prediction of gas hydrate equilibrium

Prediction of Gas Hydrate Equilibrium

by

BABAK AMIR-SARDARY

B.A.Sc., Sharif University of Technology, 2003

A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF

THE REQUIREMENTS FOR THE DEGREE OF

MASTER OF APPLIED SCIENCE

in

The Faculty of Graduate Studies

(Chemical and Biological Engineering)

THE UNIVERSITY OF BRITISH COLUMBIA

(Vancouver)

March 2012

© Babak Amir Sardary, 2012

ii

Abstract

This thesis studies the application of Statistical Association Fluid Theory (SAFT) in the

prediction of hydrate formation conditions. The main objective is to develop a robust,

reliable and purely predictive model for calculating the formation of single hydrates former

gases. The current study is based on the use of the algorithm proposed by Englezos et al.

(1991). Simplified SAFT (Fu & Sandler 1995) was employed to model the vapor and liquid

phases as well as the van der Waals-Platteew model to represent the hydrate phase.

The predictive ability of the model was investigated on single hydrate formers in the

presence of inhibitors. With this end in mind, the inhibiting effects of methanol and ethylene

glycol on methane, ethane, propane and carbon dioxide incipient hydrate forming were

studied. The calculated results were compared to the experimental data obtained from the

literature. A deviation of less than in pressure or in temperature was desired.

Additionally, the phase equilibria of water-methanol, methanol-methaen, methanol-ethane

and methanol-propane were also studied.

Excellent results were obtained from incipient hydrate calculations and the SAFT equation of

state was found to be highly capable of tackling non-ideal mixtures such as water-alcohol

and water-alcohol-hydrocarbon systems. Estimation of the SAFT pure component parameters

and the temperature range over which the SAFT parameters are estimated was found to be

crucial. To overcome this issue, several parameters were estimated over various different

temperature ranges, and the one which provided the smallest average absolute deviation was

selected.

iii

Table of Contents

Abstract ..................................................................................................................................... ii

Table of Contents ..................................................................................................................... iii

List of Tables ............................................................................................................................ v

List of Figures .......................................................................................................................... vi

Nomenclature ......................................................................................................................... viii

Acknowledgments ................................................................................................................... xi

1 Introduction ....................................................................................................................... 1

1.1 Motivation .................................................................................................................. 1

1.2 Knowledge Gap ......................................................................................................... 5

1.3 Scope of This Work ................................................................................................... 7

1.4. System of Interest ...................................................................................................... 8

2 Clathrate Hydrates ........................................................................................................... 10

2.1 Hydrate Structure ..................................................................................................... 10

2.2 Thermodynamics of Gas Hydrates ........................................................................... 13

2.2.1 Evaluation of Cell Partition Function ............................................................... 19

3 Statistical Associating Fluid Theory ............................................................................... 22

3.1 Introduction .............................................................................................................. 22

3.2. Simplified-SAFT ...................................................................................................... 24

3.2.1. Pure Components .............................................................................................. 26

3.2.2. Mixtures ............................................................................................................ 31

4 Methodology ................................................................................................................... 35

4.1 Estimating the SAFT Parameters ............................................................................. 36

4.2 Implementation of the van der Waals-Platteeuw Model into MATLAB Code ....... 37

4.3 Performing the Incipient Hydrate Formation Calculation ....................................... 40

iv

4.4 Investigating the Accuracy of the Results................................................................ 43

5 Results and Discussion .................................................................................................... 44

5.1 Estimating the Simplified-SAFT Parameters ........................................................... 44

5.2. Prediction of Vapor Liquid Equilibrium for Binary Systems .................................. 46

5.3. Evaluating the Proposed Model in Hydrate Formation Calculations ...................... 49

5.3.1. Inhibiting Effect of Ethylene Glycol ................................................................ 52

5.3.2. Inhibiting Effect of Methanol ........................................................................... 57

6 Conclusions and Recommendations ................................................................................ 61

6.1. Conclusions .............................................................................................................. 61

6.2. Recommendations .................................................................................................... 62

References............................................................................................................................... 63

Appendices: ............................................................................................................................ 70

Appendix A: Helmholtz Free Energy ................................................................................. 70

Appendix B: Driving Compressibility Factor from Helmholtz Free Energy ..................... 72

v

List of Tables

Table ‎2.1: Geometry of cages (adapted from Bagherzadeh Hosseini (2010), by permission

from the author) ...................................................................................................................... 12

Table ‎4.1: Thermodynamic reference properties for gas hydrates (Englezos et al. 1991), by

permission from the author ..................................................................................................... 38

Table ‎5.1: Simplified-SAFT parameters obtained in this work .............................................. 45

Table ‎5.2: Prediction of hydrate formation pressure .............................................................. 50

Table ‎5.3: Required parameters for the van der Waals-Platteew model selected for this work

................................................................................................................................................ 54

vi

List of Figures

Figure ‎1.1: A gas hydrate block from 1200 metres under water (source:

http://commons.wikimedia.org/wiki/File:Gashydrat_mit_Struktur.jpg) .................................. 3

Figure ‎2.1: Gas hydrate structure (source:

http://commons.wikimedia.org/wiki/File:Clathrate_hydrate_cages.jpg) ............................... 11

Figure ‎2.2: Comparison between different potential models.................................................. 20

Figure ‎3.1: Procedure for forming a molecule in SAFT (adapted from Al-Saifi (2012), by

permission from the author) ................................................................................................... 25

Figure ‎3.2: Molecular shape in SAFT (adapted from Al-Saifi (2012), by permission from the

author) ..................................................................................................................................... 25

Figure ‎4.1: Computational flow diagram (P. Englezos et al. 1991), by permission from the

author ...................................................................................................................................... 42

Figure ‎5.1: Predicted results of vapor-liquid equilibrium ( =0) by SSAFT for the methanol

(1)/ water (2) system at , , , and ..................................................... 46

Figure ‎5.2: Predicted results of vapor-liquid equilibrium ( =0) by SSAFT for the methane

(1)/ methanol (2) system at and

.................................................................................... 48

Figure ‎5.3: Predicted results of vapor-liquid equilibrium ( =0) by SSAFT for the ethane

(1)/ methanol (2) system at and

.................................................................................... 48

Figure ‎5.4: Hydrate formation prediction ( =0) by SSAFT for methane hydrate in presence

of ethylene glycol aqueous solution, , , and ...................... 52

vii

Figure ‎5.5: Hydrate formation prediction ( =0) by SSAFT for ethane hydrate in presence


Figure ‎5.6: Hydrate formation prediction ( =0) by SSAFT for propane hydrate in presence


Figure ‎5.7: Hydrate formation prediction ( =0) by SSAFT for carbon dioxide hydrate in

presence of ethylene glycol aqueous solution, , , and ....... 56

Figure ‎5.8: Hydrate formation prediction ( =0) by SSAFT for methane hydrate in presence

of methanol aqueous solution, , , , , and

................................................................................................................................................ 58

Figure ‎5.9: Hydrate formation prediction ( =0) by SSAFT for ethane hydrate in presence of

methanol aqueous solution, , , , , and .. 58

Figure ‎5.10: Hydrate formation prediction ( =0) by SSAFT for propane hydrate in presence

of methanol aqueous solution, , , and ............................ 59

Figure ‎5.11: Hydrate formation prediction ( =0) by SSAFT for carbon dioxide hydrate in

presence of methanol aqueous solution, , , and ................ 60

viii

Nomenclature

Molar Helmholtz free energy per mole of molecules

Molar Helmholtz free energy per mole of segments

Temperature-dependent segment diameter,

Number of segments

Number of association sites on the molecules

Avogadro’s number

⁄ Temperature-dependent dispersion energy of interaction between

segments,

⁄ Temperature-independent dispersion energy of interaction between

segments,

Total volume

Temperature-dependent segment volume, ⁄

Temperature-independent segment volume, ⁄

Mole fraction

Monomer mole fraction

Compressibility factor

Langmuir constant, 1/MPa

Fugacity, MPa

Enthalpy, J/mole

Boltzman’s‎constant, J/K

Number of components

Number of hydrate forming components

Pressure, MPa

Radial distance from centre of hydrate cavity, m

Universal gas constant, J/mole K

Type m spherical cavity radius, m

Temperature, K

Molar volume, /mole

ix

Cell potential function, J

Greek Letters

Volume of interaction between site and

Strength of interaction between site and ,

⁄ Association energy of interaction between site and ,

Pure component reduced density

Molar density, ⁄

Polynomial defined by equation 7

Chemical potential, J/mole

Number of cavities type m per water molecule in hydrate

Lenard-Jones segment diameter,

Subscripts

Component i

Component j

Type of cavity

Water

Superscripts

Residual

Segment

Associating

Hard-sphere

Ideal gas

Association sites

Hydrate

Liquid

Pure liquid water

x

Empty lattice

Reference conditions, 273.15 K and zero absolute pressure

Vapour

xi

Acknowledgments

I would like to thank my supervisor Professor Peter Englezos for providing me with

valuable supervision and continuous guidance and encouragement. Thank you also for the

wonderful time I had working in your research group.

I owe particular thanks to Dr. Nayef Al-Saifi for his unwavering support, valuable

input, and immense assistance in the process of my research. Thank you for your guidance

during the thesis course - it could have not been done without your backing.

I also want to thank my colleagues Alireza Bagherzadeh, Negar Mirvakili, Nagu

Daraboina, Iwan Townson and Sima Motiee for their friendship and encouragement

throughout this project.

I would like to express heartfelt thanks to my wife Sheida Sharifi for her

understanding, consideration, and companionship, as well as her consistent help and

encouragement during the course of this Master’s Program. Thank you for being there when

I needed you most. Finally, I offer sincere gratitude to my family, particularly my mother

Shahin Goodarzi, my sisters Anahita and Mandana and my parents-in-law Hassan and

Nadereh for their unconditional support and love.

http://www.linkedin.com/profile/view?id=129630714&authType=name&authToken=Ql8u&goback=%2Econ

xii

Dedicated to my parents

and

to my wife

1

1 Introduction

1.1 Motivation

Clathrate hydrates are non-stoichiometric crystalline compounds that consist of a

hydrogen-bonded network of water and encaged molecules. Davy (1811) first observed

clathrate hydrate while he was working on mixing chlorine with water. However, clathrate

hydrates were not extensively studied until Hammerschmidt (1934) found that the natural gas

pipelines could be blocked by the formation of gas hydrates. This observation raised a great

deal of attention in the oil and gas industry, prompting increasing research on the gas

hydrates of natural gas. Ever since, the hydrate formation condition and its prevention have

been of special interest to those creating chemical technology, especially in the natural gas

industry, with the overall aim of avoiding it from plugging gas pipelines.

In the early 1970s, Russian researchers reported natural gas trapped in hydrate form

in northern Russia (Makogon 1981). Following these findings, large amounts of hydrates,

mostly methane, were also discovered below the seafloor and in regions of permafrost such

as in northern Alaska, Siberia and Canada (Haq 1999; Max 2003). This amount of methane

has been estimated to be in an order of magnitude greater than the methane in all known

reservoirs all around the world (Collett 2002). It has been proposed that this gas be used as

an energy resource (Yamazaki 1997).

2

Recent discoveries show that hydrates are connected with environmental concerns.

Gas hydrates may play a crucial role with regard to global warming. Gas hydrates may

increase global warming. The increase in temperature of the planet’s‎ surface layer, which

results from increasing amounts of greenhouse gases such as methane, might cause hydrate

decomposition, consequently releasing methane into the atmosphere (Englezos, 1993).

Studies of gas hydrate thermodynamics have concentrated on measuring the pressure-

temperature conditions at which all existing phases - water, ice, solid hydrate, vapour and/or

liquid hydrocarbon - are in equilibrium. Attempts to accurately predict the equilibrium

conditions for this multi-phase system depend on the thermodynamic model which describes

each phase. Through the use of x-ray analysis in the late 1940s and early 1950s (Stackelberg

1949; Claussen 1951; Pauling & Marsh 1952), van der Waals and Platteew (1959) gained an

understanding of the structure of gas hydrate, and thus were able to derive statistical

thermodynamic equations for gas hydrates. In their work, an expression for the chemical

potential of water in any hydrate structure was developed using an approach analogous to

Langmuir adsorption. This model was later used to represent hydrate phase behavior. Some

authors have made slight changes to the van der Waals and Platteew model by, for instance,

applying modifications in the Langmuir constant approximation (Klauda & Sandler 2000;

Klauda & Sandler 2002; John & Holder 1982; John & Holder 1985; Sparks et al. 1999) or

through modifying the original assumptions1 (Ballard & Sloan Jr 2002a; Ballard & Sloan Jr

2002b; Jager et al. 2003; Ballard & Sloan, et al. 2004; Ballard & Sloan, et al. 2004). Others,

meanwhile have adapted the original model (Parrish & Prausnitz 1972; Englezos et al.

1991).

1 These assumptions will be covered in Chapter 2.

3

Subsequent to the work of van der Waals and Platteuw, many studies were conducted

to develop a predictive model adopting their model to represent the hydrate phase and an

empirical or semi-empirical model for the fluid phases. Parrish and Prausnitz (1972)

developed an iterative scheme, using the Redlish-Kwong (RK) (Redlich & Kwong 1949)

equation for the vapour phase. They used the Krichevsky and Kasarnovsky equation and

Morrison’s‎ method‎ to‎ estimate‎ gas‎ solubilities‎ during‎ the‎ liquid‎ phase.‎ Anderson‎ and‎

Prausnitz (1986) used the Redlish-Kwong equation (Redlich & Kwong 1949) for the vapor

phase and the UNIQUAC model for the liquid one. Du & Guo (1990) modeled the inhibiting

effect of methanol on the formation of gas hydrates. In their study, both the vapour and liquid

phases were modeled based on the Peng-Robinson equation of state (PR) (Peng & Robinson

1976) and‎on‎Mollerup’s‎random‎and‎non-random (PNR) theory (Mollerup 1983).

Figure ‎1.1: A gas hydrate block from 1200 metres under water (source:

http://commons.wikimedia.org/wiki/File:Gashydrat_mit_Struktur.jpg)

4

Englezos et al. (1991) proposed an algorithm for calculating incipient hydrate

formation conditions in the presence of inhibitors like methanol. They used the Trebble-

Bishnoi equation of state (Trebble & Bishnoi 1988; 1987) for the both liquid and the vapour

phases. Englezos et al. (1991) showed that with the aid of an accurate model for the vapor

and liquid phases, one could obtain a good agreement with regard to the hydrate formation

calculation‎using‎van‎der‎Waals‎and‎Platteew’s‎statistical‎thermodynamic‎model.

5

1.2 Knowledge Gap

Due to the regular structure of gas-hydrate, the van der Waals-Platteeuw statistical

thermodynamic model has been used to quantitatively represent gas hydrate thermodynamic

properties. As disscused in last section, the next essential action in developing a predictive

model for the incipient hydrate formation conditions involves utilizing a reliable model to

represent the fluid phases. Many empirical and semi emprical equation of states and activity

coefficent model such as Soave, Peng-Robinson, Redlish-Kowang, Trebble-Bishhoi,

UNIFAQ and UNIQUAQ have been used in the literature.

Despite their widespread use in industry and in the scientific community, empirical

and semi-empirical models have certain limitations which must be taken into account. For

instance, the accuracy of these equations requires a large experimental database over the

entire P-T range for which the model is intended to be used. Generally, it cannot be safely

extrapolated. Furthermore, because of the applied simplifications, it is often impossible to

reproduce experimental data with the required accuracy for these models. Moreover, these

models do not take association interactions into account, and hence fail to predict the fluid

properties of polar and hydrogen bounded fluids (Churakov & Gottschalk 2003). Therefore,

empirical and semi-empirical equations may fail to accurately calculate the dissociation

pressure of hydrates, particularly in the presence of inhibitors such as methanol and ethylene

glycol.

Molecular-based equations however, are more reliable for tackling difficult systems

of associative fluids. One of the most successful molecular-based models is the Statistical

6

Association Fluid Theory (SAFT) which has the ability to deal successfully with associative

fluids. However, it is necessary to adjust experimental data to the SAFT in order to obtain

good results for associating mixtures. In order to render the SAFT predictive for such

mixtures, Al-Saifi et al. (2008) exploited the fact that the association phenomena are result of

both hydrogen bonding as well as dipole-dipole interactions which were not considered in

the original SAFT. Thus, along with hydrogen bonding interactions, they incorporated the

dipole-dipole interactions into the SAFT. They demonstrated that their model was capable of

predicting several water-alcohol-hydrocarbon systems. Using this approach, the required

parameters were estimated solely according to pure component data.

7

1.3 Scope of This Work

The excellent results of the work of Al-Saifi et al. (2011) motivated us to employ a

new approach for calculating hydrate formation conditions in order to overcome the non-

ideal nature of the mixture and to avoid using binary interaction parameters. Based on the

algorithm presented by Englezos et al., and adopting the work of Al-Saifi et al., the new

approach is able to accurately predict the incipient hydrate formation conditions, and binary

data are not needed for this prediction. In other words, SAFT is the model used for the

vapour and liquid phases, and the solid hydrate phase is represented by the van der Waals-

Platteew model. Specifically, this work focuses on evaluating the SAFT model, as improved

by Al-Saifi et al., with regard to the conditions of incipient hydrate formation. Single

component hydrate former gases in the presence of inhibitors such as methanol and ethylene

glycol, are examined in the present work, and multi-component mixtures will be left for

future studies.

8

1.4. System of Interest

The study of gas hydrates has attracted the attention of the natural gas transportation

industry for nearly a century. Hydrates are considered an inconvenience due to pipelines

blocking, foul heat exchangers and plug columns and expanders valves (Sloan Jr. 1991). The

most common approach used in the natural gas industry to prevent the unwanted effects of

gas hydrates is to use a variety of thermodynamic inhibition techniques (Englezos et al.

1991). These techniques provide thermodynamically unstable conditions for hydrate

formation through the introduction of a less-structured water molecule organization which

results from inhibitor-water and inhibitor-hydrocarbon interactions (Englezos et al. 1991).

Natural gas consists of a flammable mixture of light hydrocarbon gases. The

composition of a natural gas may vary according to its components. However, it is made up

primarily of methane (70-90%), ethane and propane although it might also contain iso-

butane, normal butane (typical natural gas may contain 0–20% ethane, propane, normal

butane and iso-butane), iso-pentane, normal pentane and carbon dioxide (0-8%).

This study focuses primarily on systems that contain single hydrate former gases in

the presence of an inhibitor. These types of systems were selected for particular attention

because of their industrial importance as well as because of the complexity of their phase

behavior. Water-alcohol-hydrocarbon mixtures present behaviours that are far from ideal.

Additionally, some interactions, such as that of hydrogen bonding and polar interactions are

difficult to describe. The literature has not, to this date, revealed any thermodynamic model

9

that is able to provide accurate phase equilibrium of these mixtures unless the models are

correlated to experimental data.

10

2 Clathrate Hydrates

2.1 Hydrate Structure

Depending on the size of the gas molecules, natural gas hydrates are categorized

according to three basic structure classes: structure ( ) tac elberg‎ ‎ M ller‎ ,

structure ( ) (Claussen 1951), and structure ( ) (Ripmeester et al. 1987). The

common building block of each of these structures is a 12-sided pentagonal-faced

polyhedral, pentagonal dodecahedron ( ). The role of guest molecules is to stabilize the

cages which are held by the hydrogen-oxygen bonds and to prevent them from collapsing.

This cage accommodates small molecules. Depending on the guest gas molecules and

ultimately the hydrate structure, more complex cages might be present. For the sI hydrates,

tetrakaidecahedron cages that have 12 pentagonal and 2 hexagonal faces ( )

accommodate the guest molecules. For sII hydrates, hexakaidecahedron cages are formed; 12

pentagonal and 4 hexagonal faces ( ). For sH hydrates, two new cages are formed, using

the previous nomenclature for a cage, and .

Table 2.1, summarizes the number and types of cages, as well as the number of water

molecules for each structure. Under normal conditions, only one molecule can occupy each

cavity (Sloan & Koh 2007). It is obvious that size of the guest molecule should be at least

equal to, or less than, the‎cavity’s‎diameter.‎For instance, , , and form ,

and and form . As opposed to the two other structures, needs larger molecules,

like methyl cyclohexane, to fill its larger cavity and smaller molecules, such as or ,

11

to play a helping role and to occupy the medium and/or small cavities so as to stabilize the

structure and prevent it from collapsing.

Figure ‎2.1: Gas hydrate structure (source:

http://commons.wikimedia.org/wiki/File:Clathrate_hydrate_cages.jpg)

Gas hydrates are formed when a gas mixture is brought into contact with water,

generally at low temperatures and at high pressure. Studies of gas hydrate thermodynamics

have concentrated upon measuring the pressure-temperature conditions at which all existing

phases - water, ice, solid hydrate, vapour and/or liquid hydrocarbon - are in equilibrium.

Attempts at accurately predicting the equilibrium conditions for this multi-phase system are

dependent on the thermodynamic model which describes each phase.

12

Table ‎2.1: Geometry of cages (adapted from Bagherzadeh Hosseini (2010), by permission from the

author)

Hydrate crystal

structure I II H

Cavity Small Large Small Large Small Medium Large

Description

Shape

Number of

cavities/unit cell 2 6 16 8 3 2 1

Average cavity

radius (Å) 3.95 4.33 3.91 4.73 3.94 4.04 5.79

Number of water

molecules/cavity 20 24 20 28 20 20 36

Repeating Unit

13

2.2 Thermodynamics of Gas Hydrates

The statistical thermodynamic equations for gas hydrates derived by van der Waals

and Platteew are based on six assumptions:

I. The contribution of the host molecules (water) to the free energy is independent of

the mode of occupation of the cavities.

II. The encaged molecules (solute) are localized in the cavities, and a cavity never holds

more than one guest

III. The interaction of the solute molecules is neglected

IV. Classical statistics are valid

V. The solute molecules can rotate freely in their cages

VI. Based on x-ray analysis, the potential energy of a solute molecule is given by the

spherically symmetrical potential proposed by Lennard-Jones and Devonshire

As did van der Waals and Platteew, the system in this work is defined as a clathrate

crystal containing molecules of Q in equilibrium with the solutes , at

temperature T and occupying volume V. In this system,

‎2-1

In the equation 2.1, stands for the absolute activity of solute J.

The independent variables in our system are as follow:

14

‎2-2

Before we describe our system with a generalized partition function, we need to

define an ordinary partition function. If the cavities of type contain molecules of each

species , the ordinary partition function PF would be (Waals 1956):

(

)∏[

∏

]

‎2-3

in which, is the number of cavities,

‎2-4

and is the number of guest molecules occupying cavities,

∑

(∑

) ∏

‎2-5

‎2-6

15

In ‎2-3, is the free energy of the empty lattice for the system described by . The

combination2 part computes the number of distinct ways that solute (guest) molecules may

occupy the host cavities, and is the partition function of encaged molecule when

trapped in cavities of type . By replacing 2.5 and 2.6 into 2.3 and multiply it by the product

of absolute activities, we may obtain,

∏∏

‎2-7

And by summing the result over all possible values of , we obtain the grand partition

function ,

(

)∑∏[

( )

( ∑ ) ∏ ∏

]

‎2-8

Utilizing the multinomial3 theorem, we may express the above equation as,

2 This combination is a way of selecting several things out of a larger group and can be described mathematically as follows:

( )

The above equation computes k-combinations of a set that has n elements.

3 For any positive integer and any nonnegative integer ,

∑ (

) ∏

16

(

)∏( ∑

)

‎2-9

Now that we have the grand partition function of our system, we can obtain other properties

from that partition function 4 . The grand partition function is connected to other

thermodynamic properties,

‎2-10

From classical thermodynamics5, we have,

∑

‎2-11

combining 2.10 and 2.11,

∑

‎2-12

now by replacing 2.1 into 1.12 one may get,

4 For more information about Partition Function, the readers are referred to “Equilibrium Statistical

Mechanics” (Jackson 2000). 5 See Appendix A.

17

⁄ ∑

‎2-13

From equation 2.13 we can simply obtain the composition of the clathrate hydrate,

∑

(

)

∑

∑

‎2-14

therefore,

∑

∑

‎2-15

‎2-16

In ‎2-16, is the fraction of type cavities occupied by guest molecules . The chemical

potential of the host molecules , is immediately obtained from 2.9 and 2.13,

(

)

∑ ( ∑

)

‎2-17

18

, chemical potential of solvent molecules at empty lattice, comes directly from the

definition of chemical potential which is:

(

)

‎2-18

On the other hand, we have,

‎2-19

In the above equation, is the fugacity and is the molecular partition function of

solute vapour, . Now, by replacing ‎2-19 into ‎2-16 and ‎2-17 and defining a new

parameter, , as:

‎2-20

the following important equations are obtained:

19

∑

∑ ( ∑

)

‎2-21

‎2-22

By constructing these parameters, the thermodynamic behaviour of clathrate might be

predicted relative to empty lattice.

2.2.1 Evaluation of Cell Partition Function

Based on assumptions V and VI, van der Waals and Platteew showed the following

expression for the Langmuir constant ,

∫ (

)

‎2-23

In their work, the Lennard-Jones 12-6 potential was used to study the force field in the

cavity. Even though this model is suitable for monoatomic gases, it fails in the case of non-

spherical molecules like and . To overcome this problem, McKoy and inano lu

20

(1963) utilized the Kihara potential with a spherical core 6 in which the molecules are

assumed to have impenetrable (hard) cores surrounded by penetrable (soft) electron clouds

(Prausnitz et al. 1998). They summed all the guest-host interactions in the cell and obtained

the spherically symmetric cell potential as follows,

Figure ‎2.2: Comparison between different potential models

6

{

[(

)

(

)

]

-7

-5

-3

-1

1

3

5

7

0 1 2 3 4 5

SW LJ 12-6 Kihara HS LJ 28-7

21

[

(

)

(

)]

‎2-24

Where

[(

)

(

)

] ‎2-25

Comparing three different potential models - the Lennard-Jones 12-6 Potential, the Kihara

Potential and the Lennard-Jones 28-7 Potential - in order to predict hydrate dissociation

pressure, McKoy and inano lu concluded that while Lennard-Jones 12-6 Potential might be

satisfactory for hydrates of monoatomic molecules such as , and spherical molecules

like , the Kihara Potential Model is more suitable for nonspherical molecules like ,

and .

22

3 Statistical Associating Fluid Theory

3.1 Introduction

The Association Fluid Theory was first developed by Wertheim (1984a; 1984b;

1986a; 1986b). According to Prausnitz et al. (1998), Wertheim’s‎ idea‎ was‎ “brilliant but

almost incomprehensible.”‎ Wertheim demonstrated that Helmholtz free energy can be

expressed as series of integrals obtained from cluster expansion. Based on physical

arguments, Wertheim showed that many of the integrals were zero. He then used

Perturbation Theory to solve these integrals, which therefore can be simplified and truncated.

The essential result of this theory involves an expression for Helmholtz free energy which

accounts for the effect of intermolecular association and/or solvation forces, for instance,

hydrogen bonding, and the effect of molecular shapes in addition to the effects of the

repulsion and dispersion forces. Based‎ on‎ the‎Wertheim’s‎ theory,‎ Chapman‎ et‎ al. (1989;

1990) developed an equation of state model for associating fluids and named it the Statistical

Associating Fluid Theory (SAFT). The concept of the SAFT equation of states, and its ability

to deal with highly associated fluids like water and alcohols, attracted the attention of many

researchers. It was not long before a number of SAFT equation of states were developed,

e.g., CK-SAFT (Huang & Radosz 1990; 1991), PC-SAFT (Gross & Sadowski 2001; 2002)

and simplified SAFT (Fu & Sandler 1995). In this chapter, a brief description of the

simplified SAFT (SSAFT), which is utilized throughout this thesis, will be given. Readers

are referred to the work of Al-Saifi (2012) for more information about Statistical

23

Association Fluid Theory and to view a complete comparison between the different versions

of SAFT.

24

3.2. Simplified-SAFT

According to SAFT, the molecules are assembled from a chain of hard sphere

segments. The number of segments, , is not necessarily an integer due to the fact that it is

determined by fitting the SAFT equation of state into the vapor pressure and liquid density

data (Al-Saifi 2012). It is assumed that these segments are attached by covalent-like bonds.

The procedure for forming a molecule of a pure fluid in SAFT is illustrated in Figure 3.1.

Initially, the fluid is composed of hard sphere segments. Then, attractive forces are added to

these segments. A proper potential model like Lennard-Jones or Square-Well may be utilized

at this stage. In the next step, the chain sites are added to each segment and the chain

molecules appear by joining the segments through their defined chain sites. Finally, the

association sites are added to the associating compounds, for instance water and alcohols,

and molecules form association complexes. Each of these four steps contributes to the

residual Helmholtz free energy,

‎3-1

25

To replace the complicated dispersion term in earlier versions of SAFT, Fu and

Sandler (1995) proposed a new simpler dispersion term based on an attraction term

developed originally by Lee et al.(1985). The new SAFT version, therefore, was named

simplified SAFT (SSAFT).

1 2 3 m

Figure 3.1: Procedure for forming a molecule in SAFT (adapted from Al-Saifi (2012), by permission from

the author)

Figure 3.2: Molecular shape in SAFT (adapted from Al-Saifi (2012), by permission from the

author)

26

3.2.1. Pure Components

Based on Carnahan‎and‎ tarling’s‎(1969) expression, the Helmholtz free energy for

one mole of hard-sphere fluid,

, is defined (Fu & Sandler 1995) as:

‎3-2

Therefore, the hard-sphere Helmholtz free energy for one mole of pure fluid consisting of

molecules with segments is,

‎3-3

in above equation, is the reduced fluid density and can be expressed as,

‎3-4

where is the molar density and is the effective temperature-dependent segment diameter .

could also be expressed based on segment molar volume in a close-packed7 arrangement,

,

‎3-5

7 Volume occupied by closely packed segments (Huang & Radosz 1990)

27

combining ‎3-4 and ‎3-5, one may obtain,

(

)

‎3-6

Because of it is clear that is temperature-dependent. As a result, it might be

useful to introduce the temperature-independent segment molar volume analogous to

(Huang & Radosz 1990),

(

)

‎3-7

where is the segment temperature-independent diameter. The temperature-independent

parameters in the above equations are related to the following temperature-dependent

parameters (Huang & Radosz 1990),

[ (

)]

[ (

)]

‎3-8

‎3-9

in equation‎3-8 and ‎3-9, is a parameter which is set at and is the temperature-

independent square-well depth.

28

For one mole of pure fluid, the dispersion Helmholtz free energy is (Fu & Sandler

1995),

‎3-10

with

(

)

‎3-11

is the maximum coordination number which is 36, and is the molar volume of the

segment,

‎3-12

In equation ‎3-11,

(

)

‎3-13

where is the parameter that describes the segment-segment interactions, and is the

temperature-dependent depth of the square-well potential,

29

[ (

)]

‎3-14

In above equation, ⁄ is equal to for all the molecules (Fu & Sandler 1995).

The chain formation contribution and the association contribution to the residual

Helmholtz free energy, respectively, are (Fu & Sandler 1995),

‎3-15

∑ [(

)

]

‎3-16

is the number of association sites, and is the mole fraction of unbounded molecules

which is,

{ ∑

[ (

) ] }

‎3-17

and are the association volume and association energy specified to the interaction

between two energy sites, and .

30

Obtaining all the compartments of the residual Helmholtz free energy, the SAFT

equation of states are good to go by driving the compressibility factor from the volume

derivative of the Helmholtz free energy8,

‎3-18

where,

[

]

‎3-19

(

)

‎3-20

‎3-21

∑[

]

‎3-22

8 See Appendix B

31

3.2.2. Mixtures

The procedure that we considered for the pure components is also followed for the

mixtures. We shall start with equation 3-1, as before. Based on Mansoori’s‎ 7 results,

and due to the fact that in developing the hard-sphere compartment of equation ‎3-1, it has

been assumed that the hard-spheres are not bonded, the expression of the Helmholtz free

energy for the mixture of hard-sphere might be as follows (Fu & Sandler 1995),

[

[

] ]

‎3-23

with

∑

‎3-24

In the above equation, is the molecule density, is the mole fraction, is the number of

segments per molecule and is the temperature-dependent diameter of the segment.

The accounted contribution for the dispersion interaction may be obtained by

extending equation ‎3-11 for the mixtures (Fu & Sandler 1995),

(

⟨ ⟩)

‎3-25

32

is the molar volume defined in equation ‎3-12 and is expressed by the following mixing

rules,

∑

‎3-26

and

⟨ ⟩

∑ ∑ (

√ ) [ (

) ]

∑ ∑

‎3-27

where

( )√

‎3-28

‎3-29

The contribution of chain Helmholtz free energy is shown in equation ‎3-30.

, the pair correlation function, is derived from the work of Mansoori et al. (1971)

for molecules with same size segments.

33

∑ ( )

‎3-30

where

[

]

‎3-31

Finally, the association term derived by Chapman et al. (1990) can be utilized to

describe the association contribution to the residual Helmholtz free energy in a mixture (Fu

& Sandler 1995),

∑ [∑[

]

]

‎3-32

with the mole fraction of unbounded molecules, ,given by,

[ ∑∑

[ (

) ]

]

‎3-33

34

Again, the compressibility factor may be obtained from the volume derivative of the

Helmholtz free energy,

[

]

‎3-34

(⟨ ⟩

⟨ ⟩)

‎3-35

∑ (

)

‎3-36

∑ [∑(

)

]

‎3-37

35

4 Methodology

To accomplish the objectives of this work the following approach was followed in

current study:

1. Parameter estimation was performed to estimate the required SAFT parameters

from pure component vapour pressure and density data

2. A code in MATLAB was prepared to employ the van der Waals-Platteew

statistical thermodynamic model to calculate the fugacity of solid hydrate phases

3. The fugacity of each component in the aqueous and the gaseous phases was

calculated by using the SAFT in-house program developed by Al-Saifi in Gas

Hydrates Group at University of British Columbia

4. The incipient hydrate formation pressure at a given temperature was then

calculated by following the algorithm proposed by Englezos et al. (1991)

5. The calculated equilibrium hydrate formation pressures (predictions) were

compared with the experimental data. It was examined whether the predictions

are within (pressure) or (temperature) when compared to experimental

data.

36

4.1 Estimating the SAFT Parameters

The SAFT equation of state requires the following six adjustable parameters: the

segment number ( ), the segment diameter ( ), the segment dispersion energy ( ⁄ ), the

energy of association ( ⁄ ), the volume of association ( ) and the fraction of the dipolar

segment in a molecule ( ). Optimum parameter values are obtained by fitting the dipolar

SAFT to the pure component vapour pressure and liquid density data. The parameters are

optimized based solely on pure component data and the following objective function :

∑

∑

‎4-1

37

4.2 Implementation of the van der Waals-Platteeuw Model into MATLAB

Code

Based on the gas hydrates statistical thermodynamic model derived by van der Waals

and Platteeuw, the fugacity of water in the hydrate phase can be described as follows

(Englezos et al. 1991):

(

)

‎4-2

In the above equation,

(

)

‎4-3

where,

∫

‎4-4

is defined as the difference between , a property of water, in state a and in state b.

In equation ‎4-2,

, the difference between the chemical potential of water in an empty

lattice ( ) and in a hydrate lattice ( ), is formulated as follows:

38

∑ ( ∑

)

‎4-5

Table ‎4.1: Thermodynamic reference properties for gas hydrates (Englezos et al. 1991), by permission

from the author

Property Structure I Structure II References

⁄

1235 10 (Holder et al. 1980)

1297 937 (Dharmawardhana et al. 1980)

1264 883 (Parrish & Prausnitz 1972)

1299.5 10 (Holder, Malekar & Sloan 1984a)

1287 1068 (Handa & Tse 1986)

⁄

-4327 (Ng & Robinson 1985)

-4622 -4986 (Dharmawardhana et al. 1980)

-4860 -5203.5 (Parrish & Prausnitz 1972)

-4150 (Holder, Malekar & Sloan 1984b)

-5080 -5247 (Handa & Tse 1986)

⁄

-38.13 -38.13 (Parrish & Prausnitz 1972)

-34.583 -36.8607 (Holder & John 1983; John et al. 1985)

⁄

0.141 0.141 (Parrish & Prausnitz 1972)

0.189 0.1809 (Holder & John 1983; John et al. 1985)

39

where, the Langmuir constant, , account for gas-hydrate interactions and are obtained

from the following equation:

∫ (

)

‎4-6

where,

[

(

)

(

)]

‎4-7

where,

[(

)

(

)

]

‎4-8

The required hydrate structure parameters were taken from the literature and are summarized

in Table 4.1.

40

4.3 Performing the Incipient Hydrate Formation Calculation

In a mixture at phase equilibrium and at a constant temperature and pressure, the

fugacity of each component is equal in all coexisting phases. Regarding the question of

interest in this work, the phase equilibrium of a system containing solid hydrate (H), vapour

(V) and liquid (L) may be represented using the following criteria:

‎4-9

‎4-10

where, is the number of components and is the number of hydrate forming components,

including water. The SAFT model improved by Al-Saifi et al. (2008) was used to compute

the fugacities of substances in both the liquid and vapour phases.

In order to calculate the incipient hydrate formation pressure (or temperature) for a

given mixture containing hydrate, vapour and liquid phases at a constant temperature (or

pressure), the flash calculations are performed at the assumed pressure (or temperature).

First, under these conditions, the fugacity of water in the hydrate phase is calculated using

equation 4-2. This fugacity is then compared to the fugacity of water in the liquid phase

which is computed by flash calculations. If these two fugacities are equal, the assumed

pressure (or temperature) is the hydrate formation pressure (or temperature) (Englezos et al.

41

1991). The computation scheme is designed as shown in Figure 4.1. It should be noted that

the fugacities of hydrate formation gases in the vapour phase, , are incorporated in

calculating the fugacity of water in the hydrate phase by using equation ‎4-5.

42

Start

Enter T (or P), feed composition and initial guess for P (or T)

Perform TP flash

Incipient hydrate formation P (or T)

Stop

Update P or (1/T)H

wf

TOLf

fL

w

H

w

2

ln

Figure ‎4.1: Computational flow diagram (P. Englezos et al. 1991), by permission from the author

43

4.4 Investigating the Accuracy of the Results

After the completion of the programing and the procurement of the calculated data, a

comparison of the experimental data is required. Since single component hydrate former

gases in the presence of inhibitors such as methanol and ethylene glycol are studied in the

present work, our choice of system are based on hydrocarbon-water-alcohol systems. The

acceptable deviation is less than 10% in pressure or 1 in temperature.

44

5 Results and Discussion

5.1 Estimating the Simplified-SAFT Parameters

Three adjustable parameters are required for all fluids in the simplified-SAFT

equation of state-. These parameters are the segment number ( ), the segment diameter ( )

and the segment dispersion energy ( ⁄ ). Associating fluids such as water and alcohols

require two additional parameters; the energy of association ( ⁄ ), and the volume of

association ( .‎According‎to‎Jog‎and‎Chapman’s‎dipolar‎term, it is necessary to introduce

an extra adjustable parameter, the fraction of dipolar segment in a molecule ( ), when

dipolar interactions are included in SSAFT.

The required parameters for water and methanol were obtained for this study by

fitting the simplified-SAFT to pure component vapour pressure and liquid density data. At

the same time, the parameters for some of the components were collected from the literature

(Fu & Sandler 1995). It should be noted that the parameters were optimized based solely on

pure component data. Table 5-2 shows a list of fitted parameters obtained in this work. For

more information about the model parameter estimation of water and alcohols, readers are

referred to the work of Al-Saifi (Nayef Masned Al-Saifi 2012).

45

Table ‎5.1: Simplified-SAFT parameters obtained in this work

⁄ ⁄

Propane 44.10 2.710 16.744 92.935 0.08 0.1 200-340K (Glos et al. 2004)

Carbon dioxide 44.01 1.839 14.492 80.563 0.36 0.68 216.55-303K (Vargraftik 1975)

Tetrahydrofuran 72.11 2.200 24.385 143.51 0.59788 1.7 3.58 0.66 250-330 (Yaws 2003)

Water 18.015 (Wagner & Pruss

2002) SSAFT 1.500 9.1362 189.74 925.78 0.10178 0.04 0.03 273.16-330K

Dipolar-SSAFT 2.407 7.853 64.511 915.85 0.04411 0.37988 1.85

Methanol 32.04 (Smith & Srivastava 1986) SSAFT 1.250 21.981 224.54 1116.6 0.00967 0.61 0.81 212-300K

Ethylene glycol 62.07

(Yaws 2003) SSAFT 1.900 22.089 235.06 1471.2 0.00891

Dipolar-SSAFT 2.100 19.569 229.77 1433.2 0.00412 0.27378 1.7

46

5.2. Prediction of Vapor Liquid Equilibrium for Binary Systems

Figure ‎5.1: Predicted results of vapor-liquid equilibrium ( =0) by SSAFT for the methanol (1)/ water

(2) system at , , , and

The capability of SAFT in dealing with highly non-ideal solutions such as polar

solvents and hydrogen bonded compounds renders this theoretically-based equation of state a

successful model for tackling difficult systems like those of water-alcohol or water-alcohol-

hydrocarbon. These types of systems are very interesting and of great interest both from the

points-of-view of industry and academia (Nayef Masned Al-Saifi 2012). As discussed in

0

20

40

60

80

100

120

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Pre

ssu

re (

kP

a)

Mole fraction

Kurihara et al. (1995)

McGlashan and Williamson (1976)

simplified-SAFT

65

60

55

50

35

47

previous sections, it is very common in the oil and gas industry to use alcohols, for example

methanol and ethylene glycol, as inhibiting agents for preventing hydrate formation.

The phase diagram of the water-methanol system over a wide temperature range is

shown in Figure 5.1. As seen, simplified-SAFT is able to predict the phase behavior of this

system very well. In this work, it was assumed that water molecules with four association

sites interact with three association site methanol molecules through hydrogen bonding. The

SAFT parameters used in the vapor-liquid equilibrium calculation are those obtained and

used for the prediction of the incipient hydrate formation conditions.

The ability of the Statistical Association Fluid Theory (SAFT) to predict vapor-liquid

equilibrium was investigated by Al-Saifi (Nayef Masned Al-Saifi 2012). In his work, Al-

Saifi employed PC-SAFT and studied the phase behavior of several alcohol-hydrocarbon

systems including those of methanol-butane, methanol-pentane and methanol-hexane. In

spite of his excellent results, Al- aifi’s‎ predictions‎ proved‎ unsatisfactory‎ for‎ methanol-

hydrocarbon systems at low temperatures ( . In this thesis project, we studied the

phase equilibria for methanol-methane and methanol-ethane systems. Figure 5.2 and Figure

5.3 show the phase diagrams of these systems respectively. Although the simplified-SAFT

was able to predict the overall phase behavior, the quality of prediction was not as good as

the one obtained for the water-methanol system. This result could be due to the fact that the

SAFT parameters estimated in this work were suitable for temperature ranges that gas

hydrates form and that are obtained by employing the liquid density and vapor pressure of

pure compounds at low temperatures.

48

Figure ‎5.2: Predicted results of vapor-liquid equilibrium ( =0) by SSAFT for the methane (1)/ methanol

(2) system at and

Figure ‎5.3: Predicted results of vapor-liquid equilibrium ( =0) by SSAFT for the ethane (1)/ methanol

(2) system at and

0

50

100

150

200

250

300

350

400

450

500

0 0.2 0.4 0.6 0.8 1 1.2

Pre

ssu

re (

atm

)

Mole fraction

SSAFT

10

20

30

40

50

60

70

80

0 0.2 0.4 0.6 0.8 1 1.2

Pre

ssu

re (

atm

)

Mole fraction of ethane

SSAFT

49

5.3. Evaluating the Proposed Model in Hydrate Formation Calculations

As we have seen in the previous chapter, other studies have given strong

consideration to the inhibiting effects of methanol and ethylene glycol on incipient hydrate

formation calculations. The systems studied in this work are methane-water-methanol,

ethane-water-methanol, propane-water-methanol, CO2-water-methanol, methane-water-

ethylene glycerol, ethane-water-ethylene glycerol, propane-water-ethylene glycerol and CO2-

water-glycerol. Based on the computational scheme presented in Figure 4.1, the equilibrium

hydrate formation pressure was calculated at a given temperature and for a given aqueous

inhibitor (methanol and ethylene glycol) concentration. The inhibitor concentration is usually

reported as the mass fraction of the inhibitor during the water phase.

The absolute average deviation of the predicted pressure is defined as follows:

∑ [|

|]

‎5-1

Table 5.3 summarizes the results.

50

Table ‎5.2: Prediction of hydrate formation pressure

No. of data

points

Methane /Water/Methanol

0 wt% methanol 275.2-291.2 0 3.98 7 (Verma 1974)

10 wt% methanol 266.2-286.4 0 3.49 11 (H.-J. Ng & Robinson1985)

(Mohammadi & Richon 2009)

20 wt% methanol 263.3-280.2 0 4.82 11 (H.-J. Ng & Robinson 1985)

(Mohammadi & Richon 2009)

35 wt% methanol 250.9-270.1 0 10.0 11 (Robinson et al. 1986)(Mohammadi

& Richon 2009)

50 wt% methanol 232.8-259.5 0 2.86 7 (Ng et al. 1987)(Mohammadi &

Richon 2009)

65 wt% methanol 214.1-240.3 0 8.89 10 (Ng et al. 1987)(Mohammadi &

Richon 2010)

Ethane /Water/Methanol 0 wt% methanol 277.8-287.2 0 3.72 10 (Avlonitis 1988)

10 wt% methanol 268.3-280.4 0 5.82 7 (Ng & Robinson 1985)

15 wt% methanol 268.2-278.9 0 7.15 5 (Mohammadi et al. 2008) 20 wt% methanol 263.5-274.1 0 4.07 6 (Ng & Robinson 1985)

35 wt% methanol 252.6-262.2 0 4.77 4 (Ng et al. 1985)

50 wt% methanol 237.5-249.8 0 19.6 4 (Ng et al. 1985)

Propane /Water/Methanol

0 wt% methanol 274.2-278.4 0 1.66 9 (Kubota et al. 1984)

5 wt% methanol 272.1-274.8 0 4.23 5 (H.-J. Ng & Robinson 1985)

10.39 wt% methanol 268.3-271.8 0 3.72 6 (H.-J. Ng & Robinson 1985) 15 wt% methanol 266.3-269.9 0 8.43 4 (Mohammadi et al. 2008)

Carbon dioxide /Water/Methanol

0 wt% methanol 273.7-282.9 0 2.27 18 (Deaton & Frost Jr 1946) 20 wt% methanol 264-268.9 0 11.4 7 (Ng & Robinson 1985)

35 wt% methanol 242-255.1 0 12.8 5 (Robinson & Ng 1986)

50 wt% methanol 232.6-241.3 0 12.7 3 (Robinson & Ng 1986)

Methane /Water/Ethylene glycol 0 wt% methanol 274.3-285.3 0 3.98 16 (Nakamura et al. 2003)




51

No. of data

points

Ethane /Water/Ethylene glycol 0 wt% methanol 277.8-285.9 0 3.72 9 (Avlonitis 1988)

10 wt% methanol 271.1-278.5 0 9.33 4 (Mohammadi & Richon 2010)

20 wt% methanol 267.1-275.3 0 8.52 4 (Mohammadi & Richon 2010) 35 wt% methanol 262.1-269.4 0 3.74 4 (Mohammadi & Richon 2010)

Propane /Water/Ethylene glycol

0 wt% methanol 274.2-278.4 0 1.66 (Kubota et al. 1984)

10 wt% methanol 271.5-274.9 0 2.73 (Maekawa 2008)

15 wt% methanol 269.8-273.7 0 3.58 (Mohammadi et al. 2008) 20 wt% methanol 267.5-270.8 0 1.13 (Maekawa 2008)

Carbon dioxide /Water/Ethylene glycol

0 wt% methanol 273.7-282.9 0 2.27 18 (Deaton & Frost Jr 1946)

10 wt% methanol 271.4-277.8 0 11.8 4 (Mohammadi & Richon 2010)

20 wt% methanol 267.5-274.5 0 7.69 4 (Mohammadi & Richon 2010) 35 wt% methanol 261.6-267.4 0 4.33 4 (Mohammadi & Richon 2010)

52

5.3.1. Inhibiting Effect of Ethylene Glycol

Figure ‎5.4: Hydrate formation prediction ( =0) by SSAFT for methane hydrate in presence of ethylene

glycol aqueous solution, , , and

The information for methane, ethane, propane and carbon dioxide hydrate in

presence of ethylene glycol are provided in Table 5.3. Figure 5.4 to Figure 5.7 also show the

experimental data with predictions that result from the use of simplified-SAFT for these

systems respectively. It should be noted that these predictions are based on parameters

obtained from pure component data, and no binary interaction parameters ( ) have been

used. Furthermore, Figure 5.4 presents the polar SSAFT prediction results for methane gas

0

2

4

6

8

10

12

14

16

18

255 260 265 270 275 280 285 290

Pre

ssu

re (

MP

a)

Temperature (K)

Robinson and Ng (1986)

Nakamura et al. (2003)

SSAFT

dipolar-SSAFT

50 wt%

30 wt% 10 wt%

0 wt%

53

hydrates. As seen, excellent agreements were obtained through the use of simplified-SAFT

predictions with regard to hydrocarbon/water/ethylene glycol systems. However, the

predictions for the carbon dioxide/water/ethylene glycol system were not as satisfactory as

those obtained for methane, ethane and propane. In all these cases except for carbon dioxide

in presence of ethylene glycol aqueous solution ( equals to ), the

absolute average deviation ( ) was found to be less than the desired maximum target

which is deviation in pressure. In spite of satisfactory predicted results in most cases,

the absolute deviation was found to be insufficient according to the specified criteria. For

instance, the AAD for methane in the presence of ethylene glycol aqueous

solution is which satisfies the pressure criteria; however, at a temperature of

, the predicted pressure has deviation from the experimental data, and

therefore does not satisfy the pressure criteria.

The prediction results of polar SSAFT were found to be satisfactory for hydrocarbons

in the presence of low concentrations of ethylene glycol aqueous solutions. Figure 5.4 shows

the phase diagram produced by polar SSAFT along with experimental data and the calculated

results obtained by SSAFT for methane hydrates. As seen, there is satisfactory agreement

between calculated and experimental data for concentrations of up to .

Interestingly, at higher concentrations, the polar-SSAFT shows a significant deviation from

the experimental data. This is most likely due to the contribution of dipolar interaction in

SSAFT which results in an overestimation in the force field and, consequently, an

underestimation in the incipient hydrate formation prediction, most specifically at higher

54

concentrations of ethylene glycol. The same deviations were observed for ethane, propane

and carbon dioxide.

On the other hand, the parameters required by the van der Waals-Platteew model play

a role in the quality of the predictions. Various sets of parameters were extracted from the

literature and compared with each other. Table 5.4 summarizes the van der Waals-Platteew

parameters. Along with different sets of parameters, the following were selected for the

current work:

Table ‎5.3: Required parameters for the van der Waals-Platteew model selected for this work

Property Structure I Structure II

⁄ 1289.5

⁄ -4327.9

⁄

-38.13 -38.13

⁄

0.141 0.141

The above set of parameters was chosen because, when calculations were made on data

available from the literature, these were the parameters that showed the smallest absolute

average deviation ( ) .

55

Figure ‎5.5: Hydrate formation prediction ( =0) by SSAFT for ethane hydrate in presence of ethylene


Figure ‎5.6: Hydrate formation prediction ( =0) by SSAFT for propane hydrate in presence of ethylene


0

0.5

1

1.5

2

2.5

3

3.5

260 265 270 275 280 285 290

Pre

ssu

re (

MP

a)

Temperature (K)

Series1

Avlonitis (1988)

SSAFT

0

0.1

0.2

0.3

0.4

0.5

0.6

265 267 269 271 273 275 277 279

Pre

ssu

re (

MP

a)

Temperature (K)

SSAFTKubota et al. (1984)Maekawa (2008)Mohammadi et al. (2008)

35 wt% 20 wt% 10 wt%

0 wt%

20 wt%

15 wt% 10 wt%

0 wt%

56

Figure ‎5.7: Hydrate formation prediction ( =0) by SSAFT for carbon dioxide hydrate in presence of

ethylene glycol aqueous solution, , , and

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

249 254 259 264 269 274 279 284 289

Pre

ssu

re (

MP

a)

Temperature (K)

Mohammadi and Richon (2010)

Deaton and Frost (1946)

SSAFT

35 wt% 20 wt%

10 wt%

0 wt%

57

5.3.2. Inhibiting Effect of Methanol

The predicted hydrate formation conditions for methane, ethane, propane and carbon

dioxide in the presence of methanol aqueous solution are presented in Figure 5.8 to Figure

5.11 respectively. The information for these systems is also presented in Table 5.3. It

must be emphasized that these are pure predictions and that no binary data was used in either

the estimation of SSAFT parameters or the calculation of equilibrium conditions.

Outstanding agreements were obtained through the use of simplified-SAFT

predictions in the hydrocarbon/water/methanol systems. For all systems except ethane in

presence of methanol aqueous solution ( equals to ), the absolute

average deviation ( ) was found to be less than the desired maximum target of

deviation in pressure. In some cases, for instance methane in the presence of methanol

aqueous solution, the deviation of predicted results from one set of experimental

data was found to be higher than for another, leading to a higher average deviation.

In spite of obtaining excellent results for the predictions in most cases, the absolute

deviation was found to be higher than acceptable for some. For instance, the AAD for

methane in the presence of ethylene glycol aqueous solution is which

satisfies the pressure criteria; however, at a temperature of , the predicted pressure

has a deviation from the experimental data.

58

Figure ‎5.8: Hydrate formation prediction ( =0) by SSAFT for methane hydrate in presence of methanol

aqueous solution, , , , , and

Figure ‎5.9: Hydrate formation prediction ( =0) by SSAFT for ethane hydrate in presence of methanol

aqueous solution, , , , , and

0

5

10

15

20

200 220 240 260 280 300

Pre

ssu

re (

MP

a)

Temperature (K)

SSAFTVerma (1974)Ng and Robinson (1985)Robinson and Ng (1986)Ng et al. (1987)Mohammadi and Richon (2010a)Mohammadi and Richon (2010b)

0

0.5

1

1.5

2

2.5

3

3.5

235.0 245.0 255.0 265.0 275.0 285.0

Pre

ssu

re (

MP

a)

Temperature (K)

Ng and Robinson (1985)

Mohammadi et al. (2008)

Ng et al. (1985)

SSAFT

0 wt% 10 wt%

20 wt%

35 wt%

50 wt%

65 wt%

0 wt%

10 wt%

15 wt%

20 wt%

35 wt%

50 wt%

59

Figure ‎5.10: Hydrate formation prediction ( =0) by SSAFT for propane hydrate in presence of

methanol aqueous solution, , , and

The prediction for the carbon dioxide-water-methanol system, however, was not as

good as those calculated for methane, ethane and propane. In the hydrate formation

calculation of carbon dioxide in the presence of methanol, the average absolute deviation

exceeded the maximum desired values ( for , for and

for ). Despite the unsatisfactory predictions calculated for the inhibiting

effect of methanol on carbon dioxide hydrate formation, significant improvement was

obtained when compared to the results reported by Englezos et al. (1991) for the -water-

methanol system using the Trebble-Bishnoi equation of states and the van der Waals-

Platteew model.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

265.0 267.0 269.0 271.0 273.0 275.0 277.0 279.0

Pre

ssu

re (

MP

a)

Temperature (K)

Kubota et al. (1984)


Mohammadi et al. (2008)

SSAFT

15 wt% 10.39 wt%

5 wt%

0 wt%

60

Figure ‎5.11: Hydrate formation prediction ( =0) by SSAFT for carbon dioxide hydrate in presence of

methanol aqueous solution, , , and

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

200 210 220 230 240 250 260 270 280 290

Pre

ssu

re (

MP

a)

Temperature (K)

Deaton and Frost (1946)


Robinson and Ng (1986)

SSAFT

0 wt%

20 wt%

35 wt%

50 wt%

61

6 Conclusions and Recommendations

6.1. Conclusions

This thesis used the simplified SAFT (SSAFT) to in conjunction with the van der

Waals Platteeuw model to predict the incipient hydrate formation conditions for natural gas –

type systems. The predicted conditions were then compared to experimental data from the

literature. It was found that the SSAFT is highly capable of predicting the conditions of

incipient hydrate formation for the single hydrate former gases of methane, ethane, propane

and carbon dioxide in the presence of the inhibitors methanol and ethylene glycol, even

without the introduction of binary interaction parameters ( .‎Jog‎and‎Chapman’s‎dipolar‎

term was then introduced into the simplified SAFT and the results were examined. Despite

obtaining satisfactory predictions at lower alcohol concentrations (of less than ),

we were not able to obtain better agreement with polar SSAFT. Pure compound parameters

were correlated using vapor pressure and liquid density for water, methanol and ethylene

glycol. It was observed that the quality of a prediction is strongly influenced by the

temperature range on which these parameters are correlated. It should be noted that in spite

of the excellent results obtained in this thesis, the adapted computational scheme was found

to be cumbersome compared to its use with conventional equation of states because of the

heavy load of computations that are required with the SAFT equation of state.

62

6.2. Recommendations

Based on the results and outcomes of the model described above, there are few

suggestions that would be beneficial if further investigated:

1. The prediction accuracy of the SAFT modular equations which was developed for the

prediction of hydrate formation conditions for single gases (methane, ethane, propane

and carbon dioxide), should be explored for gas mixture as well.

2. The mixture of the inhibitors, methanol and ethylene glycol, which SAFT model

was used for examining the degree of inhibiting effect, should be investigated along

with the introduction of other inhibitors such as glycerol and triethylene glycol .

3. Although the use of original form of van der Waals-Platteeuw model employed in the

current study resulted in satisfactory to excellent agreements, the improvement on the

degree of accuracy by modifying the assumptions of van der Waals-Platteeuw should

be explored further.

4. The behavior of introducing quadrupole moments to the SAFT model prediction of

carbon dioxide should be examined.

63

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70

Appendices:

Appendix A: Helmholtz Free Energy

For a closed system (Smith et al. 2004):

B-1

we also have,

B-2

B-3

Substituting

B-2 and

B-3 into B-1, one may obtain,

B-4

Recalling the definition of Helmholtz free energy,

B-5

B-6

71

combining B-4 and B-6,

B-7

and finally, according to the Gibbs/Duhem equation:

∑

B-8

72

Appendix B: Driving Compressibility Factor from Helmholtz Free Energy

(

)

C-1

Substituting B-5 and B-7 into C-, we may obtain,

(

)

C-2

from the above equation,

[

⁄

]

C-3

and

[

⁄

]

C-4

Because , the compressibility factor and the Helmholtz free energy are related as,

[ ⁄

]

C-5

As an example, we show how to obtain the hard-sphere compressibility factor, ,

from the hard-sphere Helmholtz free energy, . Recalling equation ‎3-3‎3-4,

and

73

At a constant temperature, for one mole of fluid,

C-6

Therefore,

(

)

( )

C-7

Performing the volume derivative of the above expression at a constant temperature,

[ ⁄

]

[ (

(

)

( )

)

]

(

)

( )

C-8

Combining C- and C-,

[ ⁄

]

[

(

)

( )

]

(

)

( )

C-9

Finally, we get,

prediction of gas hydrate equilibrium

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