THERMODYNAMIC PROPERTIES FROM CUBIC EQUATIONS OF STATE

by

P A T R I C K C H U N G - N I N M A K

B . A . S c , The University of British Columbia, 1985

A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L M E N T O F

T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F

M A S T E R O F A P P L I E D S C I E N C E

in

T H E F A C U L T Y O F G R A D U A T E S T U D I E S

D E P A R T M E N T O F C H E M I C A L E N G I N E E R I N G

We accept this thesis as conforming

to the required standard

T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A

February 1988

© P A T R I C K C H U N G - N I N M A K , 1988

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degree at the University of British Columbia, I agree that the Library shall make it

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copying of this thesis for scholarly purposes may be granted by the head of my

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publication of this thesis for financial gain shall not be allowed without my written

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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

T c/T b - 1 and

* T/T - 1 T = for T > T

V T b " 1

where T c is the critical temperature and is the normal boiling point. The

a-function is now given by the following :

a = 1 + p(T*) q for T < T c

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 < T r ^ 1.05, the proposed equation gives rather poor results. F o r isobaric

heat capacity calculation, both Soave-Redlich-Kwong and Peng-Robinson equations

are adequate. The Soave-Redlich-Kwong equation gives the lowest overall average

R M S % 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 T r = 0.8. A n 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

A B S T R A C T ii

L I S T O F F I G U R E S vi

L I S T O F T A B L E S x

A C K N O W L E D G E M E N T xii

Chapter 1. I N T R O D U C T I O N 1

Chapter 2. L I T E R A T U R E R E V I E W 7

Chapter 3. D E V E L O P M E N T O F A N E Q U A T I O N O F S T A T E 19

3.1. Theoretical Considerations 19 3.2. Lie lmezs-Merriman Equation of State 25 3.3. Modified Lielmezs-Merriman Equation of State 32

Chapter 4. A P P L I C A T I O N S O F T H E C U B I C E Q U A T I O N O F S T A T E 45 4.1. Region I : Saturation 46 4.2. Region II : Subcritical 70 4.3. Region III : Supercritical 89 4.4. Region I V : Compressed Liquid 106 4.5. Inversion Curve 123

Chapter 5. S E C O N D V I R I A L C O E F F I C I E N T 137 5.1. Introduction 137 5.2. Theoretical Background 139 5.3. A New Correlation 141 5.4. Comparison 145

C O N C L U S I O N S 150

R E C O M M E N D A T I O N S F O R F U R T H E R S T U D Y 153

N O M E N C L A T U R E 154

R E F E R E N C E S 157

A P P E N D I X 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

i v

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. Vir ia l 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 : E r r o r distribution curves of n-pentane 63

Figure 4.4 Region I : Error distribution curves of 1-propanol 64

Figure 4.5 Region I : E r r o r 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

v i

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 S R K , PR, L M , and G C P equations 128

Figure 4.53 Inversion curve of propane calculated from the S R K , PR, and L M equations 129

Figure 4.54 Inversion curve of n-butane calculated from the S R K , PR, and L M equations 130

Figure 4.55 Inversion curve of carbon monoxide calculated from the S R K , PR, L M , and G C P equations 131

Figure 4.56 Inversion curve of carbon dioxide calculated from the S R K , PR, and L M equations 132

Figure 4.57 Inversion curve of ethylene calculated from the S R K , P R , L M , and G C P equations 133

Figure 4.58 Inversion curve of p-hydrogen calculated from the S R K , PR, and L M equations 134

Figure 4.59 Inversion curve of ammonia calculated from the S R K , P R , and L M equations 135

Figure 4.60 Inversion curve of argon calculated from the S R K , PR, L M , and G C P 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 R M S % 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 R M S % Error 53

Table 4.3 Region I - Liquid and Vapor Volume R M S % Errors 55

Table 4.4 Region I - Enthalp3 f and Entropy of Vaporization : Average Absolute Deviation 57

Table 4.5 Overall Average Errors (RMS % and A A D ) 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 R M S % E r r o r 76

Table 4.9 Region II - Pressure and Temperature R M S % 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 R M S % E r r o r 81

Table 4.15 Region III - Volume R M S % E r r o r 93

Table 4.16 Region I U - Pressure and Temperature R M S % 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 I l l

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

A C K N O W L E D G E M E N T

M y 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 ( L 1 )

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. M a n y of the

common two-constant cubic equations of state can be expressed by the general

form suggested by Schmidt and Wenzel [103] :

3

RT p =

V - b

a(T) V 2+ubV+wb 2

(1.2)

F r o m 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 E O S 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 R K equation

(SRK) [110] and the P R equation are such equations.

Since the introduction of the S R K and P R 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 M e r r i m a n (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 / * \ /V

Region I : \ \ 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 P R 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'

hard sphere equation to represent the repulsion pressure:

8

RT P =

R V-b (2.1a)

while the attraction pressure can be expressed by the following equation:

a(T) V*+ubV+wb2 (2.1b)

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) V(V+b) (2.2)

and the two constants a and b are defined by Eq.(2.2a) to (2.2c)

a(T) = fla a (2.2a)

-0.5 (2.2b) a = T 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 R K 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. T h a t 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 R K 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 T f 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:

a = [ 1 + m ( l - T r ° ' 5 ) ] 2

( 2 . 3 )

m = 0 .480 + 1 .574u - 0 .176w 2 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 R K [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 P R remains the same as before but

the attraction term is now given by the following:

a(T) P a 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],

m = 0.48508 + 1.55171w - 0.15613w2 ( 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 = - l o g (a t T r = 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 ( P y p r ) at T r = 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 H a l m and Stiel [46], O'Connell and

Prausnitz [81], and T a r a k a d 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 :

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:

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.

a = 1 + (1 - T r)(m + n/Tr) (2.8)

a = 1.0 + p(T*) (2.9)

where

(2.9a)

15

Recently Stryjek and Vera [117,118] modified the PR equation by

introducing a more complicated temperature dependent a-function:

a = [ 1 + K<1 - T r0* 5) ] 2

( 2 1 0 )

K is a complex function of reduced temperature and the acentric factor u:

K- = «•„ + [ Ki + K 2 ( K 3 ~ T r ) ( l " Tr°'5) ] •

(1 + Tr°'5)(0.7 - T r) ( 2. 1 0 a)

where »c0 is given by

K0 = 0.378893 + 1.4897153w - 0.17131848a)2

+ 0.0196544w3 (2.10b)

and K L T 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

utilizing the volume-translation technique;

correction constant b are corrected by c.

constant are now V and b', respectively:

V = V - c

b" = b - c

modification of the SRK equation by

the specific volume and the volume

The new volume and volume correction

(2.11)

(2.12)

The equation of state after translation becomes

RT a(T) P = - (o n i

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 T r = 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 . i 4 )

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 T r = 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 C 2 0. 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

S R K E O S , for saturated vapor and superheated gas phase density and fugacity

predictions for polar and nonpolar compounds. Probably the most comprehensive

review yet on E O S is the one by Y u et al [129]. Their comparison involved 14

cubic equations, both PR and S R K were included, for eight properties of straight

chain hydrocarbons from d to C 1 0 . Reasonable ranges of pressure and

temperature were covered in their stud3'.

F r o m 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 L M 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 L M 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)

p l ' p V (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

f 1 = f v

(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

E O S . 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

G 1 - G V = 0 (3.3a)

Substituting Eq.(3.7) into (3.3a) yields

A 1 - A V + P 1V 1 - P VV V = 0 (3.8)

But A^ - A V can be written as

A 1 - A V = / 3A

av dv (3.9)

and from the fundamental equations of thermodynamics,

3A 3V = -P

JT (3.10)

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(V 1 - V v) = 0 V 1 (3.11)

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

V v

J x PdV = area 1-2-3-4-5-6-7-1 ( 3 1 2 )

and the second term represents the area bound by the isobar P and the isochors

V 1(=V 7) and V V (=V6) or

P(V V - v 1) = area 1-3-5-6-7-1 ( 3 1 3 )

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 V̂ - is the saturated liquid and V v 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

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:

PV lim — P—>0 RT

= 1 (3.17)

3 P = 0 (3.18) 3V

24

9 2P av2

= 0 (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):

Z = 1 + a

b RT

1 — + V

b 2 + uab RT

b 3 + (w-u 2)ab 2

RT 1 V 3 (3.20)

The virial equation:

B C D Z = 1 + - + — + — + V V 2 y 3 (3.21)

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 -RT (3.22)

C = b 2 + uab RT (3.23)

(w-u2)ab2

D = b 3 + RT

25

(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):

* _ T c / T ' 1

where T , T̂ , 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 ( 2 9 )

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 M e r r i m a n [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 Merr iman [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 ( F K T ) [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 %)

RMS % Error = I (% e r r o r ) 2

N (3.25)

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] :

T, ln(P /P") s = -£=

l-T.

28

(3.26) br

which is the negative of the slope between the critical point and normal boiling

point of the vapor pressure equation:

lnP r= 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 Z c , 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 - 2 9 )

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 :

p = -1.1977373 + 0.39942704s - 0.026211814s2

variance = 0.00004469

(3.30)

p = 0.19011333 + 0.67187907a) - 0.85099293a)2

variance = 0.00004475

30

(3.31)

Group 2

p = -1.8772891 + 0.55686202s - 0.033057663s2 ( 3 3 2)

variance = 0.00024869

p = 0.1639189 + 1.1372604a) - 1.0533208o>2 ( 3 3 3 )

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 , T̂ , and P c are needed for

calculating s and vapor pressure at T r = 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. F o r 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 V e r a [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 T r = 0.7. O n 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 :

u = -0.65652243 + 0.079558804s + 0.0080858283s2

( 3 3 6 )

variance = 0.00000867

Group 2 :

u = -0.71488058 + 0.10168299s + 0.0058534228s2

variance = 0.00001785

(3.37)

32

Group 3 :

u = -1.7653114 + 0.5377135s - 0.03883573s2 ( 3 3 8 )

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 L M equation to the entire P V T surface, the T

coordinate system must be modified. One criteria in modifying the L M 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 R M S % error in predicting volume

by the first modification is slighty lower than the second modification. For

methane, the R M S % 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

a = 1 - p(T*) q

The two constants, p and q, are the same in all regions.

(3.40)

34

The proposed modification of the a-function has kept the simplicity of the

original L M equation and yet it is applicable to the entire P V T 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. F r o m 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 Z J-(1-B)Z1+(A-3B1-2B)Z-(AB-B1-B')=0

FOR Z V & Z

CALCULATE f V & f 1

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

3 -0.1

8.0

10.0

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 F i t t e d p = f(s) p = f(u) Fi t t e d 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 D i s u l f i d e 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 Ter t i a r y 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 F i t t e d p = f(s) p = f(u) F i t t e d Avg. s u Group

Propylene 0. , 27311 0 .26929 0 .27091 0 .83694 0 .83 6 . 17555 0. 148 1 1-Butene 0. .29085 0 .28708 0, , 28600 0 .85773 0 .83 6 .43366 0. 187 1 Water 0, .44221 0. .42842 0. .43049 0 .73237 0 .83 7 . 33080 0. 344 2 n-Deuterlum 0. .06394 0. .06245 0. ,06428 0 .61602 0, .78 4 .50830 -0.13 3 n-Hydrogen 0. ,01877 0. .01838 0. ,01724 0, ,36720 0 . 78 4 .08143 -0. 22 3 p-Hydrogen 0. ,01582 0. .01664 0. .01724 0. .30806 0. . 78 4 .06100 -0. 22 3 Nf trogen 0. 20477 0. 21681 0. 21563 0. ,81713 0. 83 5 .59774 0.040 1 Ammon1 a 0. 38595 0. 38180 0. 38240 0. 85842 0. 83 6 . .80948 0.250 2 Oxygen 0. 20734 0. 20271 0. 20385 0. 81007 0. 83 5 . .46878 0.021 1 Neon 0. 1446G 0. .14964 0. 18425 0. ,76353 0. .78 5 . .11029 0.000 3 Argon 0. 18893 0. 18789 0. 17964 0. 78649 0. .78 5 .32535 -0.004 3 Krypton 0. 19626 0. 19068 0. 18194 0. 80560 0. 78 5, .34029 -0.002 3 Xenon 0. 20060 0. 19503 0. 18658 0. 79778 0. 78 5 . .36331 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, .431 0. 749 0. , 565 ' 0. ,586--o .990 29 Ethane 0. 542 0 . 569 0, .442 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. .733 0. 194 0 . 282 0. .641--0 . 998 32 i-Butane 0. 357 0 .312 0, .731 0. 186 O. . 345 0. ,640--0 .998 30 n-Pentane 0. 678 0 . 565 0, ,500 0. 184 0. , 236 0, ,658--0 .999 27 i-Pentane 0. 501 0 .404 0. ,526 0. 164 0, , 146 0, ,654--0 .999 27 Neopentane 0. 203 0 . 140 0. .741 0. 246 0. ,426 0. ,652--0 .999 25 n-Hexane 1 . 464 1. . 136 0. ,403 0. 573 0. ,445 0. ,538--0. .995 43 n-Heptane 1 . 583 1. . 128 0. ,351 0. 455 0. , 241 0. ,545--0. .987 30 n-Octane 1 . 841 0 .894 0. ,601 0. 543 0. , 751 0. ,488--0. .976 27 Benzene 1 . 008 0, .896 0. ,745 0. 495 0. ,636 0. ,553--0. .997 47 Carbon Monoxide 0. 730 0. .699 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. ,995 32 Carbon D i s u l f i d e 3. 836 3. .841 2. 482 3. 310 3 . 761 0. 500--0. .976 26 Hydrogen Sulfide 1 . 580 1 . ,593 1 . 767 1 . 378 1 . 814 0. 565--o. ,997 30 Sulfur Dioxide 1 . 921 1 , ,723 0. 806 1 . 1 16 4 . 404 0. 593--0. .992 35 Methanol 5. 409 4 . ,618 3. 211 4 . 654 1 . 477 0. 532--0, ,996 44 Ethanol 3. 595 2 . .761 2. 940 3. 378 0. 684 0. 543--0. ,979 29 1-Propano1 5. 667 6. .876 7 . 565 6. 800 2 . 488 0. 555--0. 993 13 Tertiary Butanol 1 . 121 1 . .295 1 . 832 1 . 709 2. 945 0. 701--0. 993 12 Acetylene 2. 391 2. ,423 3. 067 2 . 668 2 . 479 0. 623--0. ,989 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. .651 0, ,707 0. . 139 0, , 168 0, .618--o. .989 26 1-Butene 0. .972 0. ,886 0, ,876 0. ,450 0, .531 0 .651--o .979 26 Water 7 .389 6. .574 2, , 1 16 7 . . 154 1 , .715 0, .422--0, .995 43 n-Deuter i urn 0, .897 1 . .647 2 , ,878 2. , 134 2 . .824 0, ,488--o. ,991 21 n-Hydrogen 0, .251 0. ,665 1 , , 135 1 . 469 0, .613 0, .615--0, .964 13 p-Hydrogen 0, .097 0. ,533 1 , ,010 1 . , 384 3 , .023 0, .615--0, .970 13 Ni trogen 0. .695 0, ,787 0, .586 0, 578 0, .449 0, .614--0, .991 18 Ammon i a 2. .053 1. .853 1 , ,664 1 . ,335 0, .811 0, .602--0, .986 29 Oxygen 0. .248 0. .408 0, .493 0. , 194 0, , 303 0, .583--0, .986 24 Neon 0. .539 0. ,797 0, ,545 0. 753 0, ,294 0, . 563--0, .968 19 Argon 0, .224 0. .442 0, ,244 0. 293 0, , 155 0, ,555--o. .994 15 Krypton 0, . 183 0. 245 0. .469 0. 243 0, .490 0. . 573--0, .984 20 Xenon 0, .225 0. ,417 0, ,492 0. 138 0, ,338 0, .569--0. .932 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. .2510 0, , 2493 0, .2486 0, . 2567 1-Pentane 6 .62562 0, . 2270 0, .2256 0 .2251 0, . 2332 Neopentane 6 .45483 0. . 1970 0, . 1939 0, . 1944 0, .2014 n-Hexane 6 .99910 0. .2960 0. , 2964 0, . 2965 0, . 3027 n-Heptane 7. , 26413 0. ,3510 0. ,3481 0. ,3488 0. . 3520 n-Octane 7 , .50371 0. 3940 0. 3957 0. 3970 0. , 3966 Benzene 6, .55501 0. 2120 0. 2124 0. ,2073 0. , 2201 Carbon Monoxide 5 . 64933 0. 0490 0. 0464 0. ,0506 0. ,0515 Carbon Dioxide 0. 2250 Carbon Dioxide 0. 2250 Carbon D i s u l f i d e 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 . 83047 0. 2510 0. 2528 0. 2501 0. 2713 Methanol 8 . 39992 0. 5590 0. 5523 0. 5424 0. 5634 Ethanol 8 . 87656 0. 6436 0. 6489 0. 6430 0. 6522 1-Propanol 8. 74510 0. 6240 0. 6220 0. 6227 0. 6277 Tertiary Butanol 8 . 73054 0. 6180 0. 6190 0. 6253 0. 6250 Acetylene 0. 1840 Acetylene 0. 1840 Ethylene 5 . 84587 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: S R K , PR,

and L M 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. Al l 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 (T c ) , normal boiling point (T^), acentric factor (<j), and the parameter

s. For carbon dioxide and acetylene, their triple point pressures are above one

atmosphere. Therefore these two compounds do not have normal boiling points.

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 ft 3/lb = 0.062428 cm3/g

0°C = 273.15 K = 32°F = 460.0 R

atm cm3

Gas constants, R = 82.05606 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

cal = 1.9872

g-mole K

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, T r 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 P R and LM(1)

equations give almost identical results. The LM(1) equation gives slightly lower

errors near the critical point. In terms of R M S % error, the S R K equation gives

almost double that of the P R or LM(1) equation. F r o m Figures 4.2-4.5, it can

be seen that the S R K equation tends to over predict liquid volumes along the

entire saturation curve for all three groups of compounds. For the P R 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 P R and L M 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. F r o m the

evidence presented up to this point, one may conclude that altering the a-function

will not dramatically improve volume predictions. B y 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 A S „ = — * (4.1) v T

Enthalpj' of vaporization is just the difference between the vapor and liquid

phase enthalpy:

AH v = (H - H°) v - (H - H 0 ) 1 (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

T r = 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 1 .70 0. .008 5 .41072 Ethane 30.068 48 .2000 305 .42 184 .47 0, ,098 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. .251 6 .75195 i-Pentane 72.1498 33 . 3700 460 .39 301 .025 0. ,227 . 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. .7003 562 .65 353 . 25 0. 212 6 .55501 Carbon Monoxide 28.010 34 . ,5264 132 .92 81 . ' 70 0. 049 5 .64933 Carbon Dioxide 44.011 72. .8062 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 . .2000 373 .07 212 . 875 0. 100 5, .95272 Sulfur Dioxide 64.066 77 . 7967 430 .65 263 .00 0. 251 6 , ,83047 Methanol 32.042 78. 5928 513 . 15 337 .696 0. 559 8 . . 39992 Ethanol 46.069 60. .5675 513 .92 351 .443 0. 6436 8 , .87656 1-Propanol 60.090 50. .2109 537 .04 370 .93 0. 624 8 , ,74510 Ter t i a r y Butanol 74.1224 41 . .7700 508 .87 356 .48 0. 618 8 , .73054 Acetylene 26.036 61 . .6494 308 .69 189 .20 0. 184 Ethylene 28.054 50. 5001 283 .05 169 .40 0. 085 5 . .84587

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. .34029 Xenon 131.30 57.6000 289.74 165.02 0.002 5. .36331

+ Physical properties that are not given in the ori g i n a l 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. .990 29 10 Ethane 0 .920 0 .420 0, .812 1 .079 0 .604- -0. .999 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. .998 32 18 i -Butane 1 , .036 0, .700 0, , 742 1 .803 0, .640--o, 998 30 19 n-Pentane 0, .991 0, .336 0. ,803 1 . 299 0, .658- -o. ,999 27 20 i-Pentane 0 .987 0, .493 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. .725 1 , .822 1 . ,325 3 . 135 0. . 538--0. 995 43 10 n-Heptane 1. .865 1 , ,034 1 . .412 1 , .754 0, , 545--o. 987 30 115 n-Octane 1, .620 4 , , 395 2. 711 3 , .374 0, .488- -0. 976 27 1 15

Benzene 1. . 297 1 . ,01 1 1 . 077 4, ,551 0. ,553--o. 997 47 10 Carbon Monoxide 0. .897 2. .455 0. 972 4 , .096 0. ,513--o. 981 24 10 Carbon Dioxide 0. .540 0. ,704 0. 385 0. .712- -0. 995 32 10 Carbon Di s u l f i d e 4 , ,490 4 , .850 2 . 317 5 , .996 0. .500- -0. 976 26 80 Hydrogen Su1f i de 1 . , 138 1 . 808 1 . 630 5 , . 524 0. .565- -0. 997 30 115 Sulfur Dioxide 2. ,244 2 . 022 1 . 263 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 . 365 6. 410 0. 701--o. 993 12 56 Acetylene 2 . 583 1 . 929 0. 415 0. 623- -0. 989 20 10 Ethylene 1 . 533 1 . 226 0. 772 1 . 627 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 . 254 0 .666 0 .699 1 . 051 0 .618--o .989 26 10 1-Butene 1 . 536 0 .923 0 .874 1 . 034 0 .651- -0 .979 26 10 Water 13.34 7 .905 5, .823 12.19 0, , 422--0, .995 43 10 n-Deuter1 urn 9. 206 5 .065 1 . . 359 2 . 801 0. 488--0 .991 21 97 n-Hydrogen 2. 096 3. .671 0. ,690 1 . 753 0. ,615- -o. .964 13 69 p-Hydrogen 2. 157 2, ,910 0. .705 1 . 655 0. 615- -o, .970 13 97 Ni trogen 1 . 503 1 . . 560 0. ,633 2 . 892 0. ,614--0, .991 18 10 Ammon i a 0. 910 0 .746 1. .465 1 . 599 0, ,602--o. .986 29 10 Oxygen 1 . 140 0 . 795 0, .643 1 . 793 0. , 583--0, .986 24 10 Neon 9 . 074 7 . .716 0. . 77 1 1 . 782 0. 563- -O, .968 19 126 Argon 1 . 371 1 . .450 0. 677 0. 683 0. 555- -0, .994 15 126 Krypton 1 . 104 1 . ,043 0. 599 1 . 277 0. 573- -0. ,984 20 126 Xenon 1 . 174 1 . ,313 0. 694 1 . 539 0. 569- -0. 932 12 1 16

t Vapor pressure, l i q u i d 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 . 523 8. 372 8. 549 2 . 765 4 .554 2 , ,710 3.767 Ethane 18 .03 9. 290 8 . 157 8 . 086 3 . 140 2 .385 0. ,842 0.946 Propane 15 .35 6 . 775 6. 109 5 . 896 0 .771 2 .863 1 . ,741 1 .410 n-Butane 22 .61 1 1 .91 10. 73 9. 883 4 . .932 2 . 280 2 . 323 2.998 1-Butane 20 .71 1C I. 53 9. 456 8. 623 6, , 244 1 .544 3. 156 4.975 n-Pentane 24 .27 13 .41 12 1.05 1 1 . 30 4 . .843 0 .981 2 . 371 3.780 i-Pentane 22 .91 12 .56 1 1 . 19 10. 53 5 . ,884 1 . 122 2 . 917 4 . 543 Neopentane 20 .57 10 i . 85 9. 842 9. 035 3 . 718 2 , , 195 1 . 015 4 . 220 n-Hexane 17 .71 7 . 179 5. 912 6 . 018 1 . 936 1 , . 763 2 . 838 4 . 376 n-Heptane 18 .20 6 . 891 5. 790 5 . 906 2 . ,660 1 , ,871 2 . 318 2.578 n-Octane 20 .08 7. 086 6. 779 6 . 929 1 . 325 3 , , 799 3. 413 3 . 724 Benzene 14 .60 6. 009 5. 144 5. 236 2 . 366 3. ,679 2. 308 5.892 Carbon Monoxide 5.819 9. 227 9. 286 9 . 458 4 . 797 3, ,618 4. 071 7.277 Carbon Dioxide 15 .78 6. 191 5. 804 2 . 293 1 . 772 1 . 867 Carbon D i s u l f i d e 9. 156 6. 861 6. 888 6 . 866 5 . 528 5. 023 8. 480 8.411 Hydrogen Sulfide 12 . 19 7 . 738 7. 378 7. 129 4 . 283 6 . 410 5. 522 8 .038 Sulfur Dioxide 18 .41 7 . 475 6 . 133 5. 734 2. 456 1 . 510 2 . 325 7.878 Methanol 41 .69 26 .02 24 .40 25 i.01 13 . 77 8 . 519 1 1 . 20 9.961 Ethanol 26 .45 12 .92 1 1 .86 12 .05 5 . 041 1 . 641 5 . 286 8.747 1-Propanol 23 . 18 1 1 .05 10 i.77 9. 147 8 . 311 10.03 5. 927 11.51 Tertiary Butanol 9.883 4 . 605 4. 700 5. 806 5. 584 3. 926 5. 051 12.99 Acetylene 14 .68 6. 344 5. 314 1 . 285 1 . 239 2 . 459 E thy1ene 9.648 7 . 246 7 . 054 6 . 971 2 . 517 2 . 068 2. 483 2.006

Table 4.3 Continued

Compound SRK L1qu1d PR

Volume LM( 1 ) LM(2) SRK

Vapor PR

Volume LM( 1 ) LM(2)

Propylene 11 .85 6.004 5.539 5 .473 2 .887 1 . 582 1 . .835 1 .876 1 -Butene 13.36 5. 147 4.727 4.555 1.615 2. 627 1 . 732 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 . 694 6 . ,286 4.311 n-Hydrogen 6.990 15.46 16 . 20 15 .84 2 .982 3. 090 2 . 316 2.352 p-Hydrogen 6.932 14. 10 14.66 14.39 2 .689 5. 333 1 . 989 4 .080 Nitrogen 9.506 8.853 8.750 9.000 3.477 5 . 198 3 . .626 4 .016 Ammonia 30.75 15.90 15.48 15.13 7 .656 6. 036 7 . 286 8.378 Oxygen 6.661 8.828 8 .873 8.688 3.860 4. 888 3 . 695 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 . 315 1 .202 Krypton 7.946 8.528 8.488 8.412 1 .849 3. 044 1 . 818 1 .772 Xenon 6.481 7.900 7 .995 7.912 1 . 335 2 . 109 0. 915 1 .437

Table 4.4 Region I - Enthalpy and Entropy of Vaporization : Average Absolute Deviation

Compound SRK Enthalpy PR

(cal/mole) LM( 1) LM(2) SRK

Entropy PR

(cal/mole-K) LM( 1 ) LM(2)

Methane 48.9 30.8 30.6 36.3 0. .333 0. . 183 0, .210 0.278 Ethane 97. 1 82. 1 56.8 51.9 0. . 356 0. , 283 0, .228 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. .406 0. 327 0. .257 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. ,302 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 D i s u l f i d e 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. 643 1 . 250 1 . 166 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 Te r t i a r y 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, .301 0.217 Water 426.8 284.5 349.5 462.8 1 , , 144 0. 729 0, .894 1 .378 n-Deuterlum 27.0 18.9 13.4 13.9 1 . ,057 0. 719 0. ,467 0.539 n-Hydrogen 9. 10 5.20 5.20 6.70 0. .386 0. 203 0. .220 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. .250 0. 254 0, , 247 0.326 Ammonia 209.8 174.5 220.5 213.2 0, ,434 0. 326 0. .489 0.443 Oxygen 39.4 23.8 25. 1 25.6 0. , 330 0. 181 0. ,221 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

P (RMS %)

V 1 ( R M S %) V v ( R M S %) A H v (AAD:cal/mole)

A S „ (AAD:cal /mole-K)

S R K P R LM(1)

2.57 2.31 1.36 15.94 10.34 9.77 4.78 3.86 3.53

123.5 100.7 103.6

0.433 0.339 0.355

L M ( 2 ) t N C N

3.24 36 933 9.92 36 933 4.99 36 933

128.6 36 933

0.432 36 933

Region II : Subcritical

Property S R K P R L M N C 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 ( A A D ) 0.0064 0.0081 0.0084 22 947 C p (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 S R K P R L M N C 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 ( A A D ) 0.0217 0.0156 0.0174 22 3417 C p (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 I V : Compressed L iqu id

Property S R K P R L M N C N

P ( R M S %) >100. >100. >100. 18 733 V ( R M S %) 14.92 7.97 7.79 20 878 T ( R M S %) 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 ( A A D ) 0.0361 0.0218 0.0228 7 261 C D (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

SOLVE FOR Z 1 & Z V

CALCULATE f V & f 1

CALCULATE AH & AS

CALCULATE ERRORS

NO

NO

YES

ADJUST P

CALCULATE AAE & RMS % ERROR

/OUTPUT Z 1, Z V, 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.

6 6

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 ° ) - R T ( Z • - 1) (4.5)

(U - U ° ) = (H -- H ° ) - R T ( 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

H° = A + BT + CT2 + DT3 ( 4 > 8)

and

S° = E + FT + GT2 + HT3 ( 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 E O S , Eq.(1.2), are shown in the Appendix A .

The results of the P V T predictions from the three E O S are tabulated in

Tables 4.8 and 4.9. A l l 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 T r = 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 T r = 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

T r = 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 T r = 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 T r = 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(10 s) 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. .15136 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, .00048150 Water -56.9968 7. . 14874 1 .36210 -0.0135023 200-1500 0. .00003452 Ammon1 a -9.23274 6. 82980 2.33560 0.704254 200-800 0. .00001325

Table 4.7 Coefficients of Ideal Gas Entropy Polynomial

Compound E F(10 !) G(10') H(10") Temp. Range Variance

n-Hexane Carbon Dioxide Sulfur Dioxide Acetylene Propylene Water Ammonia

55.8375 40.7596 48.0444 36.4355 47.4536 35.5415 33.7072

13.3747 4.04572 4.44643 4.53894 5.94086 3.84262 5.54379

-3.36104 -2.25648 -2.61622 -2.32769 -1 .58523 -2.54623 -5.65984

4.19050 5.70186 6.73076 5.58302 2.24722 7.13967 26.9856

298-1500 200-1500 200-1500 298-1500 298-1500 200-1500 200-800

00020731 01 134 01998 003852 00016728 03261 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, ,095 0, .655--0 .982 0 .021--0, .892 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, .999 0 .028- -0, .972 59 19 n-Pentane 1 .740 0 .589 0, ,708 0, .660--0, .980 0 .030- -0, .842 31 20 i-Pentane 3 .279 1 .058 0. ,659 0. ,673--0. .988 0 .030- -0, , 899 37 21 Neopentane 1 . 197 1 .211 1 . 455 0, ,669- -o, ,991 0 .032- -o, .919 28 22 n-Hexane 3 . 224 1 . 154 0. ,840 0. . 722--o. ,995 0 .033- -0, ,910 40 10 n-Heptane 0 .432 1 .425 1. ,729 0. .703--0. .963 0 .037- -0, . 731 35 122 n-Octane 0 .739 1 .778 2, 065 0. .703--0. .985 0, .041--0. .806 37 122 Benzene 0 . 335 0. .816 0. 990 i'O. .658- -0, .978 0, .021--0. 811 51 122 Carbon Dioxide 3 .904 2. .381 2. 388 0. 730--0. 986 0, .014- -0. 888 33 10 Sulfur Dioxide 1 . .453 1 . . 140 1 . 500 0. 645--0. 993 0. .013- -o. 918 56 10 Methanol 1 . .644 2. .501 2. 795 0. 702- -o. 974 0. ,013- -o. 628 51 122 Ethanol 0. .784 1 . ,650 1 . 871 0. 701- -o. 973 0. ,017- -o. 652 43 122 1-Propanol 0. ,602 1 . ,318 1 . 321 0. 708- O. 968 0. 020- -o. 590 40 122 Acetylene 6 . . 196 2 , ,961 2 . 326 o. 97 1--o. 998 0. .016- -0. 971 43 10 Propylene 2 . . 101 1 , , 256 1 . 266 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 . 199 o. 840 0. 652- -0. 995 0. O05--0. 935 93 10 Ammon i a 3 . .081 1 . 354 1. 196 0. 630- -0. 986 0. 009- -0. 856 48 10

t PVT, departure function, and fugacity c o e f f i c i e n t 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. .603 0 . 737 0 .800 0 .428 0 . 494 0 .523 1-Butane 1 , .096 0 .635 0 .684 0 . 546 0 .412 0 .506 n-Pentane 1 , .033 0 .462 0, .520 0 .647 0 . 346 0 .372 i-Pentane 1 . , 324 0, .388 0, , 336 0, .663 0. . 223 0. .230 Neopentane 0. ,693 0, .714 0. , 783 0, .480 0. . 462 0, .482 n-Hexane 2. ,030 0. 828 0, ,670 1 . .223 0. .582 0. .502 n-Heptane 0. ,349 1, 010 1, 135 0. .267 0. . 738 0. ,803 n-Octane 0. 563 1. ,263 1. ,401 0. .416 0. 911 0. ,980 Benzene 0. 170 0. 621 0. 686 0. 1 14 0. 478 0. ,511 Carbon Dioxide 2. 687 1. 732 . 1. 729 1 . 662 1. 125 1 . , 127 Sulfur Dioxide 0. 851 0. 631 0. 769 0. 536 0. 395 0. 441 Methanol 1 . 216 1. 836 2. 022 0. 765 1. 188 1. 297 Ethanol 0. 579 1. 193 1 . 328 0. 368 0. 761 0. 826 1 -Propanol 0. 474 1. 015 1 . 010 0. 327 0. 699 0. 693 Acetylene 2. 523 1. 141 0. 908 1 . 205 0. 511 0. 390 Propylene 1 . 074 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 . 206 0. 756 0. 702 0. 801 0. 560 0. 554 Ammon) a 1 . 859 0. 850 0. 795 1 . 099 0. 563 0. 546

Table 4.10 Region II - Enthalpy and Entropy Departure Functions : Average Absolute Deviation

Compound SRK Enthalpy (cal/mole)

PR LM Entropy

SRK (cal/mole-K) 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

Compound SRK Helmholtz (cal/mole)

PR LM SRK Gibbs (cal/mole)

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 1-Butene 1 .3 1 . 3 1 .3 1 . 2 4.4 4.5 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 Coef f i c i e n t 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. .995 0 .024-0, ,952 40 122 n-Butane 1 . . 1 1 . 1 1 . .9 0. ,659-0, .988 0, .027-0. .747 35 122 i-Butane 2 . .6 2 . 6 3 .3 0. .662-0. .980 0. .028-0. .861 30 122 n-Heptane 0, ,3 0. .3 0 .4 0. .703-0. .963 0. .037-0. . 731 35 122 n-Octane 0. ,2 0. .3 0 .8 0. .703-0. .985 0. .041-0. .806 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, .973 0. .017-0. .652 43 122 1-Propanol 0. 2 0. 2 0. . 3 0. , 708-0, .968 0. .020-0. , 590 40 122 1-Butene 0. .3 0. 4 0, ,4 0. 667-0, .953 0. .025-0. ,498 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

0.66

0.64

0.90

0.88

0.88 -

N 0.84

0.80

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 0.7

0.6

0.4

IS

0 85

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

< i

<

3.10

3 05

3.00

2 95

Z.B5

2 80

i -Butane Data

— SRK — PR

LM

i . i . i

4.4

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

CH

I

0.85

0.80

0.75

0.70

0.65

0.60

0.55

i -Butane • Data

SRK PR LM

0.55

0.90 0.92 0.94 0.96 0.98 1.00 1.02 Tr

0.25

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

0.82

0.80

0.78

0.78 -

0.74

0.72

a,

0.90

0.88 -

0 86

0.84

0.82 -

0.80

Figure 4.22 Region II : Fugacity coefficients of i-butane (Pr = 0.56) and water (Pr = 0.31) versus reduced temperature.

OH

1.05

1.00

0 90

0.85

0.80

0.75

0.70

0.85

1.05

1.00

0 90

0.85

0.80

0.75

0.70

Water • Data

SRK PR LM

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 . ,950 1 .007--7 . 343 0 .022--21 .83 272 122 Ethane 7. 583 5 .013 5. ,266 1 .002--1 .768 0 .021--10 i . 37 108 31 Propane 6 . 422 4 .687 4 . 938 1 .000--1 .892 0 .024--16 .66 180 122 n-Butane 9 . 791 5 .439 6 . 057 1 .002--1 .411 0 .027--18 .68 69 18 i-Bu tane 6. 377 3 . 775 3 . 989 1 .005--1 .715 0 .028--1 1 . 1 1 165 122 n-Pentane 8 . 977 7 .627 7 . ,947 1 .001--1 . 491 O .030--21 .05 144 122 i-Pentane 8. 896 4 . 382 4 . 794 1 .02 1--1 . 738 0 .030--8 . 990 160 122 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 . 687 1 .006--1 . 159 0 .033--1 . 365 174 10 n-Heptane 8. 112 3. .903 4. 332 1 .037--1 .851 0, .037--18 .28 156 122 n-Octane 10.25 5, .542 6 . 491 1 .020--1 .758 0, .041--20 .14 143 122 Benzene 5 . 155 2 , ,515 3 . 082 1 .031--1 . 777 o, ,02 1--2 . 027 132 122 Carbon Dioxide 3. 104 1 . .468 2 . 003 1 .022--4 . 492 0, ,014--2 . 804 120 10 Sulfur Dioxide 7 . 971 3 . ,476 4 . 238 1 .019-• 1 .212 0. .013--3 . 936 180 10 Methanol 1 1 . 17 7. 620 8 . 186 1 .013--1 . .949 0. 013--6 . 907 208 122 Ethanol 9 . 884 6 . 122 6 . 954 1 .012--1 . , 946 0. 017--8 . 147 182 122 1 -Propanol 9. 300 5 . 480 6 . 055 1 .043-• 1 , ,862 0. 020--9 . 878 169 122 Acety1ene is l . 60 10.63 12.34 1 .001-• 1 , ,034 0. 016-• 1 . 435 63 10 Propylene 3. 565 1 . 574 1 . 902 1 .020-•3, .745 0. 022--4 . 476 130 10 1-Butene 4 . 421 1 . 664 1 . 666 1 .049-•2 , . 383 0. 025--12 .44 195 122 Water 3 . 651 2 . 385 3 . 213 1 .038--1 , . 768 0. 005-- 1 . 7 14 194 10 Ammon i a 8 . 616 5 . 359 5 . 384 1 .04 1--1 . . 452 0. 009- - 3 . 056 120 10

t PVT, departure function, fugacity c o e f f i c i e n t data are taken from the same source.

Table 4.18 Region III - Pressure and Temperature RMS % Errors

Compound SRK Pressure PR LM SRK

Temperature PR LM

Methane 4 . 555 13.59 13 .79 1.514 7.484 7.846 Ethane 19 .68 11.14 1 1 .30 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 .25 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 . 681 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 . 720 1 .310 0.615 1 .450 n-Heptane 26 . 25 6.374 6 . 871 6 .080 1 . 709 1 .664 n-Octane 40 . 13 10.92 1 1 .92 7.882 2.360 3.312 Benzene 9. 189 3. 199 3. 684 1 .397 0.581 0.597 Carbon Dioxide 7. 425 1 . 768 2 . 324 1 .949 1 . 157 1 .638 Sulfur Dioxide 24 .58 6 . 104 6 . 954 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.. 498 2.601 2 . 714 1 .455 0.683 0.787 1-Butene 10 .68 3.778 3 . 788 3 .010 1.315 1 .358 Water 3.. 418 1 .966 2 . 521 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

Compound SRK Enthalpy (cal/mole)

PR LM Entropy

SRK (cal/mole-K) 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

Compound SRK Helmholtz (cal/mole)

PR LM SRK Gibbs (cal/mole)

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 Coe f f i c i e n t 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. .021-10.37 90 31 Propane 7 . . 1 7 . .2 6 . 6 1 .027-1 .892 0. .024-16.66 162 122 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. .041-20.14 143 122 Methano1 0. . 7 0. .6 2 . . 3 1 .013-1 .949 0. .013-6.907 208 122 Ethanol 0. 8 0. .7 2 . . 7 1 .012-1 .946 0. ,017-8.147 182 122 1-Propanol 0. 9 0. 8 0. .5 1 .043-1 .862 0. ,020-9.828 169 122 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

1.0

0.9 h

tSl 0.8

0.7

0.5

1.1

0.9

0.5

0.3

1.0 1.2 1.4 1.6 Tr

l .B Z.O

Figure 4.24 Region III : Compressibility factors of n-octane (Pr=4.03) and ethanol (Pr=1.63) versus reduced temperature.

1.9

n-Octane O Data

SRK PR

1.30

tsi

1.25 h

1.20

1.15 h

1.10

1.06

1.00

0.95

Ethanol O Data

SRK PR

O

_1_ 0.0 2.0 4.0 6.0

Pr 6.0 10.0

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

Ir-CC

< I

<

9.0

8.0

7.0

6.0

5.0

3.0

-

-

n-Octane O Data

SRK PR

1

LM

- • i — - J •

< i

10.0

9.0

8.0

7.0

6 0

5.0

4.0

-

-

Ethanol O Data

SRK PR LM

._i . 1 1 ,

1.0 12 1.6 1.0 1.2 Tr

1.4 1.8 Tr

1.8 2.0

Figure 4.30 Region III : Reduced Helmholtz departure functions of n-octane (Pr = 4.03) and ethanol (Pr=1.63) versus reduced temperature.

< I

<

12.0

10.0

8.0

6.0

4.0

20 , >

0.0 ©•

n-Octane ) O Data ) SRK ) PR > LM

1 . 1

0.0 5.0 10.0 15 0 20.0 25.0 Pr

14 0

12.0

10.0

0 . 0 ® — 1-0.0 2.0

Ethanol O Data

SRK PR LM

) —. i i -

4.0 6.0 8.0 10.0 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

8.0 h

E -

a i

o

6.0

4.0

3.0

n-Octane O Data

SRK PR

- LM

10.0

9.0 I-

8.0

o E -K 7.0

O I 6.0

O

5.0

4.0

3.0

-

Ethanol O Data

SRK PR

1

LM I , I i i

1.0 1.2 1.6 1.8 1.0 1.2 Tr

1.4 1.6 Tr

1.8 2.0

Figure 4.32 Region III : Reduced Gibbs energy departure functions of n-octane (Pr = 4.03) and ethanol (Pr=1.63) versus reduced temperature.

o E -

O I

O

12.0

10.0

8.0

6.0 o i

o

o.o ©•

14.0

12.0

10.0

8.0

6.0 -X

4.0 Ethanol O Data

I SRK 2.0 1 ) PR

<

0.0 <

)

>——1—-LM

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:

H - H" H - H° RT

( o )

+ 0)

H - H" RT.

( i )

(4.12)

where

H - H" RT,

( o )

= A 0 + A J P J . + A 2 P R * (4.13)

H - H° RT.

( i ) = B 0 + BjP r + B 2P r = (4.14)

108

The quantities Aj and Bj are functions of T r and are given by the following:

A 0 = 5.742533 + 0.743206Tr - 3.003445Tr2

A x = 0.075271 - 0.500988Tr + 0.443336Tr2

A 2 = -0.017460 + 0.054554Tr - 0.045077Tr2

B 0 = 17.334961 - 18.851639Tr + 5.325703Tr2

Bx = 0.092967 - 0.244039Tr + 0.158373Tr2

B 2 = 0.004468 + 0.001513Tr - 0.002061Tr2

This correlation is applicable in the region where 0.5<Tr̂ 0.8 and 0.2<Tr<9.0.

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.

(4.13a)

(4.13b)

(4.13c)

(4.14a)

(4.14b)

(4.14c)

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 . 610 9 . 831 1C i.01 0 . 525--0 . 944 0 .216--0 .862 86 68.126 Ethane 10 .93 6 . 410 6 . 125 0 .519--0. .994 0 .551--0 .981 48 68,82 Propane 8 . 831 4 . 943 4 . 898 0 .622--o. .919 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 .49 3. 464 3. 374 0 .718--0. .939 0 . 548--o. .822 13 68,126 n-Pentane 10 .82 2 . 292 2 . 283 0. .624--o, ,837 0, , 297--0. . 89 1 24 68,126 n-Hexane 14 .20 2 . 951 2 . 280 0. .676--0. ,951 0. .330--o. .990 23 68,126 n-Heptane 13 .70 1 . 537 1 . 267 0, .555--0, ,926 0, .037--0, . 731 49 122 n-Octane 16 .69 3 . 793 3 . 430 0 .563--0. 949 0. . 04 1--0, .806 54 122 Benzene 9 . 761 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 . 832 5 . 991 0. .670--0. 968 0. 020--0. 786 41 122 1-Butene 8 . 895 4 . 519 4 . 643 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 .' 070 8 . 709 8 . 780 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 . 872 15 .04 14 .42 0. 563--0. 901 0. 226--o. 942 20 68,126 Argon 3 .: 978 10 i . 74 10 .85 0. 563--o. 961 0. 204--o. 919 53 68,126

t PVT, departure function, and fugacity c o e f f i c i e n t data are taken from the same source

Table 4.23 Region IV - Pressure and Temperature RMS % Errors

Compound SRK Pressure

PR LM SRK Temperature

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 n-Octane Benzene Methano1 Ethanol 1-Propanol 1-Butene

71.1 101 .4 65.0 39.3 58 .6 83.2 11.5

65.5 95. 1 34.9 46.7 58 .0 78.4 18.4

68 . 1 97 .9 40. 3 64 . 7 62.7 62.0 20.9

66 . 3 94 .6 60.8 35.8 47 .6 74 .0 9.6

65 93 35 44 53 75. 18

67.8 96 . 5 41.2 62.8 58 .6 59.0 21.8

Table 4.26 Region IV - Internal Energy Departure Function and Fugacity C o e f f i c i e n t Average Absolute Deviation

Internal Energy (cal/mole) Fugacity Coef f i c i e n t 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 . . 9 2 6 0 . 0 3 7 - - o . . 731 4 9 122

n-Octane 3 . 3 2 . 8 3 . . 8 0 . . 563-- 0 . 9 4 9 0 . 0 4 1-- 0 . . 8 0 6 54 122

Methano1 3 . 2 1 . 7 2 . 0 0 . 702- - 0 . 8 5 7 O. . 0 3 8 -- 0 . 6 2 8 27 122

Ethanol 6 . 0 4 . , 2 5 . 0 0 . 701- - 0 . 9 7 3 0 . , 0 3 3 - - 0 . 8 1 5 37 122

1-Propano1 5 . 3 3 . . 6 1 0 . 3 0 . 670- - 0 . 9 6 8 O. , 0 2 0 - - 0 . 7 8 6 41 122

1-Butene 1 . 0 1 . . 3 5 . 1 0 . 572- - 0 . 9 5 3 0 . . 0 2 5 - - 0 . 7 4 6 38 122

Water 4 . 0 2 . . 9 4 . 5 0 . 5 1 5 - - 0 . 9 6 3 0 . . 2 0 3 -- 0 . 9 0 4 1 18 44

Argon 1 . 8 1 , . 1 2 . 1 0 . 5 6 3 - - 0 . 961 0 . 2 0 4 - - 0 . 9 1 9 53 126

116

tsi

0.14

0.13

0 12

0.11

0.10 -

0.09

n-Heptane eS DaU

0.105

0.095

tS3 0.085

0.075

0.065

0

Methanol 0 Data

SRK PR LM

0 0 0

-i 1 1

0.5 0.6 0.7 0.8 0.9 1.0 Tr

0.65 0.70 0.75 0.80 0.85 0.90 Tr

Figure 4.38 Region IV : Compressibility factors of n-heptane (Pr = 0.73) and methanol (Pr = 0.63) versus reduced temperature.

0.14

0.12 -

o.io

0.08

0.06

0.10

0.08

0.06

0.04

0.02

0.00

Methanol Q Data

SRK PR LM

0.0 0.2 0.4 0.6 Pr

0.8

Figure 4.39 Region IV : Compressibility factors of n-heptane (Tr = 0.67) and methanol (Tr = 0.70) versus reduced pressure.

117

7.45

7.40

O cS 7 35

ad I

£ 7.30 I

7.25

7.20

0 0 ra 0

n-Heptane Q Data — S R K

PR LM

' i — i i

0.0 0.2 0.4

Pr 0 6 0.8

OS

I PC

9.6

9.4 es 0 0

9.2 Methanol

9.0 -0 Data

SRK PR

8.8 LM

8.6

8.4 -

8.2

8.0 • 1 • i . i i .

0.0 0.1 0.2 0.3 0.4

Pr 0.5 0.6 0.7

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

Tr 0.65 0.70 0.75 0.80 0.85 0.90

Tr

Figure 4.42 Region I V : Reduced entropy departure functions of n-heptane (Pr = 0.73) and methanol (Pr = 0.63) versus reduced temperature.

11.0

10.9

C O

I C O

Y 10 8

10.7

0 0

10.6 0.0

\ C O

I C O

V

15.5

15.0

14.5 h

14.0 h

13.5

13.0

Methanol 0 Data

SRK PR

-LM

0 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 : (Tr = 0.67) and methanol (Tr =

Reduced entropy departure functions of n-heptane 0.70) versus reduced pressure.

119

0.51

0.49

0.47 O

E— OS \ ° <

I . <

0.43 -

0 41

0.39 0.0

n-Heptanc 0 Data — SRK

PR 0 0

L M

0

0

I . I . I .

0.2 0.4 0.6 Pr

o E— OS

< I

<

1.28

1.26

1.24

1.22

1.20

1.18

-

- ro

0 0

Methanol 0 0 Data

SRK PR LM

• * • • - • - • - •

0.8 0.0 0.1 0.2 0.3 0.4 Pr

0.5 0.6 0.7

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

2.0

O E- 1.0

O I

o o.o

-1.0

-2.0

-

-

n-Heptane 0 Data — S R K

PR

1

LM

3.0

2.5

2.0 o

E-QJ \ G~* 1.5 O I

O 10

0.5

0.5 0.6 0.7 0.8 0.9 1.0 Tr

0.0

-

Methanol • Data

SRK PR

1

LM 1 , i . 1...

0.65 0.70 0.75 0.60 0.85 0.90 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.

o i

o

-0.10

-0.15

-0.20

-0.25

-0.30

0

0.65

0.60

O E-\

0.55 O I

O

0.0 0.2 0.45

0 0 0

Methanol 0 Data —SRK

PR

-I 1 L

LM _ ^ 1 . _ _ L . 1 .

0.4 Pr

0.6 0.8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 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

6.80 -

O

\ C ZD I 2 - 6.70 I

6.65

6.60

es

0 0

n—Heptane 0 Data

SRK PR LM

0.0 0.2 0.4 Pr

0.6 0.8

O

I

8.8

8.6

8.4

8.2

8.0

7.8

0 0 0 Q

Methanol • 0 Data

SRK PR LM

1 , L_ 1 I 1 I [ I 1 1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

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

n-Heptane 0.6 • Data es

SRK PR

0.5 LM

0.4

0.3

0.2

0.1

0.0 2d=L—i i i

0.5

0.4

0.3

0.2

0.1

0.5 0.6 0.7 0.8 0.9 1.0 Tr

0.0

Methanol eS Data

SRK PR LM

0.65 0.70 0.75 0.80 0.85 0.90 Tr

Figure 4.50 Region IV : Fugacity coefficients of n-heptane (Pr = 0.73) and methanol (Pr = 0.63) versus reduced temperature.

0.25

0.20

0.15

o.io V

0.05

0.7

0.5

0.4

0.3 \

0.2

0.0 0.2 0.4 Pr

0.0

Methanol - eS Data

SRK PR

\ LM

t 1 0.6 0.8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

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 = 9P (4.15) 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:

P r = -36.275 + 71 .598T r - 4 1 . 5 6 7 T r

2 + 1 1 . 8 2 6 T r

3

- 1.6721T r * + 0 .091167T r

5

( 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:

C^r+uS+w) 2 + C 2 $ ( 2 $ + u ) ( £ - l ) 2 + c 3 ($- l ) 2 ($ 2 +u$+w) = 0

(4.18)

where

C i = % T r (4.18a)

° 2 = " flaa (4.18b)

C 3 = 0.0 a* (4.18c)

(4.18d) S = V/b

The reduced inversion pressure is given by the following:

RT 1 a 1 o _ r a p r " ~ 7777 TT7 77777777 (4.19) (5-1) 0^ ($2+u$+w)

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 T r = 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 . 04 Propane 12 , . 12 1 .89 3 .61 13 .42 1 . 98 4 . '21 14 . . 53 2 .31 4 . 73 n-Butane 12 . . 24 1 .84 3 .45 13 .55 1 . ,91 3. 99 15 . ,08 2 . 26 4 . 45 Carbon Monoxide 11 .87 2 .06 4 . 14 13 . 16 2 . 18 4 . 97 13 .60 2 . 49 5 . 61 Carbon Dioxide 12 . . 33 1 .80 3 . 34 13 .65 1 . 87 3 . 83 15 .68 2 . 24 4 . 28 E thy 1ene 1 1 . .95 1 .99 3 . 93 13 . 24 2 . 10 4 . 67 13 .82 2 . 4 1 5 . 26 p-Hydrogen 1 1 . 95 3 .09 7 .39 13 .70 3, , 74 10.97 13. .61 3 .83 1 1 . 37 Ammon i a 12 . 41 1 . 77 3 . , 26 13 . 74 1 . ,84 3. 72 15 . 27 2 . 19 4 . 24 Argon 1 1 . 77 2 . 17 4 . .49 13 .07 2 . , 33 5 . 52 13 . . 33 2 . .62 6 . 20

7.0

Methane

o . o I ' i i ' 1 ' 1 i 1 1 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 S R K , PR, L M , and G C P 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 equations.

to CO

Figure 4.54 Inversion curve of n-butane calculated from the SRK, PR, and LM equations.

C O

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 LM equations.

co to

Figure 4.57 Inversion curve of ethylene calculated from the S R K , PR, L M , and w G C P equations. w

Figure 4.58 Inversion curve of p-hydrogen calculated from the S R K , PR, and L M equations.

co

Figure 4.59 Inversion curve of ammonia calculated from the S R K , PR, and L M equations.

co cn

Figure 4.60 Inversion curve of argon calculated from the SRK, PR, LM, and w GCP equations. 0 5

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

r rp s

r (5.1a)

137

138 0.331 0.423 0.008

f' 1' = 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 +

v

+ ^ + " • * ( 5 - 2 )

or

Z = 1 + B'P + CP 2 (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)

B = B'RT

C = C (RT) 2 + B 2

140

(5.4a)

(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

RT

where

!Zc = _ a ( V p c p RT RT R2T 2

C C C

where

(5.5)

a(T) = a(T c) • a ( 5 6 )

The derivation of Eq.(5.5) is shown in the Appendix A. In reduced form:

(5.7)

F = a/Tr (5.8)

141

It was shown by Shaw and Lielmezs [105] that for most cubic E O S , the

F-function can be written as a power series in inverse reduced temperature. F o r

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 C 1 } C 2 , and C 3 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 E O S gives such

poor results in the low temperature region. Below T r = 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 P R form cubic E O S :

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

RT

where

a(T) = a ( T J • 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

(5.5)

Substituting a(T) and b into Eq.(5.5) and simphfying, the reduced second virial

coefficient is

BP C a — = % ~ fla — (5.11)

Rearrange Eq.(5.11), a is then given by Eq.(5.12)

143

a J

where

(5.12)

B = r BP RT (5.13)

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 + _ + .

•V 4 T 5 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 T r 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 T f 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 V i r i a l C o e f f i c i e n t : Average Absolute Deviation (cc/mole)

Compound SRK PR LM Kub i c This Work

Tsono-polous

Deviation Tr Range Est imates

N

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 , .49 15

n-Butane 74 . 1 69 . 0 67 . 2 9.2 11.8 9.6 18.2 0 . 59-- 1 . 32 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 . 07 13

Carbon Monoxide 3.9 5.5 7.6 1 . 1 1 .5 0.5 --- 2 . 05- -3 . 18 7

Carbon Di s u l f i d e 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 . 37 8

1-Butene 13.6 23. 0 23. 6 14.7 6.6 7 . 3 0. 72- -1 . 00 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<Tr<1.0.

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 P R or modification of the L M equation. For temperature

prediction, all three equations are similar in accuracy. For derivative properties,

the modification of the L M equation gives comparable prediction to the P R

equation in the region where T r > 1 . 0 5 . The S R K and P R equation are similar in

accurac.y for enthalpy, entropy, and internal energy departure function estimations.

The S R K equation gives the lowest overall average R M S % error for the

Joule-Thomson coefficient prediction.

In the compressed liquid region, it has been found that the overall

average deviation or R M S % 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 P R

and L M equations give similar results. These two equations are more accurate

than the S R K equation. For derivative properties, predictions from the three E O S

are comparable.

For inversion curve calculation, the most reliable equation of state is the

S R K . Its estimation closely follows the experimental data right up to the

m a x i m u m inversion pressure. The predictions from all three equations are alike

up to T r = 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 T r = 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

AH V 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

AS y 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

Z j ^ 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)

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APPENDIX A

In this Appendix all the necessary constants and equations are derived

from the generalized cubic equation of state [103]. The exact equations of all the

properties considered in this work are given at the end of each section for the

three cubic E O S : the Soave-Redlich-Kwong [110], the Peng-Robinson [87], and the

proposed modification. The generalized cubic E O S is given b}' the following:

RT a(T) P =

V-b V2+ubV+wb2 (A.l)

1. CRITICAL COMPRESSIBILITY FACTOR

Multiply E q . (A . l ) by ( V - b ) (V 2 +ubV+wb 2 ) and collect similar terms:

V 3 + RT

ub - b P

V 2 + RT a

wb2 - ub 2 - ub — + -P P

RT ab wb3 + wb2 — + —

P P = 0 (A.2)

A t the critical point,

( V - v c ) 3 = 0 (A.3)

Expand E q . (A.3) algebraically,

164

V3 - 3V2V + 3VV 2 - V 3 = 0 c c c

165

(A.4)

Compare the coefficients of Eq. (A.2) and (A.4), and we have the followings:

RT ub - b

P = -3V (A..5)

RT a wb2 - ub 2 - ub — + -

P P = 3V C

2 (A. 6)

RT ab wb3 + wb2 — + —

P P = V 3

c (A. 7)

Let V c = Xb, then Eq. (A.5), (A.6) and (A.7) become (A.8), (A.9), and (A. 10),

respectively.

RT b(u - 1 + 3X) = —

P (A.8)

RT a b2(w - u - 3X2) = ub —

P P (A.9)

RT a X 3b 2 -wb2 - wb — = -

P P (A. 10)

Substitute Eq. (A. 10) into (A.9) and simplify

RT b2(w - u -3X2) = b(u + w) — + wb2 - X 3b 2

P

Substitute E q . (A.8) into (A. 11) and simplify

(w - u - 3X 2) = (u - 1 + 3X)(u + w) + w - X

Collect similar terms and get E q . (A. 13):

X3 - 3X2 - 3(u + w)X + (w - uw - u 2) = 0

Define Q& and as

a P c 0_ = £

b P c RT

A t the critical point E q . ( A . l ) becomes

p = RT^ _ a(T c)

Substituting E q . (A. 16) into (A. 14)

166

(A. 11)

(A. 12)

(A. 13)

a R 2 T 2 (A. 14) c

(A. 15)

C V-b V 2+ubV +wb2 ( A - 1 6 )

c c c

167 a(T ) (a(T ) ) 2

a RT c(V c-b) R 2T c2(V c

2+ubV c+wb 2) (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

At the critical point 3P 3V

= 0 and Eq. (A. 18) becomes

RTC a(Tc)(2Vc+ub) (V c-b) 2 (Vc

2+ubVc+wb2)2 (A. 19)

Rearrange Eq. (A. 19)

RT (V 2+ubV +wb2)2

V = a ( T - ) = (A.20) (Vc-b)2(2Vc+ub)

Replace V c with Xb and substitute Eq. (A.20) into (A. 17) and simplify

= (X2-2X-u-w) a.

(X2+uX+w)2

(X-l)*(2X+u)2 (A.21)

Similarly, can be written as

(X2-2X-u-w) Qh =

(X- l) 2(2X+u)

168

(A.22)

Divide & a with fl^

J2a

(A.23) bRT c

or

T_ = bRfi„ (A.24)

Divide C2„ with f^ 2

a

v " ^ < A-2 5 )

or

P = b 2 f l (A. 26)

B y definition, Z _ is c

P V •7 - C C Z c - — (A.27)

169

Substitute Xb for v , T c with Eq. (A.24), and P c with (A.26), Eq. (A.27)

becomes

z c = x f lb (A. 2 8)

Replace with Eq. (A.22) and get

X(X2-2X-u-w) (X-l) 2(2X+u)

1.1. The Soave-Redlich-Kwong Equation

u = 1

w = 0

X = 3.8473221...

G 0 = 0.4274802...

= 0.0866403...

Z c = 0.3333333...

1.2. The Peng-Robinson Equation

u = 2

w = -1

X = 3.9513730...

Q = 0.4572355. a

(A.29)

= 0.0777960

= 0.3074013

1.3. This Work

u = 2

w = -1

X = 3.9513730.

0 = 0.4572355 3.

= 0.0777960

Z„ = 0.3074013 c

2. CONSTANTS A AND B

Let's define A and B as

aP A =

R 2 T 2

bP B = —

RT

where

a = a(T c) • a

171

b = b ( Tc > (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

0.5 .2 a = [ 1 + m(l - T r ) ]

The constant m is

m = 0.480 + 1.574CJ - 0.176w2

2.2. The Peng-Robinson Equation

0.5 ,2 a = [ 1 + K(1 - T r ) ]

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 /T b - 1)

and for T > 1.0

a = 1 - p(T*) q

» _ (T/T, - 1) (T c/T b - 1)

The constants p and q are substance-dependent constants.

3. COMPRESSIBILITY FACTOR EQUATION

Multiply Eq. (A.2) by P 3/(R 3T 3) and let

PV Z = —

RT

Collect similar terms and Eq. (A.2) becomes

173

Z 3 + (uB - B - 1)Z2 + (A + wB2 - uB - uB2)Z

- (AB + wB2 + wB3) = 0 ^ A 3 6 ^

3.1. The Soave-Redlich-Kwong Equation

Z 3 - Z 2 + (A - B - B Z ) Z — AB = 0

3.2. The Peng-Robinson Equation

Z 3 - (1 - B ) Z 2 + (A - 3B 2 - 2B)Z - (AB - B 2 - B 3 ) = 0

3.3. This Work

Z 3 - (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 in - = - ;

P RT 0

RT V

P dP (A. 3 7)

Eq. (A. 3 7) may be rearranged to give

f 1 P lnP m - = — ; vdp - ; dinp ( A 3 8 ) 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 ( A 4 0 )

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 - 4 1 )

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

S P d V = RTln v - b

V ° - b 2b61

In V+(92b

V+0 3 b

In 2b61

v + e 2 b

v»+e3b (A.42)

where

6X* = - w (A.43)

e2 = u - - 0i 2

(A.44)

6, = u 2 (A. 45)

Combining Eq. (A.38), (A.40), and (A.42), the fugacity coefficient becomes

In - = (Z - 1) - In P V-b V°-b

In 2bRT<9i

V+f?,b

V+03b

a V°+02b P + In - In —

2bRT0x V0+e93b po (A. 46)

176

By l'Hospital's rule, the term (V°+0 2b)/( 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 In - = (Z-l) - ln(Z-B) + In

P Z+t?2B Z+03B (A. 47)

4.1. The Soave-Redlich-Kwong Equation

0a = 0.5

e2 = o.o

e 3 = 1.0

z Z+B

4.2. The Peng-Robinson Equation

f A In - = (Z-l) - ln(Z-B) + - In

P B

61 = Jl

62 = 1 - y/2

e3 = i + »/2~

f A In - = (Z-l) - ln(Z-B) + In

P 2v^B Z-0.414B Z+2.414B

177 4.3. This Work

e, = 1/2

e2 = 1 - vT

t9 3 = 1 + /2

f A In - = (Z-l) - ln(Z-B) + — = — I n

P 2\/2B

Z-0.414B Z+2.414B

5. ENTHALPY DEPARTURE FUNCTION

B y definition, enthalpy is given

H = U + PV (A. 48)

Differentiate E q . (A.48) with respect to volume at constant temperature

9(PV) " 9H ' 9U 3V T 9 V

+ T 9v (A. 49)

But

dU = TdS - PdV (A.50)

Differentiate E q . (A.50) with respect to volume at constant temperature

178

au r p

" 3S "

3v T — X

3v (A.51)

F r o m Maxwell relation

" 35 ' " 3P " 3v T 3T (A.52)

T

Combining Eq. (A.49), (A.51), and (A.52) gives

r 3H 1 3P 3(PV) — = T — - P + 3V T 3T V 3V (A.53)

T

Integrating Eq. (A.53) gives the isothermal changes in enthalpy

H2 " Hx = / 3P

T( — > 3T V

dV + A(PV) (A. 5 4)

Differentiate Eq. (A.l) with respect to temperature at constant volume

3P 3T JV V-b V2+ubV+wb2

da dT (A.55)

Substitute Eq. (A.l) and (A.55) into (A.54) and simplify

179

H2 - Hx - a - T da dT

dV V2+ubV+wb2

+ A(PV)

(A. 5 6)

Integrating Eq. (A.56) gives

H 2 - H, = a - T(da/dT)

26yb

V+02b In v+e3b

+ P 2V 2 - P 1V 1

(A. 5 7)

pressure. Eq. (A.57) becomes

H - H° = RT(Z-l) + a - T(da/dT)

2c9ib

be the ideal

Z+t?2B In

Z+t93B

gas at zero

(A.58)

5.1. The Soave-Redlich-Kwong Equation

a = a(T ) • [ 1 + m(l - T r0- 5) ] :

Differentiate a with respect to T and simplify

da dT

= - ma(Tc) a

TT C J

da , T — = - ma(T )(aT )

dT c r

180

da a - T —

dT = (1 + m)

A T Z H -- H° = = RT(Z-l) + R(l+m)

B a 0 • 5 In

Z+B

5.2. The Peng-Robinson Equation

a = a(T c) • [ 1 + K(1 - T/- 5) ] 2

Differentiate a with respect to T and simplify

da — = - K3(T ) dT c

a

TT c J

da , T — = - Ka(T_)(oT_) dT c r

da a - T —

dT = (1 + K )

H - H° = RT(Z-l) + R(l+»c) A T 2v̂ 2 B a"

In Z-0.414B Z+2.414B

5.3. This Work

For T < 1.0 r

a = a(T c) • [ 1 + p(T ) q ]

T = (Tc/T - 1) (T c/T b - 1)

Differentiate a with respect to T and simplify

da dT

qa(T c)(o-l) T r(T c-T)

da T — =

dT qT ca(T c)(g-l)

(Tc-T)

da a - T —

dT = a 1 + qTr

(T c-T)

where

4> = 1 + qT qT,

(T c-T) a(T c-T)

182

For T > 1.0 r

a = a(T c) • [ 1 - p(T ) q ]

T = (T/T c - 1) (T c/T b - 1)

Differentiate a with respect to T and simplify

where

da qa(T c)(a-l) dT (T-T )

da qa(T_)T(o-l) T — = dT (T-T ) c

da T —

dT = a 1 -cjT qT

(T-T c) a(T-T c) = a • 4>

1 -qT qT

(T-T c) a(T-T c)

The enthalpy departure function is given by the following:

H H° = RT(Z-l) + RT$ A

2y/2 B In

Z-0.414B Z+2.414B

183 6. HEAT CAPACITY DEPARTURE FUNCTION

The constant pressure heat capacity departure function is given by the following

thermodynamic relation:

(C p - Cp") = 9(H-H°)

9T (A.59) 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 p 9T P 9T JP

9 a - T(da/dT) — ( ) 9T 26^

V+02b I n

. P V+03b

or

a - T(da/dT) 2exb

9 v+e2b ( — I n

9T V+0 3 b

(A. 60)

(C p - C p«) = P 9V

9T - R + x x l n

V+c?2b V+(9,b

a - T(da/dT)

2c?xb (A.61)

184

where

dT a - T(da/dT)

26xb (A.62)

Differentiating Eq. (A.62) gives

Xx = -d 2a dT2 (A.63)

and

X 2 = ( — In 9 v

v+e2b V+f93b

9V 9T (A.64)

Differentiate Eq. (A.64) and simplify, x 2 becomes

26xb X 2 =

V2+ubV+wb2

9V 9T (A.65)

JP

Substituting Eq. (A. 63) and (A.65) into (A.61) gets

(C p - Cp") = P 9V 9T

- R -

JP 26xb

a - T(da/dT) V2+ubV+wb2

d 2a V+62b In

dT 2 V+03b

9v

9T P (A.66)

Substitute Eq. (A.l) for P and simplify. Eq. (A.66) becomes

185

(C p - C p") = RT T(da/dT) V-b V2+ubV+wb2

3V

3T

T d 2a In

V+t?2b 26xh dT 2

In V+t?3b

- R (A.67)

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 + —

3P T dT

JP (A.69)

At constant volume

9V 9V 0 = — dP + —

3P T dT (A. 70)

Rearranging Eq. (A.70) and gets

186

3v 3V 3P 3T P 3P T 3T (A. 71)

O r

3V 3T

(3P/3T)

(3P/3v)r

v (A.72)

(3P/3V) T is given by Eq. (A.18) and (3P/3T)V is given by Eq. (A.55). Let

X 3 = T(da/dT) V2+ubV+wb2

RT V-b (A. 73)

Multiply x 3 by (3P/3T)v and simplify

3P

3T R2T 2RT(da/dT) +

(V-b) 2 (V-b)(V2+ubV+wb2)

da dT (V2+ubV+wb2)2 (A. 74)

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

Numerator = x :

3P 3T JV

V2+ubV+wb2 2 da b(V-b) Rb dT

(V-b) R 2b V2+ubV+wb2

da dT (A.75)

and

Denominator = 3P 9V

V2+ubV+wb2

Rb(V-b) a(2V+ub)(V-b)

R2Tb(V2+ubV+wb2)

(A. 76)

Rewrite Eq. (A.75) and (A.76) in terms of A, B, and Z

and

3P 3T JV

Z2+uBZ+wB2

B(Z-B) 2 da Rb dT

B da Rb dT

(Z-B) B(Z2+uBZ+wB2)

(A. 7 7)

3P 3V

Z2+uBZ+wB2 A(2Z+uB) (Z-B) RB(Z-B) RB Z2+uBZ+wB2 (A. 78)

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

X»OP / 3 V ) T R(M-N)2

( 9 P / 3 V ) t M2-A(2Z+uB) (A. 79)

The final equation for (C p - C p°) is then given by the following:

R(M-N)2 T d 2a In

Z+6»2B M2-A(2Z+uB) 2 0 i b dT 2

In Z+t?3B

- R

(A.80)

Or Eq. (A.80) can be written as

C - C 0

<~p l_p ' Cp - Cy " " Cp" " Cy"

R R R R

(A. 81)

189

where

C - C 0 d 2a Z+02B In

dT2 Z+t93B (A.82)

c p c v (M-N)2

M2-A(2Z+uB) (A.83)

= 1 (A.84)

6.1. The Soave-Redlich-Kwong Equation

Differentiate Eq. (A. 3 2) twice with respect to temperature and get

d 2a dT 2

a(T c)m 2TT C°• 5

m C

M = Z(Z+B) (Z-B)

B da N =

Rb dT

C - C 0 = — d 2a dT 2

(Z+B) In +

R(M-N)2

M2-A(2Z+B) - R

6.2. The Peng-Robinson Equation

Differentiate Eq. (A.32) twice with respect to temperature and get

d 2a dT2

a ( T j K

2TT 0• 5

C

a" • 5 K

+ •̂0 .5 0 . 5

C

M = Z2+2BZ-B2

Z-B

B da N =

Rb dT

C - C 0 = 2/2 b

d 2a In

Z+2. 414B dT 2

_ z-o. 414B

R(M-N)2

M2-2A(Z+B)

6.3. This Work

Differentiate Eq. (A.32) twice with respect to temperature and get

For T r < 1.0

d 2a dT 2

(2T r-q-l) da T r(T -T) dT

and for T r > 1.0

d 2a dT 2

(q-D da (T-T ) dT

191

M = Z2+2BZ-B2

Z-B

B da N =

Rb dT

C - C 0 = UP *-p 2|/2~ b d 2a Z+2 .414B

In dT2 Z-0 .414B

R(M-N)2

+ - R M2-2A(Z+B)

7. JOULE-THOMSON COEFFICIENT

The Joule-Thomson coefficient, u, is given by the relation:

-1 M =

T(3P/3T), OP/3V).

+ V (A.85)

(3P/3V) T is given by Eq. (A.18), (3P/3T) V is given by Eq. (A.55) and (C p -

C p°) is given by Eq. (A.80)

7.1. The Soave-Redlich-Kwong Equation

R 1 da V-b V(V+b) dT

-RT a(2V+b) + (V-b) 2 V 2(V+b) 2

3P 3T

3P 3V

7.2. The Peng-Robinson Equation

ap 3T

da V-b V2+2bV-b2 dT

3P av JT

-RT 2a(V+b) + (V-b) 2 (V2+2bV-b2)2

7.3. This Work

ap 3T

JV

da V-b V2+2bV-b2 dT

ap av

-RT 2a(V+b) + (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

3T JV

RT 1 da V-b V2+ubV+wb2 dT

193

Since

Let

Therefore

or

where

a = a(T c) • a (A.32)

da da T — = a(T ) • T —

dT dT (A.88)

da <j> = T —

dT (A. 89)

3P 9T

RT V-b V2+ubV+wb2

a(T W (A. 90)

9P 9T

T r P c 1 °a pc* \ <S-l) flb

2 (52+u5+w) (A. 91)

I = V/b (A.92)

9P 9V JT

-RTV aV(2V+ub) (V-b) 2 (V2+u±»V+wb2)2 (A.93)

194

or

9 P

9V -T P r c fi P a

a c S(2$+u)

JT Ojj ($-l) 2 0^ ($2+u5+w)2 (A. 94)

Adding Eq. (A.91) and (A.94) and simplify

( o - D :

° b

$(2$+u)a - (£2+u$+w)c6 ($2+u$+w)2 (A. 95)

Rearranging Eq. (A. 95) gives the inversion curve equation

C^^+uS+w)2 + C 2$(2$+u)(S-l) 2 + C,<S-l>a(Sa+u$+w) = 0

where

(A. 96)

C 2 = - R aa

C 3 = K<t>

(A. 9 7)

(A. 98)

(A.99)

The reduced inversion pressure is given by the following:

p r = RTr 1 R aa 1

($-1) f^2 (S2+u$+w) (A. 100)

195

8.1. The Soave-Redlich-Kwong Equation

0 = - m(aT r) 0- 5

C x = 0.08664Tr

C 2 = - 0.42748 [ 1 + m(l - T r0- 5) ] :

C 3 = - 0.42748 m(aT r) 0- 5

The inversion curve is

C i S C S + l ) 2 + C 2(2$+l)($-l>* + C,<5+1)<5-1>2 = 0

The inversion pressure is

T 1 0 a 1 P r " ~

8.2. The Peng-Robinson Equation

0 = - r c(aT r) 0- 5

C x = 0.07780Tr

C 2 = - 0.45724 [ 1 + /c(l - T r«- S) ] :

C 3 = - 0.45724 * c(aT r) 0- 5

196

The inversion curve is

C 1 ( S 2+2S-1) 2 + 2C 2 S($+1 ) ( S-1) 2 + C 3($ 2+2$-l)($-l) 2 = 0

The inversion pressure is given by

r % ( S - D V S 2+2$ - l

8.3. This Work

For T r < 1.0

a-1 <S> = - q

1-T r

Cx = 0.07780Tr

C 2 = - 0.45724 [ 1 + p(T*) q ]

a-1 C 3 = - 0.45724 q 1-T

r

and for T r > 1.0

T (1-a) * = - q —

Tr-1

Cx = 0.07780Tr

197

C 2 = - 0.45724 [ 1 ~ p(T*) q ]

T r(l-a) C 3 = - 0.45724 q

Tr-1

The inversion curve is

C 1 ( S 2+2S-1) 2 + 2C 2 S ( S + D ( S-1) 2 + C3($2+2$-im-l)2 = 0

The inversion pressure is given by

T 1 0 a 1 P = _E 2

Ou (s-i) n h

2 s2+2$-i

9. VIRIAL COEFFICIENTS

The generalized cubic equation is given by Eq. (A.l):

RT V-b V2+ubV+wb2

The first term, (RT/V-b), can be written as

RT RT 1 V-b V 1-b/V

(A.l)

(A. 101)

or

198

RT RT V-b V

b b 2 b 3

1 + - + — + — + V V2 V3 (A. 102)

The second term can be written as

V2+ubV+wb2 V V+ub+(wb2/V) (A.103)

or

V2+ubV+wb2

a V

1 ub (u2-w)b2

— - — + +

V V2 V3 (A. 104)

Substract Eq. (A. 104) from (A. 102) gets

Z = 1 + b -RT

b 2 + uab RT

1 V2

b 3 + (w-u2)ab2

RT V 3 (A. 105)

The virial equation:

B C D Z = 1 + - + — + — +

V V 2 V3 (A. 106)

Therefore the second virial coefficient is

r

a B = b

RT

199

(A.107)

The third virial coefficient is

uab C = b 2 + (A.108)

RT

The fourth virial coefficient is

(w-u 2)ab 2

D = b 3 + (A. 109) RT

9.1. The Soave-Redlich-Kwong Equation

a B = b

RT

ab C = b 2 + —

RT

ab 2

D = b 3 -RT

9.2. The Peng-Robinson Equation

a B - b

RT

2ab C = b 2 + RT

5ab2

D = b 3

RT

200

9.3. This Work

a B = b

RT

2ab C = b 2 +

RT

5ab2

D = b 3

RT