 383 0. 025- -12 .44 195 122 Water 3 .65 1 2 .38 5 3 .21 3 1 .038- -1 ., 768 0. 005- - 1 7. 14 194 10 Ammon i a 8 .61 6 5 .35 9 5 .38 4 1 .04 1--1 . 452 0. 009- -3 . 056 120 10 t PVT, departure function, fugacity coefficient data are taken from the same source. Table 4.18 Region III - Pressure and Temperature RMS % Errors Pressure Temperature Compound SRK PR LM SRK PR LM Methane 4 .55 5 13.59 13 .79 1.514 7.484 7.846 Ethane 19 .68 11.14 1 1.3 0 2.689 3 . 508 3 . 776 Propane 18 . 15 14.99 15 . 1 1 2.657 5.666 5 .923 n-Butane 32 . 10 15 .85 15 .98 3.754 5.740 6.131 1-Butane 17 .51 1 1 . 18 1 1.2 5 2.626 3.415 3.555 n-Pentane 23 .81 24 . 21 24 .27 3.672 10.44 10.71 1-Pentane 21 . 35 7.537 7 .68 1 3.700 1 . 724 1 . 723 Neopentane 15 .68 12 . 54 12 .66 2 . 169 4 .064 4 . 246 n-Hexane 10 .90 4.266 4 .72 0 1 .310 0.615 1 .450 n-Heptane 26 . 25 6.374 6 .87 1 6 .080 1 . 709 1 .664 n-Octane 40 . 13 10.92 1 1.9 2 7.882 2.360 3.312 Benzene 9. 189 3. 199 3.68 4 1 .397 0.581 0.597 Carbon Dioxide 7. 425 1 . 768 2 .32 4 1 .949 1 . 157 1 .638 Sulfur Dioxide 24 .58 6 . 104 6 .95 4 2.931 1 .029 1 .221 Methanol 75 .74 31.61 32 .74 6 . 258 3.774 8.754 Ethanol 44 .94 17.42 18 .53 5.745 3.211 7 . 505 1-Propano1 34 .04 12.79 13 .31 5.735 3.044 4 .001 Acetylene 68 . 1 1 30.97 32 .43 3.937 2.075 5.491 Propylene 8..49 8 2.601 2 .71 4 1 .455 0.683 0.787 1-Butene 10 .68 3.778 3 .78 8 3 .010 1.315 1 .358 Water 3..41 8 1 .966 2 .52 1 1 . 169 0.689 0.644 Ammon1 a 32 .90 14.77 15 .07 3 .066 1 .720 2 .844 Table 4.17 Region III - Enthalpy and Entropy Departure Functions: Average Absolute Deviation Enthalpy (cal/mole) Entropy (cal/mole-K) Compound SRK PR LM SRK PR LM Methane 16.5 31 .4 39.3 0.045 0.035 0.082 Ethane 44.2 59 .5 73.7 0. 134 0. 155 0. 197 Propane 51 .5 74 .6 173.6 0. 132 0. 152 0.403 n-Butane 79.4 83 .4 179.5 0. 177 0. 188 0. 379 i-Butane 66.0 77 . 3 99.0 0. 133 0. 164 0.211 n-Pentane 189.8 192.3 298.5 0. 369 0. 342 0.582 1-Pentane 108.6 80.2 90. 2 0. 173 0. 160 0. 180 Neopentane 61.8 57.4 81.9 0. 109 0.121 0. 183 n-Hexane 231 . 1 217.7 201 .2 0.313 0.316 0.277 n-Heptane 70.7 98 . 3 91.8 0. 179 O. 168 0. 156 n-Octane 90.0 123.4 129 . 7 0. 229 0.214 0. 229 Benzene 21.8 42.4 42 . 5 0.039 0.047 0.047 Carbon Dioxide 57.2 70. 1 78.9 0.091 0.093 0. 106 Sulfur Dioxide 112.1 113.4 115.1 0. 140 0. 159 0. 196 Methanol 58.6 79.3 88 . 7 0. 124 0.124 0. 140 Ethanol 66.8 78 . 7 1 10. 2 0. 144 0. 143 0. 202 1-Propanol 79.9 98.6 141.4 0. 175 0.171 0.231 Acetylene 98.4 89.6 60. 1 0. 169 0. 182 0. 130 Propylene 33.7 33.5 43.3 0.060 0.060 0.069 1-Butene 27.9 39 . 3 41.0 0.07 1 0.064 0.061 Water 62.7 83. 1 73.4 0. 103 0. 105 0. 1 16 Ammonia 78. 1 70.9 80.9 0.237 0.243 0.268 Table 4.18 Region III - Helmholtz and Gibbs Free Energy Departure Functions : Average Absolute Deviation Helmholtz (cal/mole) Gibbs (cal/mole) Compound SRK PR LM SRK PR LM Methane 12.4 8.6 10.4 9 . 7 29.7 32 . 3 Ethane 12.S 11.7 11.7 18.7 18.9 22.5 Propane 25.9 16.5 15.6 42 . 2 30.0 35.4 n-Butane 29.2 22. 1 21.8 48 .6 28 . 3 30.8 1-Butane 19.6 16.0 15.9 34 . 1 27 . 2 29.9 n-Pentane 51 .5 35. 1 35.0 58 . 2 46.8 50.4 i-Pentane 20.9 21 .6 22.9 48 .0 15.6 18.5 Neopentane 21.2 18 . 1 18.4 28.3 36 . 3 40. 3 n-Hexane 99.0 96.7 95.0 118.4 99. 1 101 . 3 n-Heptane 32 .5 26.4 26.5 83 . 2 33.8 32.9 n-Octane 40. 7 34 . 5 35.3 102.8 47.3 47 . 3 Benzene 9. 1 12.5 12.6 11.8 11.1 12.2 Carbon Dioxide 58.8 59.0 59.0 56.2 58.9 62 .0 Sulfur Dioxide 95 . 1 93. 1 94 .6 102 .0 102 . 1 102 .9 Methanol 13.0 18. 1 16.7 35.4 23. 1 24.7 Ethanol 11.1 11.2 12.0 52 . 1 31.4 39.9 1-Propanol 13.6 13.3 14.6 61.6 35. 1 44.4 Acetylene 33 . 3 30.0 28 .9 55 . 3 40.4 42.7 Propylene 31.1 29.0 29. 1 40.0 25.4 27 . 1 1-Butene 14.3 10.8 10. 7 34 .5 11.3 11.6 Water 111.1 112.4 112.8 103.5 114.2 113.2 Amnion i a 77.4 79 .6 79 .6 67 .0 77 .9 78 . 1 CO Table 4.19 Region III - Internal Energy Departure Function and Fugacity Coefficient Average Absolute Devi at Ion Internal Energy (cal/mole) Fugacity Coefficient Compound SRK PR LM SRK PR LM Methane 18.2 13.8 21.6 0.0104 0.0337 0.0381 Ethane 47 .6 55.4 69.4 0.0135 0.0156 0.0186 Propane 53.2 65.5 164.5 0.0264 0.0186 0.0225 n-Butane 66.9 77.7 170. 3 0.0242 0.0153 0.0168 i-Butane 60.9 74.9 97. 1 0.0180 0.0161 0.0178 n-Pentane 165.8 158 . 3 274.2 0.0284 0.0226 0.0249 1-Pentane 77.4 67. 1 78 . 3 0.0234 0.0096 0.0112 Neopentane 50.8 56.0 86.0 0.0135 0.0204 0.0228 n-Hexane 215.1 214.3 193.4 0.0315 0.0135 0.0157 n-Heptane 106.2 113.5 105.2 0.0426 0.0179 0.0170 n-Octane 141.5 148 . 7 153. 1 0.0521 0.0243 0.0237 Benzene 35.0 40.4 36 . 3 0.0062 0.0065 0.0071 Carbon Dioxide 73 .0 78. 1 93.3 0.0135 0.0133 0.0166 Sulfur Dioxide 96.5 102.8 103.6 0.0301 0.0324 0.0326 Methanol 93.0 99 .5 109.5 0.0187 0.0136 0.0142 E thano1 94.4 101 .9 144.3 0.0271 0.0173 0.0218 1-Propanol 116.6 125 . 3 177 .0 0.0310 0.0185 0.0239 Acety1ene 76.7 78 .6 50.0 0.0207 0.0056 0.0077 Propylene 30.4 33 . 1 40.8 0.0091 0.0076 0.0097 1-Butene 34.9 37 . 1 37.9 0.0203 0.0080 0.0083 Water 74.9 81.8 69.3 0.0055 0.0053 0.0048 Ammonla 64.3 65 .8 75.6 0.0101 0.0069 0.0076 to Table 4.20 Region III - Isobaric Heat Capacity : Average Absolute Deviation (cal/mole-K) Compound SRK PR LM Tr Range Pr Range N Ref. E thane 4 .6 4 ,8 4 . 4 1 .048-1 . 768 0..021-10.3 7 90 31 Propane 7 . 1 7 .2 6 . 6 1 .027-1 .892 0..024-16.6 6 162 122 n-Butane 9., 1 9 .3 9 .8 1 . 129- 1 .411 0 .027-18.68 32 122 i-Butane 9.. 5 9.. 7 8 .9 1 .029-1 .715 0..028 - 11.11 149 122 n-Heptane 0. 8 0,. 7 1 .. 4 1 .037-1 .851 0 .037-18.28 156 122 n-Octane 1 .. 1 1 . 0 3.. 4 1 .020-1 .758 0..041-20.1 4 143 122 Methano1 0.. 7 0.. 6 2 . 3 1 .013-1 .949 0..013-6.90 7 208 122 Ethanol 0. 8 0.. 7 2 . 7 1 .012-1 .946 0.,017-8.14 7 182 122 1-Propanol 0. 9 0. 8 0.. 5 1 .043-1 .862 0.,020-9.82 8 169 122 1-Butene 0. 3 0. 3 0. 3 1 .049-2 . 383 0. 025-12.44 195 122 Table 4.21 Region III - Joule-Thomson Coefficient RMS % Error Compound SRK PR LM Tr Range Pr Range N Ref . Methane 29.56 55.22 54.73 1.049-2.413 0.022-21.55 196 4 Ethane 10.15 14.35 12.79 1.018-1.236 0.021-0.706 35 100 Propane 4.27 4.16 8.66 1.021 0.041-0.891 12 99 Ethylene 40.93 76.61 55.87 1.053-1.495 0.020-49.51 77 23 Argon 21.22 27.44 38.58 1.065-3.799 0.021-4.137 112 98 99 1.1 1.0 0.9 0.9 h tSl 0.8 0.5 0.7 0.3 0.5 1.0 1.2 1.4 1.6 l.B Z.O Tr Figure 4.24 Region III : Compressibility factors of n-octane (Pr=4.03) and ethanol (Pr=1.63) versus reduced temperature. 1.9 1.30 n-Octane Ethanol 1.25 O Data h O Data SRK SRK PR 1.20 PR 1.15 h tsi 1.10 O 1.06 1.00 0.95 _1_ 0.0 2.0 4.0 6.0 6.0 10.0 Pr Figure 4.25 Region III : Compressibility factors of n-octane (Tr=1.76) and ethanol (Tr= 1.95) versus reduced pressure. 100 Figure 4.26 Region in : Reduced enthalpy departure functions of n-octane (Pr=4.03) and ethanol (Pr=1.63) versus reduced temperature. 10.0 Figure 4.27 Region III : Reduced enthalpy departure functions of n-octane (Tr=1.76) and ethanol (Tr=1.95) versus reduced pressure. 101 Figure 4.29 Region III : Reduced entropy departure functions of n-octane (Tr=1.76) and ethanol (Tr=1.95) versus reduced pressure. 102 9.0 10.0 8.0 - 9.0 - 7.0 8.0 Ir- CC 6.0 - 7.0 - < < I i < 5.0 6 0 n-Octane Ethanol O Data O Data SRK SRK PR 5.0 PR LM LM 1 - • i — - J • ._i . 1 1 , 3.0 4.0 1.0 12 1.6 1.0 1.2 1.4 1.8 1.8 2.0 Tr Tr Figure 4.30 Region III : Reduced Helmholtz departure functions of n-octane (Pr = 4.03) and ethanol (Pr=1.63) versus reduced temperature. 12.0 14 0 12.0 10.0 10.0 8.0 6.0 < I < 4.0 n-Octane Ethanol ) O Data O Data ) SRK SRK 20 , )> PR PR > LM LM 0.0 ©• 1 . 1 0.0®— 1- ) —. i i- 0.0 5.0 10.0 15 0 20.0 25.0 0.0 2.0 4.0 6.0 8.0 10.0 Pr Pr Figure 4.31 Region III : Reduced Helmholtz departure functions of n-octane (Tr=1.76) and ethanol (Tr=1.95) versus reduced pressure. 103 9.0 10.0 9.0 I-- 8.0 h 8.0 o E- E- K 7.0 6.0 a O i I 6.0 o O Ethanol n-Octane 5.0 O Data O Data SRK SRK 4.0 PR 4.0 PR - LM LM 1 I,I i i 3.0 3.0 1.0 1.2 1.6 1.8 1.0 1.2 1.4 1.6 1.8 2.0 Tr Tr Figure 4.32 Region III : Reduced Gibbs energy departure functions of n-octane (Pr = 4.03) and ethanol (Pr=1.63) versus reduced temperature. 12.0 14.0 12.0 10.0 10.0 8.0 o E- 8.0 6.0 O o I i 6.0 -X O o 4.0 Ethanol O Data I SRK 2.0 1 ) PR < ) LM o.o ©• 0.0 <>——1— - L__- 1_— 1 b— 5.0 0.0 2.0 4.0 6.0 8.0 10.0 Pr Figure 4.33 Region III : Reduced Gibbs energy departure functions of n-octane (Tr=1.76) and ethanol (Tr=1.95) versus reduced pressure. 104 Figure 4.34 Region III : Reduced internal energy departure functions of n-octane (Pr=4.03) and ethanol (Pr=1.63) versus reduced temperature. Figure 4.35 Region III : Reduced internal energy departure functions of n-octane (Tr=1.76) and ethanol (Tr=1.95) versus reduced pressure. 105 106 4.4. REGION IV : COMPRESSED LIQUID In the compressed liquid region (Figure 1.1), the number of compounds studied are limited due to the lack of available experimental data, especially data on departure functions. The predictions of PVT, five departure functions which include the enthalpy, entropy, Helmholtz free energy, Gibbs free energy, and internal energy; fugacity coefficient, and isobaric heat capacity obtained from the SRK, PR, and LM equations have been compared with experimental data. The Joule-Thomson coefficient is not part of this study because data are not available. The experimental data sources are listed in Table 4.22. The RMS % errors from the three EOS for PVT calculations are tabulated in Tables 4.22 and 4.23. The overall average RMS % errors are summarized in Table 4.5. For pressure prediction, all three equations give enormously high errors. Each of the three equations gives more than 100 % RMS error for all compounds considered. For temperature estimation, the magnitude of the errors from the PR and LM equations is the same in all cases while the SRK equation gives substantially higher errors than the PR or LM equation. In terms of an overall average RMS % error (Table 4.5), the errors from the PR and LM equations are about half that of the SRK equation. A very similar conclusion can also be drawn for the volume prediction. Both PR and LM equations are much more accurate than the SRK equation. Overall average RMS % error of the SRK equation shown in Table 4.5 is twice that of the PR or LM equation. Figures 4.38 and 4.39 further illustrate this observation. Figure 4.38 shows the plots of compressibility factor against reduced temperature 107 for n-heptane and methanol and Figure 4.39 shows the plots of compressibility factor against reduced pressure for the same two compounds. Both of these plots clearly suggest that both PR and LM are indeed similar in predicting compressibility factor or volume. As was mentioned earlier, experimental data on departure functions are extremely scarce in this region. The TRC data book [122] contains enthalpy and entropy departure functions data for a limited number of compounds, seven of which have been selected for comparison with the predictions from the three EOS considered in this work. Lu et al [68] proposed a correlation which can be used for calculating enthalpy departure function of compressed liquid. They claim that their correlation can be applied to both hydrocarbons and nonhydrocarbons. Therefore, in addition to the seven compounds from the TRC data book, thirteen other compounds have been added to this study. However, only enthalpy and internal energy departure functions are tested for these additional compounds. The correlation by Lu et al is given as follows: ( o ) (i) H - H" H - H° H - H" + 0) (4.12) RT RT. where (o) H - H" = A + AJPJ. + A P * 0 2 R (4.13) RT, (i) H - H° = B0 + BjPr + B2Pr = (4.14) RT. 108 The quantities Aj and Bj are functions of Tr and are given by the following: 2 A0 = 5.742533 + 0.743206Tr - 3.003445Tr (4.13a) 2 Ax = 0.075271 - 0.500988Tr + 0.443336Tr (4.13b) 2 A2 = -0.017460 + 0.054554Tr - 0.045077Tr (4.13c) 2 B0 = 17.334961 - 18.851639Tr + 5.325703Tr (4.14a) B 2 x = 0.092967 - 0.244039Tr + 0.158373Tr (4.14b) 2 B2 = 0.004468 + 0.001513Tr - 0.002061Tr (4.14c) This correlation is applicable in the region where 0.5 The results of the comparison for the five departure functions and fugacity coefficient are summarized in Tables 4.24 to 4.26. The overall average errors are listed in Table 4.5. Figures 4.40 through 4.49 are plots of the five departure functions against reduced temperature and pressure for n-heptane and methanol. These plots provide some information on how accurate the three equations are in predicting various departure functions. In general, all three equations are very similar in accuracy and none of these equations has clear advantage over the others. The accuracy of each of these EOS for this region are lower comparing to the other regions. In fact, the overall average deviation shown in Table 4.5 indicate that this region has the highest deviation comparing to the subcritical and supercritical regions for the same five departure functions and fugacity coefficient. For fugacity coefficient prediction, all three equations behave quite similarly. Figures 4.50 and 4.51 show that the difference among the three equations are minimal. 109 The results from the three EOS for predicting the isobaric heat capacity are summarized in Table 4.27. Also in Table 4.27 are the data source references and temperature and pressure ranges. Alike to the subcritical and supercritical regions, the SRK and PR equations are again very similar iri accuracy for predicting the isobaric heat capacity. The LM equation gives the highest deviations for all cases. Overall average deviation shown in Table 4.5 also shows that the LM equation is the least accurate of the three EOS. It follows that none of the above three EOS are adequate for predicting various physical and thermodynamic properties of this region. The accuracies of the PR and LM equations are similar for temperature and volume prediction. For pressure prediction, all three are poor. For departure functions and fugacity coefficient predictions, all three equations behave similarly and no noticeable differences are observed. In predicting isobaric heat capacitj', the SRK and PR equations are both adequate. In terms of overall average deviation or RMS % error, this region is the highest among the single phase regions in each of the property studied. Table 4.22 Region IV - Volume RMS % Error Compound SRK PR LM Tr Range Pr Range N Ref. t Methane 3 .61 0 9 .83 1 1Ci.0 1 0 . 525--0 . 944 0 .216--0 .862 86 68.126 Ethane 10 .93 6 .41 0 6 .12 5 0 .519--0..99 4 0 .551--0 .981 48 68,82 Propane 8 .83 1 4 .94 3 4 .89 8 0 .622--o..91 9 0 .211--0 .940 84 68,126 n-Butane 12 .33 3. 177 3. 056 0 .690--0 .925 0 .527--0 .790 16 68,126 i-Butane 1 1.4 9 3. 464 3. 374 0 .718--0..93 9 0 . 548-o- ..82 2 13 68,126 n-Pentane 10 .82 2 .29 2 2 .28 3 0..624 --o,,83 7 0,, 297-0- .. 89 1 24 68,126 n-Hexane 14 .20 2 .95 1 2 .28 0 0..676 --0.,95 1 0..330 --o..99 0 23 68,126 n-Heptane 13 .70 1 .53 7 1 .26 7 0,.555 --0,,92 6 0,.037 --0,. 731 49 122 n-Octane 16 .69 3 .79 3 3 .43 0 0 .563--0. 949 0.. 04 1-0- ,.80 6 54 122 Benzene 9 .76 1 3. 637 3. 593 0..551 --0. 942 0,.021 --0, 811 53 122 Carbon Dioxide 12 .43 3. 184 2. 962 0..799 --o. 964 0..206 --o. 824 23 25,68 Methanol 37 .36 21 .43 21 .24 0..702 --o. 857 0. 038- -0. 628 27 122 Ethanol 23 .81 9. 929 9. 324 0,.701 --0. 973 0..033 --0. 815 37 122 1-Propanol 18 .85 5 .83 2 5 .99 1 0..670 --0. 968 0. 020--0. 786 41 122 1-Butene 8 .89 5 4 .51 9 4 .64 3 0..572 --0. 953 0.,025 --0. 746 38 122 Water 37 .83 22 .18 21 . 74 0.,515 --0. 963 0.,203 --0. 904 1 18 44,68 N i t rogen 7 .'07 0 8 .70 9 8 .78 0 0.,715 --0. 953 0.,209 --0. 896 17 25,68 Ammon i a 30 .86 15 .73 15 .52 0. 764--0. 962 0. 269--0. 898 54 25,68 Neon 4 .87 2 15 .04 14 .42 0. 563--0. 901 0. 226--o. 942 20 68,126 Argon 3 .:97 8 10i . 74 10 .85 0. 563--o. 961 0. 204--o. 919 53 68,126 t PVT, departure function, and fugacity coefficient data are taken from the same source Table 4.23 Region IV - Pressure and Temperature RMS % Errors Pressure Temperature Compound SRK PR LM SRK PR LM Methane 179 1 144 1 174 2 . 792 18.29 18.47 Ethane 718 876 904 6 . 965 10.47 10.42 Propane 1 104 637 643 9.893 7.171 7 .095 n-Butane 698 171 173 10.17 3 .072 3.050 1-Butane 468 172 172 8 . 177 3.545 3.509 n-Pentane 2139 333 344 16 .09 3.624 3.606 n-Hexane 2001 132 123 16.36 1 . 758 1.612 n-Heptane 70070 1555 1417 36.30 2 .079 1 .803 n-Octane 4639 4583 45.46 6.204 5.894 Benzene 281 12 5722 5932 20.24 7.268 7.207 Carbon Dioxide 722 139 128 8 .083 1 .891 1 .771 Ethanol 97345 10358 10446 39.48 10.46 10.45 1 -Propanol 64189 4181 4248 3 1 . 85 5 .078 5 . 128 1-Butene 15773 4942 5383 16.15 8.418 8.617 N1trogen 130 485 492 2.872 8 . 757 8 .936 Ammon1 a 5505 1346 1353 32 .62 12 .95 12 .95 Neon 670 1217 1 127 9 . 881 24 .06 23 . 27 Argon 152 918 929 2.435 16.54 16 .67 Table 4.24 Region IV - Enthalpy and Entropy Departure Functions: Average Absolute Deviation Enthalpy (cal/mole) Entropy (cal/mole-K) Compound SRK PR LM SRK PR LM Methane 42 .6 10. 3 37.7 E thane 89.0 60. 3 98.7 Propane 77.8 37.5 81.8 n-Butane 68 . 2 45.7 91.4 i-Butane 68.4 47 . 4 81.9 n-Pentane 52.3 31 .9 88.7 n-Hexane 85.9 56.5 91.5 n-Heptane 186.5 238 . 2 48.8 0. 531 0. 739 0. 273 n-Octane 214.0 233.8 69.5 0.511 0.670 0.312 Benzene 122 . 1 146 . 3 59.7 0. 405 0. 447 0. 138 Carbon Dioxide 49.9 38.6 61.9 Methanol 510.0 629.2 887.6 1 . 325 1 .649 2.305 Ethanol 132.0 208 . 2 300.3 0.346 0.612 0.871 1 -Propanol 159.7 229.2 32.8 0.413 0. 657 0. 198 1-Butene 93.6 158.0 73 . 5 0. 342 0. 533 0. 247 Water 76.4 164 . 3 408 . 7 N1trogen 25 .9 14.5 14.6 Ammon1 a 71.1 51.9 87 .4 Neon 9.5 2.5 17.4 Argon 28 . 1 10. 5 14.8 Table 4.25 Region IV - Helmholtz and Gibbs Free Energy Departure Functions : Average Absolute Deviation Helmholtz (cal/mole) Gibbs (cal/mole) Compound SRK PR LM SRK PR LM n-Heptane 71.1 65.5 68 . 1 66 . 3 65 67.8 n-Octane 101 .4 95. 1 97 .9 94 .6 93 96 . 5 Benzene 65.0 34.9 40. 3 60.8 35 41.2 Methano1 39.3 46.7 64 . 7 35.8 44 62.8 Ethanol 58 .6 58 .0 62.7 47 .6 53 58 .6 1-Propanol 83.2 78.4 62.0 74 .0 75. 59.0 1-Butene 11.5 18.4 20.9 9.6 18 21.8 Table 4.26 Region IV - Internal Energy Departure Function and Fugacity Coefficient Average Absolute Deviation Internal Energy (cal/mole) Fugacity Coefficient Compound SRK PR LM SRK PR LM Methane 42.8 9.6 36 .0 E thane 86.6 58.5 96 .9 Propane 80. 1 35.7 80. 1 n-Butane 69.4 44.8 90.5 1-Butane 66.7 45.5 80.7 n-Pentane 58.8 31.8 88.2 n-Hexane 87.5 55 . 1 90.9 n-Heptane 188 . 7 238.6 49 . 1 0.0222 0.0242 0.0238 n-Octane 218.1 234 . 7 70. 2 0.0282 0.0308 0.0316 Benzene 124.4 146.6 59. 1 0.1336 0.0153 0.0156 Carbon Dioxide 49 . 3 37.5 61.4 Methano1 497.8 622 . 1 880. 7 0.0159 0.0203 0.0285 Ethanol 135.9 207.8 298. 7 0.0190 0.0223 0.0255 1 -Propanol 164 .6 231 . 1 30. 2 0.0300 0.0315 0.0248 1-Butene 93.2 158.6 72.8 0.0038 0.0083 0.0100 Water 76.8 157 . 1 395 . 7 N1trogen 25.6 13.1 13.4 Ammon1 a 72.9 51.4 78.9 Neon 9 . 3 2 . 3 18.3 Argon 27 .8 10.4 14.6 Table 4.27 Region IV - Isobaric Heat Capacity : Average Absolute Deviation (cal/mole-K) Compound SRK PR LM Tr Range Pr Range N Ref. n-Heptane 2 .8 2 .5 3 ,. 9 0. . 555--o. .926 0 .037- -o. . 731 49 122 n-Octane 3 . 3 2 . 8 3 .. 8 0. . 563--0. 949 0 . 04 1--0. . 806 54 122 Methano1 3 . 2 1 . 7 2 . 0 0. 702- -0. 857 O.. 038--0. 628 27 122 Ethanol 6 .0 4 ., 2 5 . 0 0. 701- -0. 973 0. ,033- -0. 815 37 122 1-Propano1 5 . 3 3 .. 6 10.3 0. 670- -0. 968 O.,020 - -0. 786 41 122 1-Butene 1 .0 1 .. 3 5 . 1 0. 572- -0. 953 0. .025- -0. 746 38 122 Water 4 .0 2 .. 9 4 . 5 0. 515- -0. 963 0. . 203--0. 904 1 18 44 Argon 1 . 8 1 ., 1 2 . 1 0. 563- -0. 961 0. 204- -0. 919 53 126 116 0.14 0.105 n-Heptane eS DaU 0.13 0.095 0 12 tsi tS3 0.085 0.11 Methanol 0 Data SRK 0.075 PR 0.10 - 0 LM 0 0 0 0.09 0.065 -i 1 1 0.5 0.6 0.7 0.8 0.9 1.0 0.65 0.70 0.75 0.80 0.85 0.90 Tr Tr Figure 4.38 Region IV : Compressibility factors of n-heptane (Pr = 0.73) and methanol (Pr = 0.63) versus reduced temperature. 0.14 0.10 Methanol 0.12 - Q Data 0.08 SRK PR o.io LM 0.06 0.08 0.06 0.04 0.02 0.00 0.0 0.2 0.4 0.6 0.8 Pr Figure 4.39 Region IV : Compressibility factors of n-heptane (Tr = 0.67) and methanol (Tr = 0.70) versus reduced pressure. 117 7.45 9.6 9.4 es 0 0 7.40 00 ra 0 9.2 Methanol 0 Data O 9.0 - 7 cS 35 OS SRK PR ad 8.8 LM I I £ 7.30 PC I 8.6 n-Heptane Q Data 8.4 - 7.25 —SRK PR 8.2 LM 1 • i.ii. 7.20 ' i— i i 8.0 • 0.0 0.2 0.4 0 6 0.8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Pr Pr Figure 4.41 Region IV : Reduced enthalpy departure functions of n-heptane (Tr = 0.67) and methanol (Tr=0.70) versus reduced pressure. 118 10 0 0.5 0.6 0.7 0.8 0.9 1.0 0.65 0.70 0.75 0.80 0.85 0.90 Tr Tr Figure 4.42 Region IV : Reduced entropy departure functions of n-heptane (Pr = 0.73) and methanol (Pr = 0.63) versus reduced temperature. 15.5 Methanol 0 Data 15.0 11.0 SRK PR LM 14.5 h- 10.9 \ CO CO 0 0 0 I 00 I 14.0 h CO CO Y 10 8 V 13.5 10.7 13.0 - 10.6 0.0 0.8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Pr Figure 4.43 Region IV : Reduced entropy departure functions of n-heptane (Tr = 0.67) and methanol (Tr = 0.70) versus reduced pressure. 119 0.51 1.28 n-Heptanc 0 Data 0.49 — SRK 1.26 - PR LM 00 0.47 O E— 0 o 1.24 - ro OS E— \ OS 0 0 °< < I . I 0 < < 1.22 0.43 - Methanol 0 0 Data 1.20 0 41 SRK PR LM I.I.I. 0.39 1.18 • *••-•-•-• 0.0 0.2 0.4 0.6 0.8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Pr Pr Figure 4.45 Region IV : Reduced Helmholtz departure functions of n-heptane (Tr = 0.67) and methanol (Tr = 0.70) versus reduced pressure. 120 3.0 3.0 2.5 - 2.0 - 2.0 O o E- 1.0 E- QJ \ O G~* 1.5 I O - I o o.o O 10 n-Heptane Methanol 0 Data • Data -1.0 —SRK SRK 0.5 PR PR LM LM 1 1 -2.0 0.0 1 , i . 1... 0.5 0.6 0.7 0.8 0.9 1.0 0.65 0.70 0.75 0.60 0.85 0.90 Tr Tr Figure 4.46 Region IV : Reduced Gibbs energy departure functions of n-heptane (Pr = 0.73) and methanol (Pr = 0.63) versus reduced temperature. -0.10 0.65 0 -0.15 0.60 O E- \ 0 -0.20 0.55 0 o O 0 i I o O Methanol -0.25 0 Data —SRK PR LM -0.30 0.45 -I 1 L_ ^ 1 .__L . 1 . 0.0 0.2 0.4 0.6 0.8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Pr Pr Figure 4.47 Region IV : Reduced Gibbs energy departure functions of n-heptane (Tr = 0.67) and methanol (Tr = 0.70) versus reduced pressure. 121 6.85 8.8 0 0 0 Q 8.6 6.80 - Methanol 8.4 • 0 Data es O O SRK 0 PR 0 8.2 \ LM C ZD I 8.0 2- 6.70 I I n—Heptane 7.8 0 Data 6.65 SRK PR LM 6.60 1 , L_ 1 I 1 I [ I 1 1 0.0 0.2 0.4 0.6 0.8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Pr Pr Figure 4.49 Region IV : Reduced internal energy departure functions of n-heptane (Tr=0.67) and methanol (Tr = 0.70) versus reduced pressure. 122 0.7 0.5 n-Heptane Methanol 0.6 • Data es eS Data SRK 0.4 SRK PR PR 0.5 LM LM 0.4 0.3 0.3 0.2 0.2 0.1 0.1 2d=L—i i i 0.0 0.0 0.5 0.6 0.7 0.8 0.9 1.0 0.65 0.70 0.75 0.80 0.85 0.90 Tr Tr Figure 4.50 Region IV : Fugacity coefficients of n-heptane (Pr = 0.73) and methanol (Pr = 0.63) versus reduced temperature. 0.25 0.7 Methanol - eS Data 0.20 SRK PR 0.5 LM 0.15 0.4 \ 0.3 \ o.io V 0.2 0.05 0.0 t 1 0.0 0.2 0.4 0.6 0.8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Pr Pr Figure 4.51 Region IV : Fugacity coefficients of n-heptane (Tr = 0.67) and methanol (Tr = 0.70) versus reduced pressure. 123 4.5. INVERSION CURVE Liquefaction and refrigeration are used extensively in the cryogenic and natural gas industries. To liquify gases, Joule-Thomson expansion is often used. Information concerning conditions at which cooling or heating may occur is essential. When the Joule-Thomson coefficient, u u = (4.15) 9P JH is positive, cooling occurs and when negative, heating results. The condition at which the crossover from cooling to heating occurs, is called the inversion condition, and is defined as " = 0 (4.16) The temperature at which inversion occurs, is called the inversion temperature, and the corresponding pressure is the inversion pressure. For each isenthalpic curve on a T-P diagram, there is one inversion point. These points form a unique line and this locus of points is called the inversion curve. The object of this section is to compare the inversion curve calculated from the three equations of state: SRK, PR, and LM with experimental data. A total of nine compounds are studied. The experimental data sources have been tabulated in Table 4.28. Along with the experimental data, the predictions from the three EOS are also compared with the generalized inversion correlation proposed by Gunn, Chueh, and Prausnitz (GCP) [45]. Their correlation is based on 89 experimental inversion points of fluids with acentric factors that are close to zero. These fluids include argon, xenon, nitrogen, carbon monoxide, methane, and ethylene. The GCP 124 correlation is given below: 2 3 Pr = -36.275 + 71.598Tr - 41.567Tr + 11.826Tr 5 - 1.6721Tr* + 0.091167Tr (4 1?) The three inversion curve equations derived from the SRK, PR, and LM equations are given in Appendix A. For ease of reference, the inversion equation derived from the generalized cubic equation of state, Eq.(1.2) is given below: 2 2 2 2 C^r+uS+w) + C2$(2$+u)(£-l) + c3($-l) ($ +u$+w) = 0 (4.18) where Ci = %Tr (4.18a) °2 = " flaa (4.18b) C = 3 0.a*0 (4.18c) S = V/b (4.18d) The reduced inversion pressure is given by the following: RT 1 a 1 o _ r a p 7777 TT7 77777772 7 r " ~ (5-1) 0^ ($ +u$+w) (4.19) To calculate the inversion pressure at a given reduced temperature, the dimensionless volume, $ is solved from Eq.(4.18) and is substituted along with the given reduced temperature into Eq.(4.19). 125 The results of the nine compounds tested are summarized in Table 4.28. Since the GCP correlation is only applicable for simple compounds, that is compounds with small acentric factor, no estimations are given for propane, n-butane, carbon dioxide, p-hydrogen, and ammonia. The comparison is based on RMS % error. From Table 4.28, it is clear that the SRK equation is more reliable than the PR or LM equation. The SRK equation gives the lowest RMS % error for most compounds. The large RMS % errors for carbon monoxide reported for the PR and SRK equations are due to the points that are near the high temperature ideal gas region at which the inversion pressure is close to zero but both the PR and SRK equations give large negative inversion pressures. There are two points on the inversion curve which can be used to characterize any fluid. These are the maximum inversion pressure and the maximum inversion temperature. Miller [78] has suggested a reasonable approximation for locating these two points. For compounds which follow the corresponding states principle such as hydrocarbons, the maximum inversion temperature is about 5 times the critical temperature and the maximum inversion pressure is about 11.75 times the critical pressure at the reduced temperature of 2.25. The values of the maximum reduced inversion pressure, Pr(max), and maximum reduced inversion temperature, Tr(max), from the three equations of state have been tabulated in Table 4.29. On the average, the maximum reduced inversion pressure from the SRK equation is 12.05, PR is 13.40, and from LM is 14.25. The maximum reduced inversion temperature is 4.22 for the SRK equation, 5.25 for the PR equation, and 5.80 for the LM equation. Therefore, in general, the SRK equation tends to give the closest maximum inversion pressure 126 and the PR equation gives the closest maximum inversion temperature. The LM equation overpredicts both the maximum inversion pressure and temperature for all compounds considered in this study. Figures 4.52 to 4.60 show the inversion curves of the nine compounds calculated from the three EOS. In the case of fluid which has small acentric factor, the curve calculated from the GCP correlation is also included in the plot. For the low temperature region, all equations give similar predictions. None of the EOS, however, is capable of giving reasonable representation at the high temperature region. It should be pointed out that the LM equation is not a continuous function because it is not defined at Tr=1.0. Therefore a discontinuity appears on all the graphs at Tr = 1.0. It is safe to say that the behavior of the three equations is identical at reduced temperature below 1.5. The SRK equation is probably the most accurate among the three EOS right up to the maximum inversion pressure. This observation is consistent with the findings of Dilay and Heidemann [24]. The PR and LM equations are not recommended for inversion curve predictions above Tr=1.5. Table 4.28 Inversion Pressure RMS % Error Compound SRK PR LM GCP Tr Range N Ref . Methane 3.680 12.58 10.72 1 .242 1 .102-2 .465 27 25 Propane 42 . 37 46 . 50 45.97 0 .811-2 . 189 52 39 n-Butane 24.92 30. 18 30. 13 0..823- 2 .023 52 48 Carbon Monoxide 3629 . 6704 . 119.9 23 . 79 1.. 528-5 .064 14 25 Carbon Dioxide 17 .07 22 . 39 21 .29 0. 832-1 . 391 18 25 E thy 1ene 8.912 17. 20 13.65 4.900 1. 053-1. 495 6 23 p-Hydrogen 50.98 74.67 74 . 14 0. 849-5 459 16 69 Ammon i a 108.8 106 . 2 101 .0 0. 792-1.. 233 20 25 Argon 20. 12 26.65 25.20 5.615 0..862-1 . 989 18 40 Table 4.29 Maximum Reduced Inversion Pressure and Temperature SRK PR LM Compound Pr(max) Tr Tr(max) Pr(max) Tr Tr(max) Pr(max) Tr Tr(max) Methane 11 .. 79 2 . 15 4 .40 13 .08 2 . 29 5. 38 13 . 34 2 . 58 6 .0 4 Propane 12 ,. 12 1 .89 3 .61 13 .42 1 . 98 4 .'2 1 14 . 53 2 .31 4 .7 3 n-Butane 12 . 24 1 .84 3 .45 13 .55 1 ,9. 1 3. 99 15 .,0 8 2 . 26 4 .4 5 Carbon Monoxide 11 .87 2 .06 4 . 14 13 . 16 2 . 18 4 .9 7 13 .60 2 . 49 5 .6 1 Carbon Dioxide 12 . 33 1 .80 3 . 34 13 .65 1 . 87 3 .8 3 15 .68 2 . 24 4 .2 8 E thy 1ene 1 1.9 .5 1 .99 3 . 93 13 . 24 2 . 10 4 .6 7 13 .82 2 . 4 1 5 .2 6 p-Hydrogen 1 1 9.5 3 .09 7 .39 13 .70 3,, 74 10.97 13..6 1 3 .83 1 1. 37 Ammon i a 12 .4 1 1 . 77 3 ,. 26 13 . 74 1 ,8. 4 3. 72 15 .2 7 2 . 19 4 .2 4 Argon 1 1 7.7 2 . 17 4 .4. 9 13 .07 2 ,. 33 5 .5 2 13 . 33 2 .6. 2 6 .2 0 7.0 Methane 1 1 o.o I ' i i ' 1 ' 1 i 1 1 • I 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 Reduced Pressure, Pr Figure 4.52 Inversion curve of methane calculated from the SRK, PR, LM, and GCP equations. oo 5.0 0.0 • I • 1 I > 1 . I ; 1 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 Reduced Pressure, Pr Figure 4.53 Inversion curve of propane calculated from the SRK, PR, and LM to equations. CO Figure 4.54 Inversion curve of n-butane calculated from the SRK, PR, and LM CO equations. o 6.0 Carbon Monoxide 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 Reduced Pressure, Pr Figure 4.55 Inversion curve of carbon monoxide calculated from the SRK, PR, LM, and GCP equations. 5.0 Carbon Dioxide A Expt. 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 Reduced Pressure, Pr Figure 4.56 Inversion curve of carbon dioxide calculated from the SRK, PR, and co LM equations. to Figure 4.57 Inversion curve of ethylene calculated from the SRK, PR, LM, and w GCP equations. w Figure 4.58 Inversion curve of p-hydrogen calculated from the SRK, PR, and LM co equations. Figure 4.59 Inversion curve of ammonia calculated from the SRK, PR, and LM co equations. cn Figure 4.60 Inversion curve of argon calculated from the SRK, PR, LM, and w GCP equations. 05 CHAPTER 5. SECOND VIRIAL COEFFICIENT 5.1. INTRODUCTION To account for vapor phase nonideality, the fugacity coefficient is necessary in phase equilibria calculations. Often when the pressure is moderate, density is less than half the critical, the virial equation truncated after the second term provides excellent estimate of the vapor phase fugacity coefficient. For normal fluids, probably the most popular and accurate correlation for calculating the second virial coefficient is the method suggested by Tsonopoulos [123]. This correlation is a modification of the Pitzer-Curl [90] equation which is based on the three-parameter corresponding states principle utilizing the acentric factor as the third parameter for improving the accuracy of the prediction. The Tsonopoulos correlation gives the reduced second virial coefficient as the sum of two terms: = f ( o ) + cjf ( 1 ) (5.1) where f<°> and f'1' are given by Eq.(5.1a) and (5.1b) 0.330 0.1385 0.0121 0.000607 f") = 0.1445 - T rp s r r (5.1a) 137 138 0.331 0.423 0.008 1 f' ' = 0.0637 + (5.1b) The f <0) represents the reduced second virial coefficient of a simple fluid which has zero acentricity and f <1 > is the correction term for a normal fluid. The coefficients of f <0' and f <1' were found by curve-fitting with experimental data. Argon, krypton, and xenon data were used b3' Pitzer and Tsonopoulos to obtain the function f <0 >. However, both authors did not specify which compounds were used for the f <1 > function. The data sources for the Tsonopoulos correlation were taken from the 1969 compilation of second virial coefficient by Dymond and Smith [26]. Recently, Yu et al [129] and Adachi et al [1] have studied a number of cubic equations of state for pure compound property predictions. These two studies have shown that none of the currently popular cubic equations of state are capable of giving accurate second virial coefficients except the equation proposed by Kubic [57]. The Kubic equation uses the Tsonopoulos correlation to determine one of its constants. Although the Kubic equation gives excellent second virial coefficient prediction, it fails to give reasonable approximations for the more fundamental properties such as vapor pressure and densities. Martin [73] proposed a modified Martin [72] equation specifically designed for second virial coefficient calculation. To determine the three required substance-dependent constants, the slope of the vapor pressure curve at the critical point, experimental second virial coefficients, and critical constants must be known. 139 5.2. THEORETICAL BACKGROUND As it was pointed out in the previous section, most of the cubic equations of state do not give accurate second virial coefficient predictions for the entire temperature range. However, it is generally true that predictions from the cubic equation of state are acceptable for reduced temperature above 0.8. Below this temperature the results deviate greatly from the experimental data. To understand why the cubic equation fails to give good representation of the second virial coefficient, it is essential to look at how the virial coefficients are represented by the cubic EOS. The compressibility of a gas can be expanded in Taylor series either in inverse molar volume or pressure, that is : B C + 2 = 1 + + "•* (5-2) v ^ or 2 Z = 1 + B'P + CP (5.3) Eq.(5.2) is called the Leiden form whereas Eq.(5.3) is the Berlin form of the virial equation. The coefficients B, B'... and C, C... are the second and third coefficients, respectively. The two sets of virial coefficients, B and B' and C and C, can be shown [91] to be related to each other by Eq.(5.4a) and (5.4b) 140 B = B'RT (5.4a) 2 2 C = C (RT) + B (5.4b) For pure gas, these coefficients are functions of temperature only. If the generalized cubic equation of state, Eq.(1.2): RT a(T) V-b V2+ubV+wb2 (1'2) is expanded in inverse molar volume similar to Eq.(5.2), it can be shown that the second virial coefficient is given by Eq.(5.5) a(T) B = b (5.5) RT where a(T) = a(Tc) • a (5 6) The derivation of Eq.(5.5) is shown in the Appendix A. In reduced form: a( p !Zc = _ V c p 2 2 (5.7) RT RT R T C C C where F = a/Tr (5.8) 141 It was shown by Shaw and Lielmezs [105] that for most cubic EOS, the F-function can be written as a power series in inverse reduced temperature. For example, the F-function of the Soave-Redlich-Kwong [110] and Peng-Robinson [87] equations are given by Eq.(5.9) C2 C3 F = Ci + + 7 (5-9) r r The constants C1} C2, and C3 are functions of the acentric factor. If we compare the functional dependence of F on reduced temperature with that of the Tsonopoulos correlation, it can readily been seen why the cubic EOS gives such poor results in the low temperature region. Below Tr = 0.8, the slope of the second virial coefficient increases rapidly with decreasing temperature (see Figures 5.1 and 5.2). With the dependence on reduced temperature of power of -1, it is not possible to include the rapid changes in the low temperature region. Since the data sets used by Tsonopoulos were materials found before 1968, and the 1980 compilation by Dymond and Smith [27] has included material published up to early 1979, an improved correlation was thought to be possible with these new and more accurate experimental data and this prompted the present investigation. 5.3. A NEW CORRELATION The present proposed correlation is based on the PR form cubic EOS: 142 RT a(T) p = (5.10) V-b V2+2bV-b2 Here, the a(T) function is modified such that it gives an accurate second virial coefficient. The second virial coefficient from a cubic equation of state is given earlier by Eq.(5.5) a(T) B = b (5.5) RT where a(T) = a(TJ • a(Tr , o) R2T 2 R2T 2 a(T„) = QQ — = 0.45724 — c c RT RT b = b(T ) =0. —^ = 0.07780 —* P P c c Substituting a(T) and b into Eq.(5.5) and simphfying, the reduced second virial coefficient is BPC a — = % ~ fla — (5.11) Rearrange Eq.(5.11), a is then given by Eq.(5.12) 143 (5.12) a J where BP B r = (5.13) RT With experimental Br, a can now be calculated at each reduced temperature. It is proposed here that the a-function for normal fluid be given by the following expression: a = a<°> + wa'1' (5.14) where a(0' represents a of simple fluids and a(1' corrects for the deviation of normal fluid from the simple fluid. The functional form of Eq.(5.14) is based on the three-parameter theorem of corresponding states with the acentric factor as the third parameter. In this work, all acentric factors of simple fluids are assumed to be zero even though the actual values are slightly different than the assumed. For example, argon has an acentricity of -0.004. To obtain the functional dependence of a(01 on reduced temperature, the calculated a values of the simple fluids are curve-fitted with a power series. The minimum variance is the criterion for determining the number of terms in the power series. Once an expression for a <0' is obtained, a(1' can then be 144 calculated from the following expression: a - a<°> a(i> = (5 CO where a is calculated from Eq.(5.12). The final expressions for a(0) and a'1' are given by Eq.(5.16) and (5.17): 14.360017 45.000285 78.907097 a<°> = -1.4524905 + + T T 2 T 3 r r r 79.449258 45.841959 14.078304 1.7835426 + _ + . 5 •V 4 T T I T 1 r r r r (5.16) 15.205023 20.874489 12.697209 a'1' = -4.3816022 + + rp m 2 T 3 r r r 2.5851948 T 4 (5.17) r A total of 57 experimental data points are involved in the least-squares fit of a(0) and the variance resulted is 0.0003175. The compounds included in the curve-fit are neon, argon, krypton, and xenon. The expression for a <1' is obtained from 124 points with a variance of 0.01041. The compounds included are propane, n-butane, n-pentane, isopentane, neopentane, n-hexane, n-heptane, n-octane, benzene, ethylene, and propylene. All experimental data are taken from 145 the Dymond and Smith [27] 1980 compilation of second virial coefficients. 5.4. COMPARISON The second virial coefficients of 24 normal and simple fluids have been calculated from the proposed correlation, Eq.(5.14). In order to test its validitj', the results are compared with the Tsonopoulos correlation and four other cubic EOS. The cubic EOS included are the Soave-Redlich-Kwong, Peng-Robinson, Lielmezs-Merriman, and Kubic. The results have been tabulated in Table 5.1 and it has clearly shown that none of the cubic EOS is capable of representing the second virial coefficient for a wide range of temperature except the Kubic equation with reasons mentioned earlier. Figures 5.1 and 5.2 have the results of n-butane, benzene, n-octane, and- argon. Since the LM equation gives almost identical results as the PR equation, the LM equation is not included in the figures. Similarly, the Kubic equation is partly derived from the Tsonopoulos correlation. It also is not included. All the cubic EOS considered here give satisfactory results for Tr greater than 0.8 but below this temperature, the predictions deviate greatly from the experimental data. The proposed correlation gives excellent agreement for the entire range of temperatures studied. For example, the average absolute deviation of benzene from the proposed correlation is about one-third that of the Tsonopoulos correlation and for n-heptane, it is about one-half. The average estimated deviations of the experimental data shown in Table 5.1 are taken from Dymond and Smith [27]. For the majority of the compounds studied, the 146 deviations from both the Tsonopoulos and the present proposed correlations are well within the experimental uncertainties. Even though the Tsonopoulos correlation gives better predictions than the proposed correlation for a number of compounds, the differences are small. The SRK equation seems to give slight^ better prediction than the PR equation for Tf greater than 0.8. Figures 5.1 and 5.2 show that the SRK predictions lie right on the experimental data whereas the predictions from the PR equation are slightty off. Table 5.1 Second Virial Coefficient : Average Absolute Deviation (cc/mole) Compound SRK PR LM Kub i c This Tsono- Deviation Tr Range N Work polous Est imates Methane 6.6 1 1 2. 12 . 1 1 .6 1 .4 0.8 2 . 6 0 . 58-3 . 15 16 E thane 5.9 15 . 4 16 . 6 8.6 5 . 8 5 . 7 2.4 0 . 65-1 .96 15 Propane 18. 1 24. 6 25. 0 10. 3 6 . 1 5.8 12.3 0 . 65-1 .4, 9 15 n-Butane 74 . 1 69 . 0 67 . 2 9.2 11.8 9.6 18.2 0 . 59- 1 3.2 17 i-Butane 41.2 43. 9 44 . 2 22.4 24 . 7 24 . 5 --- 0 .67- -1 . 25 1 1 n-Pentane 78 . 2 67 . 0 66 . 6 14.2 10.9 11.9 28 . 3 0 .64- -1 . 17 12 i-Pentane 103.3 72. 9 72. 0 15.2 20.0 21.3 27 . 1 0. .61--o. 98 7 Neopentane 33 .6 37 . 3 38 . 1 9.9 3 . 5 3.0 17.0 0. .69- -1 . 27 10 n-Hexane 205 . 3 176 . 5 177 . 2 28.4 19.0 14.1 27 . 2 0. 59--0. 89 9 n-Heptane 212.1 180 '.7 180 .9 46.4 15.0 29 . 1 --- 0. . 56-1 . 30 12 n-Octane 376.3 341 .0 339 .9 84 .4 30. 5 51.5 0. 53--1 . 23 12 Benzene 164 . 3 147 .9 144 .8 61.1 17.3 60. 1 16 .9 O. 52- -1 .0 7 13 Carbon Monoxide 3.9 5.5 7.6 1 . 1 1 .5 0.5 --- 2 . 05- -3 . 18 7 Carbon Disulfide 154 .0 135 .4 131 .3 26.6 39.0 28.8 20.0 0. 51--0. 79 9 Hydrogen Sulfide 15.8 21 .4 21 .9 16.5 16 . 1 16 . 2 --- 1 .00 - -1 . 32 7 Ethylene 1 .7 15. 9 17. 0 2.6 0.4 0.4 1 . 2 0. 85- -1 . 59 9 Propy1ene 6. 1 18 . 8 19 . 4 5.8 1 .5 1 . 1 4.8 0. 77- -1 .3 7 8 1-Butene 13.6 23. 0 23. 6 14.7 6.6 7 . 3 0. 72- -1 .0 0 12 N i t rogen 4.8 8.3 8.0 4.9 3 . 9 3.9 2.4 0. 60- -5 . 56 14 Oxygen 5.7 9 . 2 9.9 3. 1 2 . 1 2 . 2 3.8 0. 58- -2 . 58 1 1 Neon 1 .6 2.7 4 . 1 1 .5 0.5 1 . 5 1 .0 1 .35 - -13.51 10 Argon 8.0 8.9 9.0 2.8 0.7 1 . 7 1 . 4 0. 54- -6 . 63 18 Krypton 16. 1 15. 7 15 . 8 2 . 2 1 .9 2.0 3.2 0. 53- -3 . 34 14 Xenon 12. 1 15. 8 16. 2 3.8 1 . 4 2.2 4.8 0. 55- -2 . 24 16 148 Figure 5.1 Second virial coefficients of n-butane and benzene versus reduced temperature. 149 SRK PR o EXPT THIS WORK __T?P!i— oo 0.4 0.6 0.8 1.0 1.2 1.4 Reduced Temperature, Tr 0.5 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 Reduced Temperature, Tr Figure 5.2 Second virial coefficients of n-octane and argon versus reduced temperature. CONCLUSIONS The Lielmezs-Merriman equation of state and its modification proposed in this work have been successfully applied to calculate various physical and thermodynamic properties of pure compounds in different regions on the PVT surface. The predictions from the LM equation and the modification of it have been compared with experimental data and with the predictions from two other equations of state: Peng-Robinson and Soave-Redlich-Kwong. Along the saturated vapor-liquid equilibrium curve, it has been found that the LM equation is the most accurate equation for predicting vapor pressure and saturated liquid and vapor volumes. However, for the two derivative properties: enthalpj' and entropy of vaporization, the PR equation is the more accurate equation. In the subcritical region, the predictions from the three equations are very similar and very small differences have been observed. In the region where Tr<0.95, the PR and LM equations are practically identical in accuracy for most of the properties tested. However, the LM equation is not recommended for calculating derivative properties such as departure functions in the region 0.95 For the supercritical region, the predictions of pressure and volume from the PR and the proposed modification of the LM equation are again very similar in accuracy. The overall average RMS % error from the SRK equation is about 150 151 twice that of the PR or modification of the LM equation. For temperature prediction, all three equations are similar in accuracy. For derivative properties, the modification of the LM equation gives comparable prediction to the PR equation in the region where Tr>1.05. The SRK and PR equation are similar in accurac.y for enthalpy, entropy, and internal energy departure function estimations. The SRK equation gives the lowest overall average RMS % error for the Joule-Thomson coefficient prediction. In the compressed liquid region, it has been found that the overall average deviation or RMS % error is the highest among the three single phase regions in each of the properties studied. None of the three equations is adequate for calculating pressure. For temperature and volume predictions, the PR and LM equations give similar results. These two equations are more accurate than the SRK equation. For derivative properties, predictions from the three EOS are comparable. For inversion curve calculation, the most reliable equation of state is the SRK. Its estimation closely follows the experimental data right up to the maximum inversion pressure. The predictions from all three equations are alike up to Tr = 1.5. None of the three equations of state tested in this work is capable of calculating the second virial coefficient accurately for the region below Tr = 0.8. The proposed a-function especially designed for second virial coefficient calculation has been found to be equal, if not better, in accuracy than the Tsonopoulos 152 correlation. The proposed a-function is based on the latest and the most accurate experimental data sources. The deviation from the proposed correlation has been found to be well within the estimated uncertainty given in the data sources. As shown throughout this work, seldom does an equation of state perfom equally well in every region. Among the three equation of state studied, the preferred equation varies, depending on what region or what type of compound is considered. One equation may be particularly suitable for representing saturated properties. Another may be superior in representing PVT data in the single phase region. However, none of the three equations studied in this work can satisfactorily represent all properties over an extended range of conditions. RECOMMENDATIONS FOR FURTHER STUDY 1. There remain many compounds such as heavy hydrocarbons and alcohols for which the p and q constants for the Lielmezs-Merriman equation of state have not been developed. Further work in the development of these constants is necessary. 2. Since the Peng-Robinson and Lielmezs-Merriman equations are so similar in accuracy for a number of properties studied, similar work should be done for the Soave-Redlich-Kwong and the Lielmezs-Howell-Campbell equations to see if similar conclusions to those found in this work can be drawn. 3. The volume-translation technique has been steadily gaining popularity since its introduction by Peneloux et al [85]. Perhaps, such a technique can be incorporated into the LM equation for improving the volume prediction. 4. The proposed a-function designed for second virial coefficient calculation can only apply to nonpolar compounds. Further development of the proposed a-function for polar compounds is essential. 5. Since the proposed a-function for calculating second virial coefficient is based on the Peng-Robinson equation and it was arbitrarily chosen, a similar a-function based on the Soave-Redlich-Kwong equation can also be correlated. 6. So far only pure compound properties have been studied, binary and multicomponent mixtures should be studied in the future. 153 NOMENCLATURE a parameter in the attraction pressure term of the cubic equation of state defined by Eq.(2.2a) A molar Helmholtz free energy A dimensionless constant defined by Eq.(A.30) AAD average absolute deviation defined by Eq.(4.3) b parameter in the cubic equation defined by Eq.(2.2c) b' translated parameter in the cubic equation defined by Eq.(2.12) B second virial coefficient B dimensionless constant defined by Eq.(A.31) c volume correction constant C third virial coefficient Cp isobaric heat capacity D fourth virial coefficient f fugacity F number degrees of freedom F function defined by Eq.(5.8) G molar Gibbs free energy H molar enthalpy AHV enthalpy of vaporization m constant in Eq.(2.3) m number of components MW molecular weight n constant in Eq.(2.8) 154 155 N number of data points NC number of compounds p substance-dependent parameter in the Lielmezs-Merriman equation of state P pressure q substance-dependent parameter in the Lielmezs-Merriman equation of * state R universal gas constant RMS root mean square defined by Eq.(3.25) s characterization parameter defined by Eq.(3.26) S molar entropy ASy entropy of vaporization T absolute temperature * T dimensionless temperature coordinate defined by Eq.(2.9a) and (3.39) u generalized cubic equation of state parameter U molar internal energy V molar volume V translated volume defined by Eq.(2.11) w generalized cubic equation of state parameter x mole fraction Z compressibility factor Zj^ Rackett compressibility factor Subscripts A attraction 156 b normal boiling point c crtiical property r reduced property R repulsion Superscripts 1 liquid phase v vapor phase ° reference state: ideal gase state Greek Symbols a temperature dependence of the parameter a in the cubic equation of state K constant defined by Eq.(2.10a) u chemical potential u Joule-Thomson coefficient co acentric factor 7r number of phases Oa coefficient of a defined by Eq.(2.2a) coefficient of b defined by Eq.(2.2c) $ dimensionless volume defined by Eq.(4.18d) REFERENCES 1. 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(A.l) by (V-b) (V2+ubV+wb2) and collect similar terms: RT RT a V3 + ub - b V2 + wb2 - ub2 - ub — + - P P P RT ab 3 2 wb + wb — + — = 0 (A.2) P P At the critical point, (V -v )3 = c 0 (A.3) Expand Eq. (A.3) algebraically, 164 165 V3 - 3V2V + 3VV 2 - V 3 = 0 c c c (A.4) Compare the coefficients of Eq. (A.2) and (A.4), and we have the followings: RT ub - b = -3V (A..5) P RT a 2 2 2 wb - ub - ub — + - = 3VC (A. 6) P P RT ab 3 2 3 wb + wb — + — = V (A. 7) P P c Let Vc = Xb, then Eq. (A.5), (A.6) and (A.7) become (A.8), (A.9), and (A. 10), respectively. RT b(u - 1 + 3X) = — (A.8) P RT a 2 2 b (w - u - 3X ) = ub — (A.9) P P RT a 3 2 2 X b -wb - wb — = - (A. 10) P P Substitute Eq. (A. 10) into (A.9) and simplify 166 RT 2 2 2 3 2 b (w - u -3X ) = b(u + w) — + wb - X b (A. 11) P Substitute Eq. (A.8) into (A. 11) and simplify 2 (w - u - 3X ) = (u - 1 + 3X)(u + w) + w - X (A. 12) Collect similar terms and get Eq. (A. 13): 3 2 2 X - 3X - 3(u + w)X + (w - uw - u ) = 0 (A. 13) Define Q& and as aPc 0_a = R2T £2 (A. 14) c bPc (A. 15) RT At the critical point Eq. (A.l) becomes p = RT^ _ a(Tc) C V-b V 2+ubV +wb2 (A-16) c c c Substituting Eq. (A. 16) into (A. 14) 167 a(T ) (a(T ))2 a 2 2 2 2 RTc(Vc-b) R Tc (Vc +ubVc+wb ) (A. 17) Differentiate Eq. (A.l) with respect to volume at constant temperature 9P RT a(2V+ub) — = - + (A IR) 3V „ (V-b)2 (V2+ubV+wb2)2 3P At the critical point = 0 and Eq. (A. 18) becomes 3V RTC a(Tc)(2Vc+ub) 2 2 2 2 (A. 19) (Vc-b) (Vc +ubVc+wb ) Rearrange Eq. (A. 19) RT (V 2+ubV +wb2)2 a(T ) = = -V 2 (A.20) (Vc-b) (2Vc+ub) Replace Vc with Xb and substitute Eq. (A.20) into (A. 17) and simplify (X2+uX+w)2 2 = (X -2X-u-w) (A.21) a. (X-l)*(2X+u)2 Similarly, can be written as 168 (X2-2X-u-w) Qh = 2 (A.22) (X-l) (2X+u) Divide &a with fl^ J2a (A.23) bRT c or T_ = (A.24) bRfi„ Divide C2„ with f^2 a v " ^ or P = 2 (A. 26) b fl By definition, Z_ is c P V •Z7c - —C C (A.27) 169 Substitute Xb for v , Tc with Eq. (A.24), and Pc with (A.26), Eq. (A.27) becomes zc = xflb (A. 2 8) Replace with Eq. (A.22) and get X(X2-2X-u-w) (A.29) (X-l)2(2X+u) 1.1. The Soave-Redlich-Kwong Equation u = 1 w = 0 X = 3.8473221... G0 = 0.4274802... = 0.0866403... Zc = 0.3333333... 1.2. The Peng-Robinson Equation u = 2 w = -1 X = 3.9513730... Q = 0.4572355. a = 0.0777960 = 0.3074013 1.3. This Work u = 2 w = -1 X = 3.9513730. 0 = 0.4572355 3. = 0.0777960 Z„c = 0.3074013 2. CONSTANTS A AND B Let's define A and B as aP A = R2T2 bP B = — RT where a = a(Tc) • a 171 b b T = ( c> (A.33) Substituting a and b into Eq. (A.30) and (A.31), A and B can be rewritten as A = ®a — a (A.34) B = % — (A.35) r 2.1. The Soave-Redlich-Kwong Equation 0 .2 a = [ 1 + m(l - Tr .5 ) ] The constant m is m = 0.480 + 1.574CJ - 0.176w2 2.2. The Peng-Robinson Equation 0.5 ,2 a = [ 1 + K(1 - Tr ) ] The constant K is K = 0.37464 + 1.54226u - 0.26992u2 2.3. This Work For T < 1.0 a = 1 + p(T*)q (T /T - 1) T = - (T /Tb - 1) and for T > 1.0 a = 1 - p(T*)q » _ (T/T, - 1) (Tc/Tb - 1) The constants p and q are substance-dependent constants. 3. COMPRESSIBILITY FACTOR EQUATION Multiply Eq. (A.2) by P3/(R3T3) and let PV Z = — RT Collect similar terms and Eq. (A.2) becomes 173 Z3 + (uB - B - 1)Z2 + (A + wB2 - uB - uB2)Z 2 3 - (AB + wB + wB ) = 0 ^A 36^ 3.1. The Soave-Redlich-Kwong Equation Z3 - Z2 + (A - B - BZ)Z — AB = 0 3.2. The Peng-Robinson Equation Z3 - (1 - B)Z2 + (A - 3B2 - 2B)Z - (AB - B2 - B3) = 0 3.3. This Work Z3 - (1 - B)Z2 + (A - 3B2 - 2B)Z - (AB - B2 - B3) = 0 4. FUGACITY COEFFICIENT Fugacity coefficient for a pure compound can be computed by the following equation : 174 f 1 P RT in - = - ; V dP (A. 3 7) P RT 0 P Eq. (A. 3 7) may be rearranged to give f 1 P lnP m - = — ; vdp - ; dinp (A38) P RT P° lnP 0 where P° is the reference state pressure of a pure substance in the ideal gas state with molal volume V° at some temperature T. The molal volume V° is given by RT The first term of the right hand side of Eq. (A.38) can be integrated by parts, that is / VdP = A(VP) - / PdV (A40) The second term of the right hand side of Eq. (A.40) can be evaluated by substituting the generalized cubic equation, Eq. (A.l), for P dV adV J PdV = RT J / (A-41) V-b V2+ubV+wb2 175 The upper limit of the integral is V and the lower limit is V°. Integrating Eq. (A.41) gives v-b V+(92b S PdV = RTln In V°-b 2b61 V+03b v+e2b In (A.42) 2b61 v»+e3b where 6 * = X - w (A.43) u e = 2 - - 0i (A.44) 2 u 6, = (A. 45) 2 Combining Eq. (A.38), (A.40), and (A.42), the fugacity coefficient becomes V-b V+f?,b In - = (Z - 1) - In In P V°-b 2bRT<9i V+03b a V°+02b P + In - In — 0 (A. 46) 2bRT0x V +e93b po 176 By l'Hospital's rule, the term (V°+02b)/( V+t^b) approaches 1 as P approaches zero. The fugacity coefficient for a pure compound is then given by Eq. (A.47): f A Z+t?2B In - = (Z-l) - ln(Z-B) + In (A. 47) P Z+03B 4.1. The Soave-Redlich-Kwong Equation 0a = 0.5 e2 = o.o e3 = 1.0 f A z In - = (Z-l) - ln(Z-B) + - In P B Z+B 4.2. The Peng-Robinson Equation 61 = Jl 62 = 1 - y/2 e3 = i + »/2~ f A Z-0.414B In - = (Z-l) - ln(Z-B) + In P 2v^B Z+2.414B 177 4.3. This Work e, = 1/2 e2 = 1 - vT t93 = 1 + /2 f A Z-0.414B In - = (Z-l) - ln(Z-B) + —=—In P 2\/2B Z+2.414B 5. ENTHALPY DEPARTURE FUNCTION By definition, enthalpy is given H = U + PV (A. 48) Differentiate Eq. (A.48) with respect to volume at constant temperature " 9H ' 9U 9(PV) + (A. 49) 3V 9V T T 9v But dU = TdS - PdV (A.50) Differentiate Eq. (A.50) with respect to volume at constant temperature 178 au " 3S " rp — X 3v 3v (A.51) T From Maxwell relation " 35 ' " 3P " (A.52) 3v 3T T T Combining Eq. (A.49), (A.51), and (A.52) gives r 3H 1 3P 3(PV) — = T — - P + (A.53) 3V 3T 3V T V T Integrating Eq. (A.53) gives the isothermal changes in enthalpy 3P H2 " Hx = / T( — > dV + A(PV) V (A. 5 4) 3T Differentiate Eq. (A.l) with respect to temperature at constant volume 3P da (A.55) V-b V2+ubV+wb2 3T JV dT Substitute Eq. (A.l) and (A.55) into (A.54) and simplify 179 da dV H2 - Hx - a - T + A(PV) dT V2+ubV+wb2 (A. 5 6) Integrating Eq. (A.56) gives a - T(da/dT) V+02b H2 - H, = In + P2V2 - P1V1 26yb v+e3b (A. 5 7) be the ideal gas at zero pressure. Eq. (A.57) becomes a - T(da/dT) Z+t?2B H - H° = RT(Z-l) + In (A.58) 2c9ib Z+t93B 5.1. The Soave-Redlich-Kwong Equation 0 5 : a = a(T ) • [ 1 + m(l - Tr - ) ] Differentiate a with respect to T and simplify da a = - ma(Tc) dT TTC J 180 da , T — = - ma(T )(aT ) dT c r da a - T — = (1 + m) dT A T Z H -- H° == RT(Z-l) + R(l+m) In B a0 •5 Z+B 5.2. The Peng-Robinson Equation 5 2 a = a(Tc) • [ 1 + K(1 - T/- ) ] Differentiate a with respect to T and simplify da a — = - K3(T ) c dT TT c J da , T — = - Ka(T_)(oT_) dT c r da a - T — = (1 + K) dT R(l+»c) A T Z-0.414B H - H° = RT(Z-l) + In 2v^2 B a" Z+2.414B 5.3. This Work For T < 1.0 r q a = a(Tc) • [ 1 + p(T ) ] (Tc/T - 1) T = (Tc/Tb - 1) Differentiate a with respect to T and simplify da qa(Tc)(o-l) dT Tr(Tc-T) da qTca(Tc)(g-l) T — = dT (Tc-T) da qTr a - T — = a 1 + dT (Tc-T) where qT qT, 4> = 1 + (Tc-T) a(Tc-T) 182 For T > 1.0 r q a = a(Tc) • [ 1 - p(T ) ] (T/Tc - 1) T = (Tc/Tb - 1) Differentiate a with respect to T and simplify da qa(Tc)(a-l) dT (T-T ) da qa(T_)T(o-l) T — = dT (T-Tc ) da cjT qT T — = a 1 - = a • 4> dT (T-Tc) a(T-Tc) where qT qT 1 - (T-Tc) a(T-Tc) The enthalpy departure function is given by the following: RT$ A Z-0.414B H H° = RT(Z-l) + In 2y/2 B Z+2.414B 183 6. HEAT CAPACITY DEPARTURE FUNCTION The constant pressure heat capacity departure function is given by the following thermodynamic relation: 9(H-H°) (C p - Cp") = (A.59) 9T JP The enthalpy departure function is given by Eq. (A.58) and differentiating Eq. (A. 5 8) gives 9(H-H°) 9(PV) 9(RT) 9T 9T 9T p P JP 9 a - T(da/dT) V+02b — ( ) In 9T 26^ . P V+03b a - T(da/dT) 9 v+e2b ( — In 2exb 9T V+03b (A. 60) or 9V V+c?2b (Cp - Cp«) = P - R + xxln 9T V+(9,b a - T(da/dT) (A.61) 2c?xb 184 where a - T(da/dT) (A.62) dT 26xb Differentiating Eq. (A.62) gives d2a Xx = - (A.63) dT2 and 9V v+e2b X = — 2 ( In (A.64) 9v V+f93b 9T Differentiate Eq. (A.64) and simplify, x2 becomes 26xb 9V X2 = 2 2 (A.65) V +ubV+wb 9T JP Substituting Eq. (A. 63) and (A.65) into (A.61) gets 2 9V d a V+62b (Cp - Cp") = P - R - In 2 9T 26xb dT V+03b JP a - T(da/dT) 9v (A.66) 2 2 9T V +ubV+wb P 185 Substitute Eq. (A.l) for P and simplify. Eq. (A.66) becomes RT T(da/dT) 3V (Cp - Cp") = V-b V2+ubV+wb2 3T 2 T d a V+t?2b In - R 2 (A.67) 26xh dT V+t?3b Volume can be written as a function of pressure and temperature: V = f(P,T) (A.68) Take the total differential of Eq. (A.68) av 3V dV = — dP + — dT (A.69) 3P T JP At constant volume 9V 9V 0 = — dP + — dT (A. 70) 3P T Rearranging Eq. (A.70) and gets 186 3v 3V 3P 3P 3 (A. 71) 3T P T T Or (3P/3T) 3V v (A.72) 3T (3P/3v)r (3P/3V)T is given by Eq. (A.18) and (3P/3T)V is given by Eq. (A.55). Let T(da/dT) RT X = 3 (A. 73) V2+ubV+wb2 V-b Multiply x3 by (3P/3T)v and simplify 3P R2T 2RT(da/dT) + 3T (V-b)2 (V-b)(V2+ubV+wb2) da 2 2 2 (A. 74) dT (V +ubV+wb ) The numerator of the first term of the right hand side of Eq. (A. 6 7) is Eq. (A. 7 4) and the denominator is (3P/3V)T. Now multiply this fraction by the fraction: (V-b)(V2+ubV+wb2) -R2Tb 187 and get 3P V2+ubV+wb2 2 da Numerator = x: 3T JV b(V-b) Rb dT (V-b) da 2 2 2 (A.75) R b V +ubV+wb dT and 3P V2+ubV+wb2 a(2V+ub)(V-b) Denominator = 9V Rb(V-b) R2Tb(V2+ubV+wb2) (A. 76) Rewrite Eq. (A.75) and (A.76) in terms of A, B, and Z 3P Z2+uBZ+wB2 2 da B da (Z-B) 3T B(Z-B) Rb dT B(Z2+uBZ+wB2) JV Rb dT (A. 7 7) and 3P Z2+uBZ+wB2 A(2Z+uB) (Z-B) 2 2 (A. 78) 3V RB(Z-B) RB Z +uBZ+wB 188 Let Z2+uBZ+wB2 M = (Z-B) and B da N = Rb dT Substituting M and N into Eq. (A.77) and (A.78). The first term of right hand side of Eq. (A.67) becomes 2 X»OP/3V)T R(M-N) 2 (A. 79) (9P/3V)t M -A(2Z+uB) The final equation for (Cp - Cp°) is then given by the following: 2 2 R(M-N) T d a Z+6»2B In - R 2 2 M -A(2Z+uB) 20ib dT Z+t?3B (A.80) Or Eq. (A.80) can be written as C - C 0 ' Cp - Cy " " Cp" " Cy" <~p l_p R R R R (A. 81) 189 where 0 2 C - C d a Z+02B In 2 (A.82) dT Z+t93B 2 cp cv (M-N) (A.83) M2-A(2Z+uB) = 1 (A.84) 6.1. The Soave-Redlich-Kwong Equation Differentiate Eq. (A. 3 2) twice with respect to temperature and get 2 d a a(Tc)m m 2 5 dT 2TTC°• C Z(Z+B) M = (Z-B) B da N = Rb dT d2a (Z+B) R(M-N)2 C - C 0 = — In + - R dT2 M2-A(2Z+B) 6.2. The Peng-Robinson Equation Differentiate Eq. (A.32) twice with respect to temperature and get 2 5 d a a(TjK a" • K + 0 5 dT2 2TT • ^•0 .5 0 . 5 C C Z2+2BZ-B2 M = Z-B B da N = Rb dT d2a Z+2.414 B R(M-N)2 C - C 0 = In 2/2 b dT2 z-o. 414B M2-2A(Z+B) _ 6.3. This Work Differentiate Eq. (A.32) twice with respect to temperature and get For Tr < 1.0 2 d a (2Tr-q-l) da 2 dT Tr(T -T) dT and for Tr > 1.0 d2a (q-D da dT2 (T-T ) dT 191 Z2+2BZ-B2 M = Z-B B da N = Rb dT d2a Z+2 .414B R(M-N)2 0 CU - C = In + - R P *-p 2 2 2|/2~ b dT Z-0 .414B M -2A(Z+B) 7. JOULE-THOMSON COEFFICIENT The Joule-Thomson coefficient, u, is given by the relation: -1 T(3P/3T), M = + V (A.85) OP/3V). (3P/3V)T is given by Eq. (A.18), (3P/3T)V is given by Eq. (A.55) and (Cp - Cp°) is given by Eq. (A.80) 7.1. The Soave-Redlich-Kwong Equation 3P R 1 da 3T V-b V(V+b) dT 3P -RT a(2V+b) + 3V (V-b)2 V2(V+b)2 7.2. The Peng-Robinson Equation ap da 3T V-b V2+2bV-b2 dT 3P -RT 2a(V+b) + av (V-b)2 (V2+2bV-b2)2 JT 7.3. This Work ap da 3T V-b V2+2bV-b2 dT JV ap -RT 2a(V+b) + av (V-b)2 (V2+2bV-b2)2 8. INVERSION CURVE The inversion condition, u = 0, is given by the equation ap ap T — + V — = 0 3T V av T where 3P RT 1 da 3T V-b V2+ubV+wb2 dT JV 193 Since a = a(Tc) • a (A.32) da da T — = a(T ) • T — (A.88) dT dT Let da Therefore 3P RT a(T W (A. 90) 9T V-b V2+ubV+wb2 or 9P T P 1 p r c °a c* 2 2 (A. 91) 9T \ where I = V/b (A.92) 9P -RTV aV(2V+ub) 2 2 2 2 (A.93) 9V (V-b) (V +u±»V+wb ) JT 194 or 9P -T P fi P a S $+u) r c (2 a c 2 2 2 (A. 94) 9V Ojj ($-l) 0^ ($ +u5+w) JT Adding Eq. (A.91) and (A.94) and simplify $(2$+u)a - (£2+u$+w)c6 2 2 (A. 95) : ($ +u$+w) (o-D °b Rearranging Eq. (A. 95) gives the inversion curve equation 2 2 a a C^^+uS+w) + C2$(2$+u)(S-l) + C, (A. 96) where (A. 9 7) C2 = - Raa (A. 98) K The reduced inversion pressure is given by the following: RTr 1 Raa 1 p r = 2 2 (A. 100) ($-1) f^ (S +u$+w) 195 8.1. The Soave-Redlich-Kwong Equation 0 5 0 = - m(aTr) - Cx = 0.08664Tr 0 5 : C2 = - 0.42748 [ 1 + m(l - Tr - ) ] 0 5 C3 = - 0.42748 m(aTr) - The inversion curve is 2 2 CiSCS+l) + C2(2$+l)($-l>* + C,<5+1)<5-1> = 0 The inversion pressure is T 1 0 a 1 Pr " ~ 8.2. The Peng-Robinson Equation 0 5 0 = - rc(aTr) - Cx = 0.07780Tr S : C2 = - 0.45724 [ 1 + /c(l - Tr«- ) ] 0 5 C3 = - 0.45724 *c(aTr) - 196 The inversion curve is 2 2 2 2 2 C1(S +2S-1) + 2C2S($+1)(S-1) + C3($ +2$-l)($-l) = 0 The inversion pressure is given by r % (S-D V S2+2$-l 8.3. This Work For Tr < 1.0 a-1 Cx = 0.07780Tr q C2 = - 0.45724 [ 1 + p(T*) ] a-1 C3 = - 0.45724 q 1-T r and for Tr > 1.0 T (1-a) * = - q — Tr-1 Cx = 0.07780Tr 197 q C2 = - 0.45724 [ 1 ~ p(T*) ] Tr(l-a) C3 = - 0.45724 q Tr-1 The inversion curve is 2 2 2 2 2 1 2 C (S +2S-1) + 2C S(S+D(S-1) + C3($ +2$-im-l) = 0 The inversion pressure is given by T 1 0 a 1 P = _E 2 2 2 Ou (s-i) nh s+2$-i 9. VIRIAL COEFFICIENTS The generalized cubic equation is given by Eq. (A.l): RT (A.l) V-b V2+ubV+wb2 The first term, (RT/V-b), can be written as RT RT 1 (A. 101) V-b V 1-b/V or 198 RT RT b b2 b3 1 + - + — + — + 2 3 (A. 102) V-b V V V V The second term can be written as 2 2 2 (A.103) V +ubV+wb V V+ub+(wb /V) or 2 2 a 1 ub (u -w)b — - — + + 2 3 (A. 104) V2+ubV+wb2 V V V V Substract Eq. (A. 104) from (A. 102) gets uab 1 Z = 1 + b - b2 + RT RT V2 (w-u2)ab2 3 b + (A. 105) RT V3 The virial equation: B C D Z = 1 + - + — + — + (A. 106) V V2 V3 Therefore the second virial coefficient is r 199 a B = b (A.107) RT The third virial coefficient is uab C = b2 + (A.108) RT The fourth virial coefficient is (w-u2)ab2 D = b3 + (A. 109) RT 9.1. The Soave-Redlich-Kwong Equation a B = b RT ab C = b2 + — RT ab2 D = b3 - RT 9.2. The Peng-Robinson Equation a B - b RT 2ab C = b2 + RT 200 5ab2 D = b3 RT 9.3. This Work a B = b RT 2ab C = b2 + RT 5ab2 D = b3 RT = - q 1-T r