Measuring and Modelling of the Thermodynamic Equilibrium

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Measuring and Modelling of the Thermodynamic

Equilibrium Conditions for the Formation of TBAB and

TBAC Semi-Clathrates formed in the Presence of

Xenon and Argon

Garcia Mendoza, Marlon Ilich

Garcia Mendoza, M. I. (2015). Measuring and Modelling of the Thermodynamic Equilibrium

Conditions for the Formation of TBAB and TBAC Semi-Clathrates formed in the Presence of Xenon

and Argon (Unpublished master's thesis). University of Calgary, Calgary, AB.

doi:10.11575/PRISM/26854

http://hdl.handle.net/11023/2223

master thesis

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UNIVERSITY OF CALGARY

Measuring and Modelling of the Thermodynamic Equilibrium Conditions for the

Formation of TBAB and TBAC Semi-Clathrates Formed in the Presence of Xenon and

Argon

by

Marlon Ilich Garcia Mendoza

A THESIS

SUBMITTED TO THE FACULTY OF GRADUATE STUDIES

IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE

DEGREE OF MASTERS OF SCIENCE

GRADUATE PROGRAM IN CHEMICAL ENGINEERING

CALGARY, ALBERTA

April 2015

© Marlon Ilich Garcia Mendoza 2015

ii

Abstract

Semiclathrates are crystalline compounds similar in nature to gas hydrates. Like

gas hydrates, semiclathrates can trap small gas molecules inside a molecular framework of

water molecules. Quaternary ammonium salts (QAS) semiclathrates hydrates, such as

tetra-n-butyl ammonium bromide (TBAB) and tetra-n-butyl ammonium chloride (TBAC),

are ionic compounds that have a stabilizing effect on the framework of water molecules.

TBAB and TBAC semiclathrates formed in the presence of a gas can form at much milder

conditions than gas hydrates. Thus, there has been much interest, in recent years, on the

possible use of TBAB and TBAC semiclathrates in the storage and separation of gases. To

date, the majority of research in the area has been directed towards experimental studies

and only a handful of studies have attempted to model the equilibrium conditions of semi-

clathrate formation. The present study aims to measure and correlate equilibrium

dissociation conditions for semiclathrates formed from aqueous solutions of TBAB and

TBAC, in combination with argon and xenon.

In the experimental part of this study, a constant-volume reactor was used for

measuring the solid–vapor–liquid equilibrium conditions of semiclathrates formed in

aqueous solutions of TBAB and TBAC. The TBAB and TBAC semiclathrates were formed

from pure argon and pure xenon. The experimental temperatures ranged from (284 to 303)

K, the experimental pressures ranged from (266 to 6114) kPa, the weight fraction of TBAB

ranged from wTBAB = (0.05 to 0.20), and the weight fraction of TBAC ranged from wTBAC =

(0.05 to 0.20). As expected, at a given temperature, the pressure required to form TBAB

and TBAC semiclathrates with argon and with xenon was much lower than the pressure

that are required to form pure gas hydrates. From the equilibrium data, the enthalpy of

formation was estimated to be between (133 and 188) kJ·mol–1 for semiclathrates formed

from argon and (55 and 127) kJ·mol–1 for semiclathrates formed from xenon.

For modeling the experimental data obtained in the present study, the PSRK

equation of state is used to describe the vapour phase, the LIFAC activity coefficient model

is used to describe the aqueous phase and the van der Waals and Platteeuw theory combined

with the model of Paricaud was employed to describe the solid semiclathrate phase. The

new model differs from previous modeling efforts in that it does not neglect the solubility

of the gas in the aqueous phase or the presence of water in the vapour phase. Rather, the

iii

solubility of the gas and molar fraction of water in vapour phase are computed from a flash

calculation. The new model also computes the Langmuir constants from the Kihara

potential rather than from an empirical correlation. The model is capable of describing the

solid-liquid equilibrium for the semiclathrate in the absence of gas molecules. The new

modeling approach is applied to TBAB and TBAC semiclathrates that are formed from

xenon and argon. For both gases, new Kihara potential parameters were regressed from the

experimental data. Further testing of the new modelling approach was conducted by

correlating available data for TBAB/TBAC semiclathrates formed in the presence of pure

methane (CH4), carbon dioxide (CO2), nitrogen (N2), and hydrogen (H2). It was found that

the new approach was able to correlate the experimental data to a high degree of accuracy

with fewer adjustable parameters for all but one of the existing modelling attempts.

iv

Acknowledgements

The author would like to express his honest gratitude to his supervisor Dr. Matthew

A. Clarke for his patience, continuous support, mentorship, encouragement and supervision

of this thesis.

The author also expresses his appreciation to the members of the examining

committee for their valuable comments.

The author also thanks Dr. Amitabha Majumdar for his assistance during the

configuration of the experimental setup.

The author wants to mention his research group members: Emmanuel Bentum,

Fahd Mohamad Alqahtani, and Su Wang for their comments regarding the experimental

setup and the thermodynamic modeling.

Financial support provided by the Natural Sciences and Engineering Research Council of

Canada (NSERC) is greatly appreciated by the author.

Finally, the author thanks the financial support provided by the department of

Chemical and Petroleum Engineering for completing the present study.

v

Dedication

This thesis is dedicated to my family.

vi

Table of Contents

Approval Page ..................................................................................................................... ii

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

Acknowledgements ............................................................................................................ iv Dedication ............................................................................................................................v Table of Contents ............................................................................................................... vi List of Symbols, Abbreviations and Nomenclature ......................................................... xiv

INTRODUCTION ..................................................................................1

1.1 Clathrate hydrates ......................................................................................................1 1.2 Hydrate structures ......................................................................................................2

1.2.1 Structure I ..........................................................................................................3

1.2.2 Structure II .........................................................................................................3 1.2.3 Structure H .........................................................................................................3

1.3 Natural gas hydrates ...................................................................................................4

1.3.1 Hydrate inhibition ..............................................................................................6 1.4 Hydrate promoters .....................................................................................................7

1.4.1 Semiclathrate hydrates .......................................................................................9 1.4.1.1 Semiclathrates of TBAB and water .......................................................10 1.4.1.2 Semiclathrates of TBAC and water .......................................................13

1.5 Potential applications of gas hydrates and semiclathrates .......................................13 1.5.1 Natural gas storage and transportation ............................................................14

1.5.2 Separation processes thorough gas hydrate formation ....................................15 1.5.2.1 Separation of carbon dioxide .................................................................15

1.5.2.2 Separation of methane ...........................................................................19 1.5.2.3 Separation of hydrogen sulfide ..............................................................21

1.5.2.4 Separation and storage of hydrogen .......................................................22 1.5.2.5 Separation of nitrogen ............................................................................23

1.6 Review of gas hydrate and semiclathrate phase equilibrium models ......................24

1.6.1 Models to calculate the phase equilibria of hydrates .......................................24 1.6.2 Models to calculate phase equilibria of semiclathrates of hydrates ................25

1.7 Scope of current study .............................................................................................32

EXPERIMENTAL APPARATUS, METHODOLOGY AND

RESULTS .................................................................................................................33 2.1 Apparatus .................................................................................................................33

2.2 Materials ..................................................................................................................35 2.3 Experimental procedure ...........................................................................................36

2.3.1 Preparation and start up of the experiments ....................................................36 2.4 Experimental results ................................................................................................38

2.4.1 Incipient equilibrium conditions for TBAB and TBAC semiclathrates

formed from pure argon gas .............................................................................39 2.4.2 Incipient equilibrium conditions for TBAB and TBAC semiclathrates

formed from pure xenon gas ............................................................................45 2.4.3 Heat of dissociation for TBAB and TBAC semiclathrates in the presence of

argon and xenon ...............................................................................................52

vii

PRESENTATION OF THE THERMODYNAMIC MODEL ........54 3.1 Motivation for the new thermodynamic model .......................................................54 3.2 Development of a new modelling approach for semiclathrates. ..............................56

3.2.1 Liquid-solid equilibrium ..................................................................................58

3.2.2 Vapour–liquid-hydrate equilibrium .................................................................65 3.3 Parameter regression ................................................................................................69

3.3.1 Validation of PSRK for predicting the solubility of gases in water ................69 3.3.2 Verification of LIFAC model for computing the mean activity and osmotic

coefficient of dissolved salts ............................................................................80

3.3.3 Validation of the extension of PSRK for usage in aqueous electrolyte

solutions ...........................................................................................................84 3.3.4 Regression of parameters for modelling solid-liquid equilibrium involving

TBAB and TBAC in the absence of gases. ......................................................88 3.3.5 Kihara Parameter regression and correlation of SVLE for systems

containing TBAB/TBAC semiclathrates and a gas .........................................91

3.4 Results and discussion .............................................................................................94 3.4.1 Xenon semiclathrates .......................................................................................94

3.4.2 Argon semiclathrates .......................................................................................97 3.4.3 Methane semiclathrates: ..................................................................................99 3.4.4 Carbon Dioxide semiclathrates ......................................................................102

3.4.5 Nitrogen semiclathrates. ................................................................................105 3.4.6 Hydrogen semiclathrates. ..............................................................................107

CONCLUSION AND RECOMMENDATIONS ............................110 4.1 Conclusions ............................................................................................................110

4.2 Recommendations ..................................................................................................111

References………………………………………………………………………………112

APPENDIX A: ISOTHERMAL ISOBARIC FLASH COMPUTATION……………...135

APPENDIX B: PSRK EQUATION OF STATE……………………………………….138

APPENDIX C: ANALYTICAL SOLUTION TO CUBIC EQUATIONS……………..148

APPENDIX D: ERROR CALCULATION IN THE PARAMETER ESTIMATION….150

APPENDIX E: PRIVATE COMMUNICATION WITH KWATERSKI…………….....151

viii

List of Tables

Table 1.1: Geometry of cages (Reproduced from Koh and Sloan [6, p. 55]). ................... 2

Table 1.2: Experimental studies on TBAB for determining hydration number and

congruent melting points. .......................................................................................... 11

Table 1.3: Structural data on TBAB semiclathrates hydrates. ......................................... 12

Table 1.4: Structural data on TBAC semiclathrates hydrates structures. ........................ 13

Table 1.5: Experimental studies on clathrate/semiclathrate hydrate for carbon dioxide

systems in combination with hydrate promoters....................................................... 16

Table 1.6: Experimental studies on clathrate/semiclathrate hydrate for methane


Table 1.7: Experimental studies on semiclathrate hydrate for hydrogen sulfide systems

in combination with hydrate promoters .................................................................... 21

Table 1.8: Experimental studies on clathrate/semiclathrate hydrate for hydrogen


Table 1.9: Experimental studies on clathrate/semiclathrate hydrate for nitrogen


Table 1.10: Summary of thermodynamic models currently available for semiclathrates

hydrates ..................................................................................................................... 29

Table 2.1: TBAB and TBAC physical properties ............................................................ 36

Table 2.2: Incipient Equilibrium Conditions for TBAB semiclathrates formed from

pure argon gas. .......................................................................................................... 40

Table 2.3: Incipient equilibrium conditions for TBAC semiclathrates formed from pure

argon gas ................................................................................................................... 42

Table 2.4: Incipient Equilibrium Conditions for TBAB Semiclathrates Formed from

Pure xenon Gas ......................................................................................................... 46

Table 2.5: Incipient Equilibrium Conditions for TBAC Semiclathrates Formed from

Pure xenon Gas ......................................................................................................... 48

Table 2.6: Estimated heat of dissociation ∆Hdis for TBAB semiclathrates in the

presence of argon and xenon. .................................................................................... 53

Table 2.7: Estimated heat of dissociation ∆Hdis for TBAC semiclathrates in the

presence of argon and xenon. .................................................................................... 53

ix

Table 3.1: Summary of stoichiometric coefficients for TBAB and TBAC ion

constituents used in the present work. ...................................................................... 68

Table 3.2: Newly optimized PSRK parameters for argon+H2O and xenon+H2O

regressed from experimental data [125], [126].The errors are at the 95%

confidence uncertainties. ........................................................................................... 72

Table 3.3: Summary of PSRK predictions of vapour-liquid equilibria for the systems:

Ar+H2O, Xe+H2O, N2+H2O, H2+H2O, and CO+H2O. ............................................. 76

Table 3.4: Summary of predictions of mean activity coefficient by means of electrolyte

NRTL for the systems TBAB+H2O and TBAC+H2O. ............................................. 82

Table 3.5: Summary of predictions of osmotic coefficient of water in TBAB and

TBAC solutions by means of electrolyte NRTL at T=298.15 K and atmospheric

pressure. .................................................................................................................... 84


CH4+H20+NaCl, CO2+H2O+NaCl,and CO2+H2O+TBAB. .................................... 87

Table 3.7: Semiclathrate parameters to compute solid-liquid equilibria of the systems

H2O+TBAB and, H2O+TBAC at atmospheric pressure .......................................... 89

Table 3.8: Regressed Kihara potential parameters for xenon, argon, CH4, CO2, H2, and

N2, in TBAB aqueous solutions. ............................................................................... 93

Table 3.9: Regressed Kihara potential parameters for xenon, argon, CH4, and CO2 in

TBAC aqueous solutions .......................................................................................... 94

Table 3.10: Model results for prediction of the equilibrium conditions of TBAB and

TBAC+H2O+gas (CH4, CO2, H2, N2, xenon, argon) systems. ................................ 109

Table B.1: Critical constants of argon, xenon, CH4, CO2, N2, H2, CO, and H2O required

in the PSRK model .................................................................................................. 140

Table B.2: Molecular and group parameters for UNIFAC and LIFAC ......................... 141

x

List of Figures and Illustrations

Figure 1.1: Cavities in gas clathrates hydrates: (a) pentagonal dodecahedron (512), (b)

Tetrakaidecahedron (51262), (c) hexakaidecahedron (51264), (d) irregular

dodecahedron (435663), and (e) icosahedron (51268) (Reproduced from Sloan and

Koh [6, p. 54]) ............................................................................................................. 3

Figure 1.2: Common gas hydrate structures (sI, sII, sH) and the water cage types that

compose the hydrate structure. (Reproduced from Koh and Sloan [9]) ..................... 4

Figure 1.3: Gas hydrate plug recovered from a subsea pipeline close to the coast

Atlantic of Brazil [13] ................................................................................................. 5

Figure 1.4: Methane-Propane double hydrate (Reproduced from Janda [14]) .................. 6

Figure 1.5: Qualitative effect of inhibitors on the hydrate equilibrium conditions.

Dashed curve represent the hydrate curve with inhibitor and solid curve represent

the hydrate curve without inhibitor. ............................................................................ 7

Figure 1.6: Qualitative effect of promoters on the hydrate equilibrium conditions. Solid

line represents the pure hydrate curve, and the dashed curve is the equilibrium

curve with a promoter. ................................................................................................ 9

Figure 1.7: Cavity in semiclathrate hydrates: Pentakaidecahedron (51263) (Reproduced

from Walsh et al [29]) ............................................................................................... 10

Figure 1.8: Semiclathrate structure type B structure for TBAB (Reproduced from

Shimada et al [27]) .................................................................................................... 12

Figure 1.9: Structure around the TBA+ cation in the structure type B for TBAB.

(Reproduced from Shimada et al [27]) ..................................................................... 12

Figure 1.10: Complete process of transporting natural gas in the form gas hydrates.

(Reproduced from Giavarini et al [38, p. 145]) ........................................................ 14

Figure 2.1: Schematic drawing from the reactor (Reproduced from Portz et al [112]) ... 34

Figure 2.2: Flow diagram of the experimental apparatus (Reproduced from Portz et al

[112]) ......................................................................................................................... 35

Figure 2.3: A typical pressure versus temperature trajectory. (Reproduced from

Meysel et al [47]) ...................................................................................................... 37

Figure 2.4: Dissociation point determination for a gas semiclathrate from Argon on

aqueous solution of 5 wt. %TBAB ........................................................................... 38

Figure 2.5: Incipient equilibrium conditions for TBAB semiclathrates formed from

argon in aqueous solutions of TBAB. ....................................................................... 41

xi

Figure 2.6: Incipient equilibrium conditions for TBAC semiclathrates formed from

argon in aqueous solutions of TBAC. ....................................................................... 43

Figure 2.7: Incipient equilibrium conditions for TBAB and TBAC semiclathrates

formed from argon. ................................................................................................... 44


xenon in aqueous solutions of TBAB. ...................................................................... 47


xenon in aqueous solutions of TBAC. ...................................................................... 49

Figure 2.10: Comparison of ln(P) versus 1/T for TBAB and TBAC semiclathrates

formed from argon in aqueous solution of TBAB, and TBAC respectively. ........... 50


formed from xenon in aqueous solution of TBAB, and TBAC respectively. ........... 51

Figure 3.1: Overall process diagram ................................................................................ 58

Figure 3.2: Solubility of argon (1) in H2O (2) at various temperatures conditions, and

total P=0.1 MPa. ....................................................................................................... 73

Figure 3.3: Solubility of argon (1) in water (2) at various temperatures conditions, and

total P=0.1 MPa. ....................................................................................................... 74

Figure 3.4: Methane (1) solubility in water (2) at various temperature and pressure

conditions. Symbols stands for experimental data and curves are predictions by

PSRK. ........................................................................................................................ 75

Figure 3.5: Solubility of carbon dioxide (1) in water (2) at various temperature and

pressure conditions. ................................................................................................... 77

Figure 3.6: Solubility of nitrogen (1) in water (2) at various temperature and pressure

conditions. ................................................................................................................. 78

Figure 3.7: Solubility of hydrogen (1) in water (2) at various temperature and pressure

conditions. ................................................................................................................. 79

Figure 3.8: Mean activity coefficient of TBAB and TBA+H2O solutions at T=298.15

K and atmospheric pressure. ..................................................................................... 81

Figure 3.9: Osmotic coefficient of water in TBAB and TBAC solutions at T=298.15

K and atmospheric pressure. ..................................................................................... 83

Figure 3.10: Solubility of CH4 (1) in H2O (2)+NaCl (3) at various total pressure

conditions and T=324.7 K.□,NaCl molality=1.0 mol/kg;+, NaCl molality =4.0

xii

mol/kg. Symbols stands for experimental data [133] and curves are the predictions

by PSRK. ................................................................................................................... 85

Figure 3.11: Solubility of CO2 (1) in H2O (2) + NaCl (3) at various pressure conditions

and fixed NaCl molality=4.0 mol/kg. □,T=313.15K;+,T =333.15K.Symbols

stands for experimental data [134] and curves are the predictions by PSRK. .......... 86

Figure 3.12: Solubility of CO2 (1) in H2O (2) +TBAB (3) at various pressure conditions

and T=283.15K;○,wTBAB=0.09 [56]; Curves are the predictions by PSRK. ............. 87

Figure 3.13: Temperature composition diagram of the H2O+TBAB mixture. The liquid

composition is expressed in terms of TBAB weight fraction (wTBAB).■ is the

experimental data from [135].The solid line represents SLE curve assuming

structure type A, dashed line represents SLE curve assuming structure type B. ...... 89

Figure 3.14: Temperature composition diagram of the H2O+TBAC mixture. The liquid

composition is expressed in terms of TBAC weight fraction (wTBAC). .................... 90

Figure 3.15: Flow diagram for computing the dissociation pressure of semiclathrates

at a given temperature. .............................................................................................. 92

Figure 3.16: Dissociation conditions of clathrate/semiclathrate hydrates for the

xenon+water/TBAB aqueous solution systems. ....................................................... 95

Figure 3.17: Dissociation conditions of clathrate/semiclathrate hydrates for the xenon+

water/TBAC aqueous solution systems. ................................................................... 96

Figure 3.18: Dissociation conditions of clathrate/semiclathrate hydrates for the argon+

water/TBAB aqueous solution systems. ................................................................... 97

Figure 3.19: Dissociation conditions of clathrate/semiclathrate hydrates for the argon+

water/TBAC aqueous solution systems. ................................................................... 98


methane +water/TBAB aqueous solution systems. ................................................ 100


methane +water/TBAC aqueous solution systems. ................................................ 101

Figure 3.22: Dissociation conditions of clathrate/semiclathrate hydrates for the carbon

dioxide +water/TBAB aqueous solution systems. .................................................. 103

Figure 3.23: Dissociation conditions of clathrate/semiclathrate hydrates for the carbon

dioxide +water/TBAC aqueous solution systems. .................................................. 104


nitrogen + water/TBAB aqueous solution systems. ................................................ 106

xiii


hydrogen + water/TBAB aqueous solution systems. .............................................. 108

Figure A.1: Scheme of vapor-liquid isothermal flash.................................................... 135

Figure A.2: Algorithm for solving the isothermal flash (Reproduced from Elliot and

Lira [146, p. 617]) ................................................................................................... 137

xiv

List of Symbols, Abbreviations and Nomenclature

Abbreviations Definition

A Anion

Br- Anion Bromide

C Cation

C2H6 Ethane

C3H8 Propane

CA Salt molecule/ Cation-Anion

CCS Carbon capture and storage

CH4 Methane

Cl- Anion Chloride

CO2 Carbon dioxide

DTAC Dodecyl trimethyl ammonium chloride

DTAC dodecyl trimethyl ammonium chloride

e-NRTL Activity coefficient model

EoS Equation of state

F- Anion Fluoride

GHG Greenhouse gases

H2 Hydrogen

H2O Water

H2S Hydrogen sulfide

LIFAC Activity coefficient model

LNG Liquefied natural gas

xv

MEA Monoethanol amine

MW Molecular weight [g∙mol-1]

N2 Nitrogen

O2 Oxygen

PRO II Commercial chemical process simulator

PSRK Predictive-Soave-Redlich-Kwong Equation of state

QAS Quaternary ammonium salts

SAFT-VRE

Statistical associating fluid theory with variable range for

electrolytes

SDC Sodium dodecyl sulfate

SLE Solid-liquid equilibria

S-L-V Solid-liquid-vapour

SRK Soave-Redlich-Kwong Equation of State

TBAB Tetra-n-butyl ammonium bromide

TBAC Tetra-n-butyl ammonium chloride

TBANO3 Tetra-n-butyl ammonium nitrate

TBAOH Tetra-n-butyl ammonium bromide

TBAX Tetra-n-butyl ammonium salts.X=Br-,Cl-,F-

TBPB Tetra-n-butyl phosphonium bromide

THF Tetrahydrofuran

UNIFAC Activity coefficient

vdWP van der Waals and Platteuw

VLE Vapour-liquid equilibrium

xvi

yxenon Molar fraction of xenon

yargon Molar fraction of argon

Symbols Definitions

wTBAB TBAB weight fraction [-]

wTBAC TBAC weight fraction [-]

Å angstrom [10-10 m]

sI Structure I in hydrates

sII Structure II in hydrates

sH Structure H in hydrates

𝑛𝑖𝑚𝑖 Nomenclature for polyhedra

ni Number of edges in a face type i

mi Number of faces with ni edges

512 Pentagonal dodecahedron

51262 Tetrakaidecahedron

51264 Hexakaidecahedron

435663 Irregular dodecahedron

51268 Icosahedron

51263 Pentakaidecahedron

MW Molecular weight [g∙mol-1]

pT Pressure vs. Temperature

w Weight fraction [-]

∆𝐻𝑑𝑖𝑠 Enthalpy of dissociation [kJ∙mol-1∙K-1]

K(T) Standard constant [-]

xvii

𝑧𝑐+ Charge of the cation [-]

𝑧𝑎− Charge of the anion [-]

𝜈𝑤 Number of water molecules[-]

𝑎ℎ𝐻 Activity of the hydrate in the solid phase[-]

𝑎𝑤𝐿 Activity of water in the liquid phase[-]

𝑎𝑐𝐿 Activity of the cation in the liquid phase[-]

𝑎𝑎𝐿 Activity of anion in the liquid phase[-]

𝛾𝑤𝐿 Activity coefficients in the liquid phase of water[-]

𝛾𝑐𝐿 Activity coefficients in the liquid phase of cation[-]

𝛾𝑎𝐿 Activity coefficients in the liquid phase of anion[-]

𝜇ℎ0,𝐻

Standard chemical potential of the semiclathrate in the hydrate

phase [J∙mol-1]

𝜇𝑤0,𝐿 Standard chemical potential of water in the liquid phase [J∙mol-1]

𝜇𝑐0,𝐿

Standard chemical potential of cation in the liquid phase [J∙mol-1]

𝜇𝑎0,𝐿

Standard chemical potential of anion in the liquid phase [J∙mol-1]

∆𝑑𝑖𝑠𝐺0(𝑇) Standard Gibbs energy of dissociation [J∙mol-1]

𝑉𝑤𝐿 Molar volume of pure liquid water and semiclathrate [m3∙mol-1]

𝑉ℎ𝐻 Molar volume of semiclathrate in solid phase [m3∙mol-1]

𝑉𝑐∞,𝐿

Partial molar volume of the cation at infinite dilution of the salt

[m3/mol]

𝑉𝑎∞,𝐿

Partial molar volume of the anion at infinite dilution of the salt

[m3/mol]

xviii

𝑥𝑤 Molar fraction of water in liquid phase[-]

𝑥𝑐 Molar fraction of cation in liquid phase[-]

𝑥𝑎 Molar fraction of anion in liquid phase[-]

𝐶𝑝,𝑖𝐿 Isobaric heat capacity of compound i in the liquid phase [J∙mol-1

K-1]

∆𝑑𝑖𝑠𝑉0(𝑇) Change in volume accompanying the dissociation reaction of the

semiclathrate [m3∙mol-1]

∆𝑑𝑖𝑠𝐻0(𝑇) Change in enthalpy accompanying the dissociation reaction of the

semiclathrate [J∙mol-1]

∆𝑑𝑖𝑠𝐶𝑝0(𝑇) change in heat capacity accompanying the dissociation reaction of

the semiclathrate [J∙mol-1 K-1]

𝑇𝑐𝑔𝑟 Temperature of congruent point [K]

𝜇ℎ𝐻,𝛽

Chemical potential per salt molecule in the empty metastable phase

β [J∙mol-1]

Yij Occupancy fraction of cavities type i by the gas molecule of type j

Cij Langmuir constant of the gas molecule of type j and cavities type i

[MPa-1]

k Boltzmann’s constant [1.38 10-23 J∙K-1]

𝑅𝑐𝑒𝑙𝑙 Radius of the cavity [10-10 m]

𝑎𝑖 Radius of spherical core of component i [10-10 m]

r Distance of the guest molecule from the center of the cavity [10-10

m]

xix

𝑤(𝑟) Potential energy function for the interaction between the guest

molecule and the molecules constituting the cavity [J∙mol-1]

a Radius of spherical molecular core [10-10 m]

σ Collision diameter [10-10 m]

ε Minimum energy [J]

P Pressure [MPa]

T Temperature [K]

A1 PSRK constant [-]

Tc,i Critical temperature of i compound [K]

Pc,i, Critical pressure of i compound [MPa]

ωi Acentric factor of i compound [-]

c1, c2, and c3 Adjustable parameters regressed from vapour pressure

experimental data in PSRK [-]

am,k, bm,k, and cm,k Interaction parameters in UNIFAC [-]

F Objective function

NP Number of points

𝑥𝑔𝑎𝑠𝑐𝑎𝑙𝑐 Calculated molar fraction of the gas in liquid phase

𝑥𝑔𝑎𝑠𝑒𝑥𝑝

Experimental molar fractions of the gas in liquid phase

𝛾𝑖 Activity coefficient of compound i in the mixture [-]

𝛾𝑖𝐿𝑅 Long range activity coefficient of compound i in the mixture[-]

𝛾𝑖𝑀𝑅 Middle range activity coefficient of compound i in the mixture[-]

𝛾𝑖𝑆𝑅 Short range activity coefficient of compound i in the mixture[-]

Tcalc Calculated dissociation temperatures of the mixture [K]

xx

Texp Experimental dissociation temperatures of the mixture [K]

1

Introduction

This chapter presents the fundamental background and relevant literature review

associated with the formation of both clathrate hydrates and semiclathrates. A description

of their different structures also is presented in this chapter. Application of gas hydrates as

source of energy and in separation processes through clathrate hydrate and semiclathrate

formation are also presented in this chapter. Finally, a review of the different

methodologies for computing dissociation conditions –pressure and temperature– by

means of thermodynamic models, are briefly discussed in the final sections of this chapter.

1.1 Clathrate hydrates

While the present study is focused on semiclathrates, the fundamental properties of

gas hydrates will be discussed so that semiclathrates can be introduced by way of

comparison to the more well-known clathrates, also known as gas hydrates.

Sir Humphry Davy [1] was the first researcher to investigate hydrate compounds,

when he noticed that and ice-like solid was formed from a mixture of chlorine gas and

water at temperatures greater than the freezing point of water [1]. Eventually, Michael

Faraday in further experiments, was able to report the composition of the chlorine hydrate

[2].

In the most general sense, a hydrate is a compound containing water [3, p. 5]. The

trapping of a molecule in a crystalline structure composed by molecules of another different

compound usually produces clathrates. Clathrates hydrates are therefore, solid crystalline

structures that are composed mainly of water molecules which are referred to as the “host”

molecules. The solid matrix of water molecules form cavities where small guest molecules

can be trapped (“enclathrated”), the guest molecule is typically referred to as “former” [3,

p. 5]. The water molecules are attached to each other by means of hydrogen bonds, which

form a host lattice that is suitable for trapping the host molecule.

The stability of the hydrate structure results from the interaction of the guest

molecule and the water lattice by means of the van der Waals forces. For this reason,

hydrates are unstable in the absence of the guest molecule [4]. As mentioned by Carroll [3,

p. 9], there is no bonding between the host and the guest molecules. Thus, the guest

2

molecule is free to rotate inside the cage. One of the most notable characteristics of hydrates

is that they can be formed at temperatures above the normal freezing point of water [5].

The following conditions need to be meet in order to form hydrates [3, p. 23]:

1. Small guest molecules (<9 Å diameter)

2. The presence of water

3. Relatively low temperature (typically < 323 K)

4. Elevated pressure

1.2 Hydrate structures

Hydrates are known to exist in one of three crystal structures; structure I (sI),

structure II (sII), and structure H (sH). Depending on the arrangement of the water

molecules in the crystal. Structures I and II consist of two types of cavities and structure H

consists of three type of cavities. Table 1.1 presents some of the characteristics of the

different hydrate structures [6, p. 55]. The three different hydrates structures are composed

of five polyhedra as shown in Figure 1.1. Jeffrey [7] proposed a nomenclature for these

polyhedral is: 𝑛𝑖𝑚𝑖,where ni is the number of edges in a face type i, and mi is the number of

faces with ni edges. For example, dodecahedron cages can be seen as twelve-sided

polyhedron with a pentagon for each face (512).

Table 1.1: Geometry of cages (Reproduced from Koh and Sloan [6, p. 55]).

Hydrate crystal

structure

I II H

Cavity Small Large Small Large Small Medium Large

Description 512 51262 512 51264 512 435663 51268

Number of

cavities

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

No. of water

molecules/cavity

20 24 20 28 20 20 36

3

Figure 1.1: Cavities in gas clathrates hydrates: (a) pentagonal dodecahedron (512),

(b) Tetrakaidecahedron (51262), (c) hexakaidecahedron (51264), (d) irregular

dodecahedron (435663), and (e) icosahedron (51268) (Reproduced from Sloan and

Koh [6, p. 54])

1.2.1 Structure I

The unit cell of structure I consists of 46 water molecules, two small 512 (pentagonal

dodecahedron) cavities, and two large cavities 51262 (tetrakaidecahedron), as illustrated in

Figure 1.2. The structure can be filled up with small size guest size molecules (less than 3

Å in molecular radius) such as methane, ethane, carbon dioxide, and hydrogen sulfide [8,

p. 2]. One important thing about cavity type 512 is that it is common to all three type of

clathrate structures and also common to semiclathrates.

1.2.2 Structure II

The unit cell of structure II is formed by 146 water molecules, sixteen small 512

cavities, and eight large cavities 51264 (hexakaidecahedron) as presented in Figure 1.2.The

structure can be filled up with both small, and larger sizes molecules. For example, propane

and isobutane are trapped in the large cavities and molecules such as nitrogen can be

enclathrated in both small or large cavities [8, p. 2].

1.2.3 Structure H

Structure H unit cell consists of 34 water molecules, three small 512 cavities, two

medium size 435663 (irregular dodecahedron) cavities, and one large size 51268

4

(icosahedron) cavity. The small guest molecules usually are enclathrated in small and

medium cavities, whereas molecules larger than 7.4 Å such as 2-methylbutane, 2,2-

methylbutane,neohexane, and cycloheptane enter the large cavity [8, p. 2]. Structure H

requires a small helper molecule such as CH4 or hydrogen sulfide (H2S).

Figure 1.2: Common gas hydrate structures (sI, sII, sH) and the water cage types

that compose the hydrate structure. (Reproduced from Koh and Sloan [9])

1.3 Natural gas hydrates

Formation of clathrate hydrates from natural gas constituents can be seen in the oil

and gas industry. Water is usually associated with natural gas in subsurface rock reservoirs,

therefore, the natural gas that is produced comes out of the reservoir along with water [10].

In general, natural gas is by definition a mixture of hydrocarbons (such as methane, ethane,

propane, etc.) and a few non-hydrocarbons such as hydrogen sulfide (H2S), carbon dioxide

(CO2) nitrogen (N2) etc., and water (H2O) [3, p. 2]. Low-temperatures, high-pressure and

the presence of small gas molecules are the necessary conditions for occurrence of hydrates

during the production of oil and gas.

5

Gas hydrates became relevant to the oil and gas industry in 1934, when

Hammerschmidt discovered that gas hydrates instead of ice were responsible for plugging

pipelines in Canada [11]. Nowadays, gas hydrates still continue to affect the industry as

they cause a risk of blockage in oil and gas pipelines, both onshore and offshore. The main

industrial interest in hydrates resides in preventing their formation and subsequent plugging

of gas lines [12].

Figure 1.3: Gas hydrate plug recovered from a subsea pipeline close to the coast

Atlantic of Brazil [13]

Hydrate plugs grow and harden within minutes, and once the plug is formed it can result

in days or even sometimes weeks of lost production so this is a situation that engineers

want to avoid and that is the reason why hydrates rank as the main flow assurance problem

in oil and gas transportation. If a plug is formed in a pipeline, the problem can be solved

by one of the methods listed below [13]:

Depressurization of the pipeline

Injection of inhibitors at the plug face/annulus

Thermal heating of the pipeline.

Mechanical removal using coiled tubing.

6

Figure 1.4: Methane-Propane double hydrate (Reproduced from Janda [14])

1.3.1 Hydrate inhibition

In order to overcome plugging of gas lines pipes, a common practice in the oil and

gas industry is the usage of inhibitors such as alcohols, glycols and electrolytes [15], whose

main purpose is to lower the chemical potential of water. The effect of inhibitors on the

equilibrium hydrate condition is depicted qualitatively in Figure 1.5. At a specific pressure,

the temperature at which hydrates will form be lower with inhibitors. Similarly, at a

specific temperature, the pressure at which hydrates will form will be higher with inhibitors

than without them. There is also effect due to the concentration of the inhibitor: the more

concentrated it is, the more the curve will shift to left as indicated by the direction of the

arrow in Figure 1.5.

7

Figure 1.5: Qualitative effect of inhibitors on the hydrate equilibrium conditions.

Dashed curve represent the hydrate curve with inhibitor and solid curve represent

the hydrate curve without inhibitor.

1.4 Hydrate promoters

Gas hydrate promoters are chemical additives used in hydrate formation that permit

the hydrate to be formed more rapidly or at more moderate conditions such as lower

pressure and/or higher temperature. The latter type of promoters are known as

thermodynamic promoters. Examples of thermodynamic promoters include

tetrahydrofuran (THF) and cyclopentane. The other fundamental purpose of promoters is

to stimulate crystal growth at higher rates to address the slow kinetics formation of gas

hydrates. This type of promoters are known as kinetic promoters. In addition to accelerate

the hydrate formation, kinetics additives can perform their job without changing the

thermodynamic equilibrium [16]. Surfactant molecules such as sodium dodecyl sulfate

(SDC) have been reported to dramatically increase the kinetics of methane hydrate

formation [17]. Additionally, promoters can be grouped into two groups according to their

impact in the hydrate structure as explained below:

Promoters with no effect on the hydrate structure and are usually

enclathrated in the large cavities of structure II or structure H; e.g.

tetrahydrofuran (THF), cyclobutanone, cyclohexane, and

methylcyclohexane [18].

Pre

ssure

Temperature

Increasing concetration

of inhibitor

8

Promoters that change the hydrate structure by becoming part of it.

Usually, the hydrate structure is broken in order to encage the promoter

molecule, organic salts are the most common type of semiclathrates

promoters, e.g. tetra-n-butyl ammonium salts (TBAX) such as tetra-n-

butyl ammonium bromide (TBAB), tetra-n-butyl ammonium chloride

(TBAC), and tetra-n-butyl ammonium fluoride (TBAF).

One case of the use of promoters for potentially storing in hydrate form is hydrogen

(H2). In this case, pure hydrogen hydrate is formed at very high pressures, however, the

addition of a promoter molecule such as tetrahydrofuran (THF) helps to reduce the pressure

by two orders of magnitude when compared to a pure hydrogen hydrate at the same

temperature [6, p. 72]. The structure formed is type sII, where the hydrogen occupies the

small cavity, and the THF is encaged in the large cavity [19]. On the other hand, if an

organic salt, such as a quaternary ammonium salt (QAS) is employed as a promoter, the

pressure can be reduced from 200 MPa (H2 pure hydrate) to a value quite close to

atmospheric pressure (0.13 MPa) with the use of TBAB at 12°C [20].

Another important difference between THF hydrates and QAS semiclathrates lies

in the fact that THF is very volatile which makes it less attractive for practical applications

because it would require some other secondary separation process to obtain a guest free of

traces of THF [21]. QAS, on the other hand, are non-volatile, pose no fire hazard, and no

known risk to the environment [22, p. 12]. The effect of a hydrate promoter on the

equilibrium hydrate condition is depicted qualitatively in Figure 1.6. At a specific pressure,

the temperature at which hydrates form will be higher with promoters. Similarly, at a

specific temperature, the pressure at which hydrates will form will be lower with promoters

than without them. There is also an effect due to the concentration of the inhibitor: the more

concentrated it is, the more the curve will shift to the right as indicated by the direction of

the arrow in Figure 1.6. The current study will focus on the measurement and prediction of

the depicted promotion effect in the presence of TBAB/TBAC.

9

Figure 1.6: Qualitative effect of promoters on the hydrate equilibrium conditions.

Solid line represents the pure hydrate curve, and the dashed curve is the equilibrium

curve with a promoter.

1.4.1 Semiclathrate hydrates

In the current study, the promoting agents in use are TBAB and TBAC, which leads

to the formation of semiclathrates. Semiclathrates hydrates are crystalline materials similar

in structure to clathrate hydrates in which some cavities encage the promoter molecule, and

the remaining cavities are suitable to trap guest molecules. In 1940, semiclathrates hydrates

were first reported in the literature by Fowler et al [23] when they discovered that the

addition of some quaternary ammonium salts (QAS), such as tetra-n-butyl ammonium

fluoride with water, could form crystals at room temperature. In a further effort, McMullan

and Jeffrey [24] used x-ray diffraction to report dimensions of the unit cells, hydration

numbers, and crystal symmetry. But it wasn’t until 1969, when Jeffrey [25] studied the

structure of these crystals thorough crystallographic and x-ray structural analysis, finally

decided to call them semiclathrates because part of the cage structure is broken in order to

enclose the large tetra-n-butyl ammonium cation while the halogen anion such as bromide

(Br-), chloride (Cl-), fluoride (F-), construct the framework along with the water molecules

through hydrogen bonding [26, 27]. The opposite occurs in clathrate hydrates, where the

Pre

ssure

Temperature

Increasing

concentration

of promoter

10

guest molecule is not physically attached to the lattice but it is rather held by van der Waals

interactions [21].

One important feature related to the structural composition of semiclathrates which

makes them different from pure hydrates is the fact they can form crystals by themselves.

That is, without the need of the stabilizing effect of a guest gas molecule. The polyhedral

cavities share faces, thus forming the following types of cages [28]:

Pentagonal dodecahedron (512), as shown in Figure 1.1

Tetrakaidecahedron (51262), as shown in Figure 1.1

Pentakaidecahedron (51263), as shown in Figure 1.7

Figure 1.7: Cavity in semiclathrate hydrates: Pentakaidecahedron (51263)

(Reproduced from Walsh et al [29])

1.4.1.1 Semiclathrates of TBAB and water

The liquid-solid phase behaviour, in the absence of gases, of the system

TBAB+H2O has been investigated by several authors. Those studies aim to determine the

type of crystalline structures formed and two important characteristics of those structures;

the first one is the determination of the hydration number, which is the number of water

molecules per salt molecule and the second one is the measurement of the congruent

melting point, which is the melting temperature point of the semiclathrate structure at the

stoichiometric composition and atmospheric pressure.

A summary of the results of the studies carried out on the TBAB+H2O system is

presented in Table 1.2. As can be seen from the literature survey, the number of structures

observed varies from study to study. For example, Gaponenko et al [30] reported four

different types of structures whereas Nakayama [31] reported the appearance of a single

structure. However, the most cited study among researchers is the one presented by Oyama

et al [32] , in which two types of structures were identified and named as type A and type

11

B. In the current study, it will not be possible, due to equipment, to verify the observed

structures.

Table 1.2: Experimental studies on TBAB for determining hydration number and

congruent melting points.

Authors Structure name Hydration

number

Congruent melting

temperature/°C

Nakayama [31] Type A 24 12.9

Gaponenko et al [30] TBAB∙24H2O

TBAB∙26H2O

TBAB∙32H2O

TBAB∙36H2O

24

26

32

36

12.4

12.2

11.6

9.5

Oyama et al [32] Type A

Type B

26

38

12

9.9

Shimada et al [27] Type B 38 N/A

The ideal unit cell of type B is composed of six cages type 512, four of type 51262,

and four of type 51263 [27]. Figure 1.8 shows the structure of the type B semiclathrates,

which contains two TBA+cations, 76 water molecules, and two Br- anions that construct

the cage structure along with the water molecules [27]. Figure 1.9 shows that each TBA+

cation is at the centre of a cavity constituted by four cages: two 51262 and two 51263. For

structure type A, the ideal unit cell is formed of ten cages type 512, sixteen of type 51262,

and four of type 51263 [26]. Structure type B allocates six cations, six anions, and 172 water

molecules. In Figure 1.8 and Figure 1.9 the TBA+ cation is represented by the grey four leg

molecule. The guest molecules can be seen in Figure 1.9, where they are presented as

golden spheres. A summary of the main characteristics of structures type A and B is

presented in Table 1.3.

12

Table 1.3: Structural data on TBAB semiclathrates hydrates.

Number of molecules per

structure

Cavity type and number

per structure

Structure Hydration

number

number

H2O TBA+ Br- 512 51262 51263

Type A 26 172 7 7 10 16 4

Type B 38 76 2 2 6 4 4

Figure 1.8: Semiclathrate structure type B structure for TBAB (Reproduced from

Shimada et al [27])

Figure 1.9: Structure around the TBA+ cation in the structure type B for TBAB.

(Reproduced from Shimada et al [27])

According to Shimada et al [27], the dodecahedral cavity (512) can encage small

gas molecules in both type of structures (type A and type B). Figure 1.9 presents the

structure around the TBA+ cation, which is located at the centre of four cages, two of them

51262, and two 51263. The two dodecahedral 512 cages depicted on the right side of the

figure, are filled with two shaded spheres that represent the guest molecules.

13

1.4.1.2 Semiclathrates of TBAC and water

The liquid-solid phase behaviour (in the absence of gases) of the system

TBAC+H2O has been investigated by Aladko and Dyadin [33].The authors report three

hydrate structures, with hydration numbers equal to 32,30,24, and the corresponding

congruent melting points at 14.7°C, 15.1°C, and 15.1°C,respectively. The structure of

TBAC has also been analyzed by Rodionova et al [28] by X-ray diffraction studies, they

reported that all the three structures are formed by ten cages type 512, sixteen of type 51262,

and four of type 51263. A summary of the different type of semiclathrates formed from

TBAC semiclathrates is presented in Table 1.4:

Table 1.4: Structural data on TBAC semiclathrates hydrates structures.

Cavity type and number per structure

Structure

name

Hydration # Congruent

melting

temperature/°C

512 51262 51263

Type I 32 14.7 10 16 4

Type II 30 15.1 10 16 4

Type III 24 15.1 10 16 4

Rodionova et al [28] also reported that small guest molecules can be encaged in the

dodecahedral small cavities (512) for the three type of structures (I, II, and III).

1.5 Potential applications of gas hydrates and semiclathrates

Gas hydrates are often viewed as a problem in the oil and gas industry due to

blockage of gas transmission pipelines. Formation of hydrates contributes to the reduction

of cross sectional area in pipelines, which leads to higher pressure drop in the pipelines

thus increasing cost of compression. Or, in the worst case, hydrate formation leads to the

complete blockage of the pipelines. Many positive application of gas hydrates and

semiclathrates have also been studied and reported in the open literature. These include

energy recovery from natural gas hydrate deposits, the use of hydrates as means of gas

14

storage and transportation, carbon dioxide capture and sequestration, refrigeration cycles,

and separation and recovery of toxic agents and pollutants (H2S and chlorinated agents).

The potential applications that are believed to also be applicable to semiclathrates will be

presented in the following sections.

1.5.1 Natural gas storage and transportation

The idea of storing and transporting gases such as hydrogen or natural gas in the

form of hydrate, or semiclathrate seems to be attractive due to their ability to concentrate

gas. In fact, a cubic meter of hydrate can store up to 160 m3 of methane at standard

conditions [34]. Additionally, the gas in hydrate form is flammable but not explosive which

is an important safety advantage over liquefied natural gas (LNG) and compressed natural

gas (CNG). Javanmardi et al [35] have probed that lower investment in infrastructure and

equipment than LNG, and Kanda et al [36] have verified that the transportation costs of

natural gas as hydrate is 20% lower when compared to LNG for distances lower than 6000

km. Figure 1.10 presents an schematic process flow diagram of production of gas hydrates,

transportation, and final gasification.

Another type of gas molecule than can be stored in the form of hydrates is hydrogen.

Strobel et al presented a literature review paper [37] indicating that hydrogen hydrates has

two clear advantages that make it a potential storage material, and these are:

Water is the only by-product upon dissociation of hydrogen, additionally

water is benign, recyclable, and compatible with hydrogen fuel cells.

The formation and decomposition kinetics of hydrogen hydrates are rapid,

on the order of minutes

Figure 1.10: Complete process of transporting natural gas in the form gas hydrates.

(Reproduced from Giavarini et al [38, p. 145])

15

1.5.2 Separation processes thorough gas hydrate formation

In this section of the present work, the most important experimental studies that

deal with the separation of gases by means of formation of hydrates and semiclathrates are

discussed. The list of gases in the discussion includes: carbon dioxide, methane, hydrogen

sulfide, hydrogen, and nitrogen. Some of them belong to the group of gases called

Greenhouse gases (GHG) which are gaseous compounds that absorb radiation from the

Earth’s surface, the atmosphere itself, and by clouds; this property is hypothesized to

increase the temperature of the atmosphere [39]. Water vapour, carbon dioxide, nitrous

oxide, methane, and ozone are the most important greenhouse gases in the atmosphere.

GHG can be produced by the combustion of fossil fuels, thus there is general interest in

separating these gases from flue gas streams.

1.5.2.1 Separation of carbon dioxide

Capture and sequestration of carbon dioxide (CCS) has become an important

research topic in the scientific community and also in the industry. Several separation

technologies have been studied to separate carbon dioxide from combustion flue gases and

landfills, these include post-combustion processes such as gas absorption with

monoethanol amine (MEA), gas adsorption, and membrane separation. However, a novel

alternative to the aforementioned processes for gas separation is thorough gas hydrate

crystallization. Due to the difference in affinity of several hydrate formers to be

enclathrated, carbon dioxide can be trapped selectively in the hydrate/or semiclathrate

phase, whereas the concentration of other gases can be increased in the in the flue gas

current. Once the capture process is completed, the CO2 in the hydrate phase can be

recovered by dissociating the crystals. The survey review presented in Table 1.5 only

focuses on experimental studies carried out to date on separation of CO2 via

clathrates/semiclathrates in the presence of promoters. One interesting study from Duc et

al [40] consisted in the simulation of a steady state plant for CO2 capture with TBAB

hydrates. The study was performed using a commercial process simulator (PRO II); six

separation stages were used, and each stage was composed of one compressor and one

crystallizer, the temperature in each crystallizer was held constant at 283 K and the pressure

16

was varied from 7.5 bar to 50 bar in the last stage. The molar fraction of CO2 at the inlet

and at the outlet of the plant was 35% and 3%, respectively. According to the authors the

cost of production per ton of CO2 was estimated to be US $25.

Table 1.5: Experimental studies on clathrate/semiclathrate hydrate for carbon dioxide

systems in combination with hydrate promoters.

Author(s) Gas system promoter Focus of study

Linga et al [41] CO2+N2 THF Dissociation conditions.

Hydrate formation rates.

Fan et al [42] CO2+CH4

CO2+C2H6

CO2+N2

CO2+CH4+C2H6+N2

ethylene

glycol

Dissociation conditions.

Fan et al [43] CO2+N2 TBAB

TBAF

Hydrate Formation Rate.

Separation efficiency

studies.

Kang et al [44] CO2+N2 THF Dissociation conditions

Li et al [45] CO2+N2 TBAB+DTAC Initial pressures on the

induction time of the

hydrate formation.

Separation efficiency.

Ma et al [46] H2+CH4,H2+N2+CH4,

CH4+C2H4

THF Dissociation conditions.

Vapour and hydrate phase

compositions.

Meysel et al

[47]

CO2+N2 TBAB Dissociation conditions.

Vapor phase composition.

Kim et al [49] CO2+H2 TBAB Dissociation conditions.

Hydrate formation kinetics.

Raman Spectroscopy.

17


Li et al [48] CO2+H2 TBAB Dissociation conditions.

Li et al [50] CO2+N2 cyclopentane Hydrate formation kinetics.

Vapour and hydrate phase

compositions.

Mohammadi et

al [51]


Belandria et al

[52]


Mayoufi et al

[53]

CO2 TBAB

TBAC

TBANO3

TBPB


Dissociation Enthalpies.

Kang et al [54] CO2+N2 THF Dissociation conditions.

Thermodynamic modeling.

Seo et al [55] CO2 THF

Propylene

oxide

1,4-dioxane


Duc et al [40] CO2+N2 TBAB Dissociation conditions.

Lin et al [56] CO2 TBAB Dissociation conditions.

Mohammadi et

al [57]

CO2,N2,CH4,H2 TBAB Dissociation conditions.

Lin et al [58] CO2 TBAB+TBPB Dissociation conditions.

Li et al [59] CO2 TBAB Hydrate formation rate.

Mayoufi et al

[60]

CO2 TBPB Dissociation conditions.

18


Ye et al [61] CO2 TBAC

TBPC


Visual morphology of

structures.

Equilibrium data of pure

semiclathrates at

atmospheric pressure.

Wang et al [62] CO2+CO

CO2+CO+H2

TBAB Dissociation conditions.

Vapour phase composition

at equilibrium.

Deschamps et

al [63]

N2,CO2

N2+CO2

CH4+CO2

TBAB Dissociation conditions.

Ricaurte et al

[16]

CO2+CH4 SDC,THF

SDC+THF


Kinetic studies.

Kumar et al

[64]

CO2 SDC

DTAC

Kinetic studies.

Sun et al [65] CO2 TBAC Dissociation conditions.

Summary comments from the literature review:

Linga et al [41] reported that there is a considerable reduction in the equilibrium

condition pressures in the presence of THF compared to hydrate formation in pure water

and proposed a three stages separation. The process operates at 2.5 MPa and 273.75 K. The

authors comment that the operating pressure is less than the pressure required in the

absence of THF and hence the compression costs are reduced from 75 to 53% of the power

produced for a 500 MW power plant.

Fan et al [43] reported that TBAB and TBAF accelerate semiclathrate formation

and reduce the pressure at the same temperature. The semiclathrate formation rate is higher

with TBAF than TBAB. The authors report that CO2 can be enriched to 90.4 mole % from

19

a feed concentration of 16.6 mole % in a binary mixture of CO2 and N2 when TBAF is used

as promoter and two separation stages are used.

Kang et al [44] presented a hydrate-based gas separation process for recovering

CO2 from flue gas with THF as hydrate promoter. The authors report that it was verified

that the process makes it possible to recover more than 99 mole % of CO2 from the flue

gas.

Li et al [45] reported that CO2 can be purified from 17.0 mol % to 99.2 mol % with

the two-stage hydrate separation process that involves Tetra-n-butyl ammonium bromide

(TBAB) + dodecyl trimethyl ammonium chloride (DTAC) as semiclathrate promoting

agents. The pressures range for the separation stages were 0.66 MPa–2.66 MPa and

temperature range of 274.95 K–277.15 K.

Li et al [48] carried out experiments that probe that TBAB can reduce the hydrate

formation pressure of CO2+H2 hydrate. The authors also report that the dissociation

pressure of the semiclathrate formed, decreases with the increase in concentration of the

TBAB at a certain temperature.

Belandria et al [52] conducted experiments with the mixture CO2+N2+TBAB and

reported that experimental dissociation pressures were lower for forming semiclathrates

than those for gas hydrates at a given temperature and they generally decreased as the

TBAB concentration increased in the range studied their work.

Mohammadi et al [57] measured semiclathrates of CO2, N2, CH4, and H2 in the

presence of TBAB and reported a comparison between their experimental data with

publications and found some disagreements.

1.5.2.2 Separation of methane

Methane is also greenhouse gas which can remain in the atmosphere up to 15 years

[39] and is 20 times more effective in trapping heat than carbon dioxide [39]. Methane can

be emitted from hydrates/semiclathrates reservoirs, landfills, animal waste, enteric

fermentation, rice cultivation, animal waste, coal mining, and natural gas

mining/production. Therefore, use of gas hydrates or semiclathrates for capturing CH4, is

20

an alternative which has been investigated. Table 1.6 summarizes the most relevant

experimental studies conducted

Table 1.6: Experimental studies on clathrate/semiclathrate hydrate for methane systems in

combination with hydrate promoters


Zhang et al [66] CH4+N2+O2 THF Dissociation conditions.

Hydrate composition.


Kondo et al [67] CH4+C2H6+C3H8 THF Dissociation conditions.

Sun et al [68] CH4+C2H6 THF Dissociation conditions.

Raman spectroscopy.


Ma et al [69] CH4+C2H6 THF Dissociation conditions.

Vapour and hydrate composition.


Sun et al [70] CH4+N2 TBAB

TBAB+SDS


Gas storage capacity.

Acosta et al [39] CH4+CO2 TBAB Dissociation conditions.

Compositions vapour phase.

Fan et al [71] CH4+CO2 TBAB

TBAC

TBAF


Sun et al [72] CH4 TBAB Dissociation conditions.

Sun et al [73] CH4 TBAC Dissociation conditions.

Zhong et al [74] CH4+N2+O2 TBAB Dissociation conditions.

Vapour phase composition.

21


Zhang et al [66] conducted experiments which used THF to promote CH4 hydrates.

The authors found that formation conditions were shifted to lower pressures. The higher

the promoter concentration, the more pronounced the promotion of THF. THF was probed

to change the high pressure conditions of pure hydrate to milder ones.

Sun et al [70] studied the dissociation conditions of the TBAB+CH4+N2. The

results show that TBAB reduced the formation pressure of the semiclathrate. The

experiments showed that CH4 can be concentrated from CH4 and N2 mixed gas via

semiclathrate formation. The addition of sodium dodecyl sulfate (SDS) to the TBAB

solution resulted in decreased the reaction time when compared to TBAB solutions.

Fan et al [71] studied the semiclathrate dissociation conditions of tetra-n-butyl

ammonium halide (TBAB, TBAC, TBAF)+CO2+CH4 and reported that best promoter was

TBAF at 285 K. The dissociation pressures of the gas mixture in TBAB/TBAC/TBAF were

3.74 MPa, 2.76 MPa and 0.94 MPa, respectively, compared to 7.30 MPa of the mixture in

pure water at the same temperature.

1.5.2.3 Separation of hydrogen sulfide

Hydrogen sulfide (H2S), which is a gaseous compound typically present in biogas,

or in waste streams during the processing of gas natural, can be corrosive and toxic gas.

Thus, it is also susceptible to be captured by means of hydrate technology. Table 1.7

presents a brief summary of experimental research regarding semiclathrates of TBAB

formed with H2S.

Table 1.7: Experimental studies on semiclathrate hydrate for hydrogen sulfide systems in


Author(s) Gas system promoter Study

Kamata et al [75] H2S+CO2+CH4 TBAB Dissociation conditions.

Mohammadi et al [76] H2S,CH4 TBAB Dissociation conditions.

22


Kamata et al [75] found in their experiments that up to 90% of the H2S in the vapor

phase was removed during the semiclathrate formation. Mohammadi et al [76] showed

with their experiments that there is a significant promotion effect of TBAB for the

formation of semiclathrates of H2S.

1.5.2.4 Separation and storage of hydrogen

Gas hydrates are very attractive for storage, and transportation of hydrogen because

of their ability to store large volumes of gas. However, the formation pressure of the pure

hydrate is extremely high. Therefore, the possibility exists for using hydrate promoters

such as TBAB and TBAC in order to reduce the pressure from values which are extremely

high e.g., 200 MPa at 0°C, to values lower than 0.5 MPa [77]. The most important

experimental studies carried out on separation of H2 are listed in Table 1.8.

Table 1.8: Experimental studies on clathrate/semiclathrate hydrate for hydrogen systems

in combination with hydrate promoters


Florusse et al [78] H2 THF x-ray diffraction.

Raman spectroscopy.

Deschamps et al [79] H2 TBAB

TBPB

TBAC


Hashimoto et al [20] H2 TFH

TBAB

Raman spectroscopic studies.


Karimi et al [80] H2 TBAOH Dissociation conditions.

Hashimoto et al [81] H2 TBAB Raman spectroscopic studies.


Du et al [82] H2 TBANO3 Dissociation conditions.

Fujisawa et al [83] H2 TBPB Dissociation conditions.

Raman spectroscopic studies.

Hashimoto et al [77] H2 TBAB Raman spectroscopic studies.


23


Deschamps et al [79] demonstrated through experimental work that the dissociation

temperatures of TBAC+H2 and tetra-n-butyl phosphonium bromide (TBPB)+H2

semiclathrates were very close to ambient at 15.0 MPa. Their results demonstrated that the

amount of hydrogen that could be stored in TBAC and TBPB semiclathrates was twice

higher than that stored in TBAB system.

Du et al [82] measured the phase conditions of Tetra-n-butyl Ammonium Nitrate

(TBANO3)+H2 semiclathrates. It was found that the addition of TBANO3 favored the

dissociation conditions of the semiclathrate to shift to higher temperatures and lower

pressures.

Hashimoto et al [77] completed experiment to determine experimental dissociation

conditions and Raman spectroscopy for semiclathrates of TBAB+H2 and concluded that

only the empty small cages of TBAB semiclathrates were occupied by one H2 molecule.

Additionally the authors demonstrated that the semiclathrate required lower pressure

conditions to form when compared with gas hydrates.

1.5.2.5 Separation of nitrogen

Nitrogen is one the components of flue gas emissions from conventional

combustion processes and is also a component that sometimes needs to be removed from

natural gas. Hydrates or semiclathrates can also be used to separate nitrogen from stack

emissions. Table 1.9 summarizes experimental studies carried out with hydrate promoters.

Table 1.9: Experimental studies on clathrate/semiclathrate hydrate for nitrogen systems in



Lee et al [84] N2 TBAB

TBAF

Dissociation conditions

Arjmandi et al [85] N2 TBAB Dissociation conditions

24


Lee et al [84] reported in their paper that the presence of TBAB or TBAF shifted

the equilibrium conditions of the semiclathrates to higher temperature and lower pressure

regions when compared with those of the pure N2 hydrate.

1.6 Review of gas hydrate and semiclathrate phase equilibrium models

In the present section, the most important thermodynamic models for predicting the

dissociation conditions of hydrates (in section 1.6.1) and dissociation conditions of

semiclathrates will be discussed. In general, all the model have to satisfy the criteria for

phase equilibrium in a solid-liquid–vapour system which is given by the following

conditions [6, p. 285]:

Temperature and pressure of the phases involved in the equilibria are

equal.

Chemical potential of each component in each of the phases are equal.

The Gibbs free energy is a minimum.

1.6.1 Models to calculate the phase equilibria of hydrates

Due to the regular structure of gas hydrates, the distribution and number of cavities

is uniform in each unit cell of a particular structure, thermodynamic properties can be

represented by statistical thermodynamics [86]. The first and most well-known

thermodynamic model for prediction of phase equilibrium involving gas hydrates was

derived by van der Waals and Platteuw (vdWP) [87]. Their model is similar to that of

Langmuir for gas adsorption into a solid. In the vdWP model the following assumptions

are made:

It is assumed that the encaged gas molecule can rotate and vibrate freely in

the cavity.

Each cavity can only entrap one guest molecule.

The interaction between entrapped gas molecules can be neglected.

Cavities are spherical.

25

The guest molecules are small enough to prevent distortion of the hydrate

lattice.

Van der Waals and Platteuw [87] used the Lennard-Jones-Devonshire to describe the guest

molecule-cavity interaction. The authors calculated dissociation pressures for nine

different gases at 273 K as reported in their original paper [87].

The vdWP theory was modified by Parrish and Prausnitz [12] in 1972 when they

presented a methodology for computation of gas-hydrate equilibria in multicomponent

systems. The methodology proposed by Parrish and Prausnitz also uses the Kihara potential

for describing the interaction between the guest and host molecules. It also requires a

reference gas for computations of dissociation pressures for different lattice structures and

different temperatures. In their model, the fugacity coefficients for the gases were

calculated with the modified Redlich-Kwong EoS [88]

Later, Holder et al [89] presented a modification to the method proposed by Parrish

and Prausnitz [12] so that the reference hydrate curve is eliminated from the model by

introducing reference properties for each type of hydrate structure. The model proposed by

Holder has produced a standard methodology for most of the further thermodynamic

models which predict the phase behaviour of gas hydrates. The mathematical details for

the vdWP model will be given in Chapter 3.

1.6.2 Models to calculate phase equilibria of semiclathrates of hydrates

Models for calculation of phase equilibria of semiclathrates hydrates have been

only recently undertaken. The first attempt was done by Mohammadi et al [90] who

proposed a model for semiclathrates. The model is a feed-forward artificial neural network

which is able to estimate clathrate and semiclathrate hydrate dissociation conditions for

H2+H2O (pure hydrate) and H2+TBAB (semiclathrate) systems. In the publication, the

authors state that the predicted and the experimental data are in acceptable agreement.

Being based in neural-networks. Their model is not straight-forward to incorporate into

further computational routines.

Joshi et al [91] published a model for representing equilibria of semiclathrates of

CH4, CO2, and N2 +TBAB based on a thermodynamic model developed by Chen and Guo

26

[92, 93], and initially used for gas hydrates systems. Joshi et al [91] assumed that

semiclathrate formation is a two-step mechanism, the first step is a quasi-chemical reaction

to form the semiclathrate structure followed by a second step which involved the adsorption

of the guest molecule in the semiclathrate cavity. The methodology does not rely on the

vdWP theory directly but on a semi empirical correlation which calculates the occupancy

factor. The proposed methodology is able to reproduce satisfactorily experimental data in

the literature. However, the authors do not report any type of numerical error for their

simulations.

Eslamimanesh et al [94] proposed a model for predicting dissociation conditions of

semiclathrates hydrates of CO2, CH4 and N2+TBAB. For modeling the hydrate phase, the

authors used the vdWP theory combined with the empirical correlation developed by

Parrish and Prausnitz [12] for the computation of the Langmuir constants. The Peng-

Robinson EoS [95], along with the Mathias-Copeman alpha function [96] was used for the

calculation of fugacity coefficients of the vapour phase. The activity coefficient of water

was determined by the use of Non-Random Two-Liquid (NRTL) activity coefficient model

and an empirical correlation for calculating the activity coefficient of TBAB. The authors

report that discrepancy between calculated and experimental values were in the range of

8% to 11% measured as %AARD (average relative deviation). The model of Eslamimanesh

et al [94] contains a large number of adjustable parameters as well as a correlation for the

vapour pressure of TBAB that does not seem to have any physical meaning. During early

stages of the present research work, the model proposed by Eslamimanesh was attempted

to reproduce in early stages of the present work but no results were obtained when using

their fitting parameters. Despite of the popularity of Eslamimanesh model and numerous

citations in other peer reviewed journals, the model seems to have several faults. For

example, Verrett et al [97] also have reported that Eslamimanesh’s model does not seem

to produce adequate results.

Liao et al [98] have also presented a study for modeling phase behavior of

semiclathrates hydrates of TBAB+CH4, TBAB+CO2, TBAB+N2, TBAB+CO2+N2, and

TBAB+CH4+N2. The authors proposed a model which is based on the work of Chen and

Guo [92, 93], which is a two-step hydrate formation mechanism that is similar to that

27

proposed by Joshi et al [91]. The fugacity of the vapour phase is calculated with the Patel-

Teja [99] EoS, the activity coefficient of TBAB is calculated with an empirical correlation

proposed by Eslamimanesh et al [94], and the activity coefficient of water is calculated

with another empirical correlation. Results from the simulations are between 5% and 9%

AARD when compared to experimental data.

Shi and Liang [100] proposed a thermodynamic model based on the vdWP theory

for semiclathrates formed with TBAB, TBAC, TBAF aqueous solutions and two different

single guest gases: CH4 and CO2. The Peng-Robinson EoS and electrolyte-Non-Random

Two-Liquid (e-NRTL) were used to compute the fugacity of the vapour phase compounds

and the activity coefficients of species in the aqueous phase, respectively [94]. The

proposed model is very similar to the one proposed by Eslamimanesh et al [94]. The authors

report that AARD% is between the range of 5% and 12%.

Paricaud [101] has recently presented a thermodynamic model which is based on

reaction equilibrium and it is used to determine the dissociation temperatures of

semiclathrates hydrates of TBAB+CO2. The model incorporates the statistical associating

fluid theory with variable range for electrolytes (SAFT-VRE) [102] for modeling all fluid

phases (liquid and vapor) involved in the calculation and the vdWP theory for describing

the semiclathrate hydrate phase. Paricaud’s model is derived from the minimization of the

Gibbs energy under the premise that the composition of the semiclathrate is fixed. The

Langmuir constants in Paricaud’s model [101] are calculated with an empirical correlation

developed originally by Parrish and Prausnitz [12]. This approach is different from the

rigorous way of calculation of the Langmuir constants. In Paricaud’s model, the

dissociation temperature of the semiclathrate hydrate is calculated by performing a vapor-

liquid-hydrate three-phase equilibrium calculation at given pressure and feed composition.

The average relative deviation (% AARD) obtained with the Paricaud [101] model is about

10%.

Fukumoto et at [103] have applied Paricaud’s model [101] to predict the

dissociation conditions of semiclathrates of CO2 made with TBAB, TBAC, TBAF, and

TBPB. The SAFT-VRE EoS [102]was used to describe the properties of the fluid phases.

28

The major contribution of Fukumoto’s paper is the development of a methodology to

predict the fusion enthalpies and the congruent melting point of semiclathrates hydrates.

Kwaterski and Herri [104] proposed a semiclathrate hydrates based on the model

previously presented by Paricaud [101]. In this method, the authors used the Soave-

Redlich-Kwong [105] EoS for computing the fugacities in the vapor phase, Henry’s law

for determining the molar fraction of the gas in the liquid phase and the electrolyte NRTL

[106] activity coefficient model to describe the liquid phase. The semiclathrate hydrate

phase is calculated with the vdWP theory in combination with the Kihara potential [107]

for modeling the guest-host interaction in the semiclathrate hydrate structure. Kwaterski

and Herri’s model was applied to the TBAB+CH4 system. The deviation of the calculated

values from the experimental data reported by the authors as average relative deviations

(%AARD) was found to be in the range from 8% to 44%. In a private communication

(Appendix E), Kwaterski admitted to having made a mistake in the computation of one

their adjustable parameters.

Verrett et al [97] have recently published a paper modeling the phase equilibria of

CO2/CH4 +TBAB semiclathrates based on the model presented by Eslamimanesh [94]. The

model uses the Trebble-Bishnoi [108] EoS for computing the gas fugacity, the e-NRTL

model [106] to calculate the activity coefficients in the liquid phase, and the vdWP theory

[87] for modeling the hydrate phase. The authors reported an average absolute relative error

(% AARE) of 5% in the case of CO2, whereas, in the case of CH4, calculated error was

22%. As was the case with the original work of Eslamimanesh et al [94], Verrett et al [97]

includes empirical parameters that seem to lack physical meaning.

Babaee et al [109] have recently presented a model to predict semiclathrate

dissociation conditions of the system argon+TBAB which is based on the model of Joshi

et al [91]. The model uses 9 adjustable parameters to make predictions. The authors report

a value of AARD equal to 0.2%.

A summary of the literature review presented in the field of thermodynamic

modeling of semiclathrates is presented in Table 1.10:

29

Table 1.10: Summary of thermodynamic models currently available for semiclathrates hydrates

Author

Thermody

namic

principle

Models used to describe phases

Treatment of

Vapor-liquid

equilibrium

Method for computation

of Langmuir constants

Total

Adjustable

parameters

Joshi et

al [91]

Phase

equilibria

Vapour phase: SRK EoS [105]

Liquid phase: empirical correlation for water

activity

Solid phase: Chen & Guo model [93]

Not calculated Antoine type expression

based on work of Chen &

Guo model [93]

7

Paricaud

[101]

Reaction

equilibria

Vapour and liquid phase : SAFT-VRE EoS

[102]

Solid phase: vdWP theory [87]

Vapour-liquid

flash solved

Empirical correlation based

on Parrish & Prausnitz

expression [12]

4

Kwaters

ki et al

[104]

Reaction

equilibria

Vapour phase: SRK EoS [105]

Liquid phase: e-NRTL activity coefficient

model [106]


Henry’s

constants for

solubility

calculation in

the liquid

phase

Kihara potential [110] 5

30

Eslamim

anesh et

al [94]

Phase

equilibria

Vapor phase: PR EoS [95]

Liquid phase: NRTL model for non-

electrolyte compounds. Empirical correlation

for TBAB activity coefficient.


Henry’s

constants for

solubility

calculation in

the liquid

phase



expression [12]

9

Liao et

al [98]

Phase

equilibria

Vapour phase: PT EoS [99]

Liquid phase: empirical correlation for water

activity

Solid phase: Chen & Guo model [93]

Vapour-liquid

flash solved

Antoine type expression

based on work of Chen &

Guo model [93]

9

Shi et al

[100]

Phase

equilibria

Vapour phase: PR EoS [95]

Liquid phase: e-NRTL activity coefficient

model


Henry’s

constants for

solubility

calculation in

the liquid

phase

Modified Parrish &

Prausnitz expression [12]

10

Verrett

et al

[97]

Phase

equilibria

Vapour phase: TB EoS [108]

Liquid phase: e-NRTL [106] activity

coefficient model


Vapour-liquid

flash solved



expression [12]

7

31

Babaee

et al

[109]

Phase

equilibria

Vapour phase: Patel-Teja EoS [99]

Liquid phase: Empirical correlation Solid

phase: Chen & Guo model [93]

Henry’s

constants for

solubility

calculation in

the liquid

phase


Present

work

Reaction

equilibria

Vapour phase: PSRK EoS [111]

Liquid phase: LIFAC activity coefficient

model


Vapour-liquid

flash solved


32

1.7 Scope of current study

There are two main objectives to be achieved in this present study, the first one is

to measure the semiclathrate equilibrium conditions for pure xenon and pure argon in the

presence of pure aqueous solutions of TBAB and TBAC at different concentrations. The

second one is to develop and use a thermodynamic model to predict the equilibrium

conditions of the mixtures involved in the present study.

Thus, the main focus of this research are:

To measure the solid–vapor–liquid equilibrium conditions of semiclathrates formed

in aqueous solutions of TBAB of xenon and argon at two different concentrations

(wTBAB = 0.05 and 0.20), and TBAC using the same gas and the same concentration

conditions.

To correlate the experimental data with a thermodynamic model, that has the

following characteristics:

Minimum number of fitting parameters.

Rigorous treatment of the vapour-liquid equilibrium.

Capability of representing the solid-liquid equilibria (SLE) of the

TBAB+H2O and TABC+H2O systems in the absence of gases.

Rigorous treatment of the Langmuir constants in the computation of the

triple phase equilibrium (solid-liquid-vapour).

To produce a thermodynamic model easily extendable to mixtures (not part

of the scope of this work)

33

Experimental apparatus, methodology and results

In the present chapter a description of the material and apparatus used to form and

dissociate semiclathrates from TBAB+xenon, TBAB+argon, TBAC+xenon, and

TBAC+argon are presented. Additionally, the experimental procedure used to form and

dissociate the crystals is presented as well. Finally, the method to determine the three phase

equilibrium point (solid-liquid-vapour) for semiclathrates is discussed. Results of this

chapter have already been published in a peer reviewed journal [34], and will subsequently

be used in Chapter Three in order to validate the thermodynamic model developed in the

present work.

2.1 Apparatus

The experimental apparatus itself consists of an isochoric sapphire cell, which was

manufactured by Insaco, Inc. of Pennsylvania, and is 114.3 mm in length, inner diameter

of 19.1 mm, and 31.8 mm of outer diameter [34]. All surfaces of the tube are polished to a

80/50 finish, considered a standard optical polish [22, p. 35]. The sapphire tube has also

been rated to operate at working pressures of 20 MPa at temperatures in the range of 250-

400 K. The sapphire tube is placed inside a concentric Plexiglas cylinder. Both plexiglass

cylinder and sapphire tube are held on the top and bottom by stainless steel flanges. Three

spanning studs are used to fix the sapphire tube and the two flanges together. The reactor

has been built in such way that only minimal axial and radial stress is transmitted to the

sapphire cell when it is assembled [112]. In order to provide agitation inside the sapphire

tube, a magnetic stirred rod is placed on the bottom flange, and it is driven by a rotating

magnet, which is placed under the reactor [112]. Figure 2.1 presents a mechanical drawing

of the cell and flanges.

34

Figure 2.1: Schematic drawing from the reactor (Reproduced from Portz et al

[112])

The space between the sapphire tube and the outer Plexiglas cylinder is filled with

an aqueous ethylene glycol solution (50/50 %v/v), which circulates at constant volumetric

flow rate between the reactor assembly and a programmable cooling bath (Refrigerated

Circ unit, Model 1267P distributed by VWR Inc.). The reactor is insulated with fiber glass

to minimize heat transfer from the surroundings. Temperatures in both the reactor and the

cooling jacket are measured using two type-T thermocouples (Omega Engineering Inc.).

The accuracy of the temperature measurement is ± 0.1 K. The pressure in the reactor is

measured using a 4-20 mA differential pressure transducer (Alphaline Pressure

Transmitter, Rosemount Instruments LTD, Calgary, Canada) with a span of 11 MPa and

an uncertainty of 0.25% of the full span, or ±27 kPa. The signals from the pressure

transducer and the thermocouples are fed through a Fieldpoint data acquisition unit

(National Instruments FP-TC-120 for thermocouples and FP-AI-110 for the pressure

transducer) from which they are subsequently transmited via a USB connection (National

Instruments FP-1601 10/100 Mbps Ethernet Interface) into the computer. The pressure

transducer was calibrated against a dead-weight tester (Chandler Engineering), and the

thermocouples were also calibrated in the range of 269-293 K using a precision

35

thermometer (F250 Honeywell). The isochoric reactor, the cooling bath, and the stirring

mechanism are held on the top of a granite table whose design minimizes all vibrations due

to its mass. Figure 2.2 shows a schematic for the experimental apparatus.

Figure 2.2: Flow diagram of the experimental apparatus (Reproduced from Portz

et al [112])

2.2 Materials

The ultrahigh purity argon (supplier stated purity: yargon ≥ 0.99999) and ultralight

purity xenon (supplier stated purity: yxenon ≥ 0.99999) were purchased from Praxair

Technology Inc. 50 wt. % tetra-n-butyl ammonium bromide solution (TBAB) aqueous

solution, analytical grade , and solid tetra-n-butyl ammonium chloride (TBAC) with a

purity x ≥ 0.99,were purchased from Aldrich. Deionized water (Millipore Simplicity water

purification system, which produces ultrapure water with a resistivity of 18.2 MΩ at 298

K) was used to prepare the aqueous solutions of (0.05 and 0.20) wTBAB and (0.05 and 0.20)

wTBAC, respectively. Table 2.1 list some physical properties for TBAB and TBAC. Aqueous

solutions were prepared following the gravimetric method [22, p. 33], using an analytical

balance (mass uncertainty ± 0.0001 g).

36

Table 2.1: TBAB and TBAC physical properties

Name TBAB TBAC

Formula C16H36NBr C16H36NCl

Linear formula [CH3(CH2)]4N(Br) [CH3(CH2)3]4N(Cl)

Structural formula

MW/g∙mol-1 322.37 277.92

2.3 Experimental procedure

The current study uses the well-established isochoric method for determining the

SLV equilibrium [112, 47, 39, 62]. The equilibrium point in an isochoric reactor can be

determined by either visual observation or by monitoring pressure versus temperature (pT)

chart. In the present work, the equilibrium point will be determined via pT plot because it

has been successfully tested during previous experiments that have been conducted with

the present experimental apparatus [112, 47, 39, 62].

2.3.1 Preparation and start up of the experiments

The reactor is rinsed three times with deionized-distilled water in order to clean the

sapphire tube and the magnetic stirred rod and then twice with the experimental liquid

solution. Then, the system is purged with the experimental gas three times at 0.2 MPa to

ensure no gas or air is left from previous experiments. After cleaning, approximately 15

mL of liquid solution is injected into the sapphire cell and allowed to equilibrate to the

initial system temperature. At this point, the magnetic stirred is turned on to agitate the

solution. Subsequently the experimental gas is added to the system until the point (a) in

Figure 2.3 is reached. The fully loaded system (experimental gas+liquid solution) is left

for 12 hours at 298 K to guarantee that the liquid solution is completely saturated with gas.

The saturation process can be observed when the pressure falls until the aqueous solution

is saturated with the experimental gas. This condition is confirmed when the pressure

37

remains unchanged for at least two hours. At this point (point (b) in Figure 2.3), the cooling

begins a rate of -10 K/h until the semiclathrate is initiated (point (c) in Figure 2.3). The

formation of semiclathrates can be inferred by an abrupt change in the slope of the pT

trajectory that is displayed on the data acquisition computer, and it is also confirmed

visually from a color change in the liquid solution from clear to white.

Semiclathrate formation is allowed to continue for approximately one hour which

is a period of time where the pressure inside the reactor stabilizes after the semiclathrate

structure has been filled with gas (point (d) in Figure 2.3). Afterwards, the system is heated

at rate of +1K/h until the temperature is within 2 K of the expected equilibrium dissociation

point (point (e) in Figure 2.3). At this point, the heating rate is reduced to 0.20 K/h. The

slow heating is continued until temperature has returned to the initial system temperature.

The slow heating starts in point (e) and ends in point (f) as shown in Figure 2.3. The point

at which the slope of the pT trajectory changes sharply is considered to be the equilibrium

point (point (f) in Figure 2.3).

Figure 2.3: A typical pressure versus temperature trajectory. (Reproduced from

Meysel et al [47])

38

The exact location of the dissociation point is determined by finding the coordinates

of the intersection of the cooling and heating curves, which are usually assumed to be

straight lines [113, 39, 114].An example of dissociation of point determination from a pT

trajectory curve is shown in Figure 2.4.

Figure 2.4: Dissociation point determination for a gas semiclathrate from Argon

on aqueous solution of 5 wt. %TBAB

2.4 Experimental results

Previous experimental studies conducted on the apparatus used in the present study

have demonstrated that determination of equilibrium conditions of semiclathrates can be

accurately measured [39, 47, 62]. Thus, there was no need to validate the experimental set

up to determine its suitability regarding the formation of semiclathrates. In particular, the

appropriate heating rate [39]. For all the four sets of data (argon+TBAB, argon+TBAC,

xenon+TBAB, and xenon+TBAC), the concentration of TBAB and TBAC was limited to

no more than w=0.20. This is due to the earlier observations that the magnetic stirrer is

unable to provide sufficient agitation at higher salt concentrations because viscosity

increases dramatically as indicated in previous studies in the apparatus used in the present

study [22, p. 82].

39

2.4.1 Incipient equilibrium conditions for TBAB and TBAC semiclathrates formed from

pure argon gas

The incipient equilibrium conditions for the formation of semiclathrates from argon

in the presence of 5, and 20 wt. % aqueous solutions of TBAB and TBAC are reported in

Table 2.2, and Table 2.3 respectively. The equilibrium values for TBAB and TBAC are

plotted in Figure 2.5 , and Figure 2.6, respectively. In the aforementioned charts, the

experimental results for pure hydrates, obtained by Marshall et al [115], are also presented.

The results obtained for argon in combination with the two promoters are consistent

with trends observed in previous studies related to semiclathrates [39, 47, 62]. At a given

temperature, the pressure required to form semiclathrates is lower than the pressure

required to form pure hydrates, and the pressure to form semiclathrates at a given

temperature decreases as the salt concentration increases. 5 wt. % TBAB semiclathrates

form at roughly 1/10th of the pressure, given the same temperature, whereas 20 wt. %

TBAC semiclathrates form at roughly 1/24th of the pressure required to form a pure hydrate.

Additionally, if the comparison is made between the two promoters at the same

concentration, it is observed that TBAC semiclathrates form at a slightly lower pressure

than TBAB semiclathrates formed with argon as shown in Figure 2.7.

40

Table 2.2: Incipient Equilibrium Conditions for TBAB semiclathrates formed from pure

argon gas.

wTBAB T/K (± 0.1 K) P/kPa (± 27 kPa)

0.05 284.0 2429

285.1 3052

285.9 3532

286.4 3972

286.9 4386

287.4 4684

0.20 286.5 2082

287.2 2504

287.7 3086

287.9 3303

288.5 3573

289.0 4183

290.8 6114

41


argon in aqueous solutions of TBAB. ●,wTBAB=0.05; ■,wTBAB=0.20; +,argon gas

pure hydrates [115].

1000

10000

100000

278 280 282 284 286 288 290 292 294

P/k

Pa

T/K

42

Table 2.3: Incipient equilibrium conditions for TBAC semiclathrates formed from pure

argon gas

wTBAC T/K (± 0.1 K) P/kPa (± 27 kPa)

0.05 283.5 1892

284.2 2084

284.8 2355

285.6 2870

286.3 3454

287.2 4126

0.20 288.1 1569

289.0 2044

289.9 2471

290.0 2446

290.1 2836

290.7 3282

291.6 4003

43


argon in aqueous solutions of TBAC. ●,wTBAC=0.05; ■,wTBAC=0.20; +,argon gas

pure hydrates [115].

1000

10000

100000

278 280 282 284 286 288 290 292 294

p/k

Pa

T/K

44

Figure 2.7: Incipient equilibrium conditions for TBAB and TBAC semiclathrates

formed from argon. ●,wTBAB=0.05; ■,wTBAB=0.20; ▲, wTBAC=0.05; ▬,

wTBAC=0.20;+,argon gas pure hydrates [115].

900

9000

90000

282 284 286 288 290 292 294

p/k

Pa

T/K

45

2.4.2 Incipient equilibrium conditions for TBAB and TBAC semiclathrates formed from

pure xenon gas

Incipient equilibrium conditions for the formation of semiclathrates from xenon in

the presence of 5, and 20 wt. % aqueous solutions of TBAB and TBAC are reported in

Table 2.4, and Table 2.5, respectively. The equilibrium values for TBAB and TBAC are

plotted in Figure 2.8, and Figure 2.9, respectively. Experimental results for pure hydrates

of xenon in water in two different ranges of temperatures [116, 117] are also presented in

the plots for the sake of the comparison. At a given temperature, the pressure required to

form semiclathrates is lower than the pressure to form pure hydrates. Compared to the

results for semiclathrates formed from argon, the results for semiclathrates formed in the

presence of xenon exhibit a much smaller degree of hydrate promotion. In general, at a

given temperature they form at roughly 1/2 of the pressure required to form pure hydrates.

Also, when formed in the presence of xenon Figure 2.8 and Figure 2.9 show that the effect

of TBAB and TBAC concentration is relatively weak. In a previous study, Jin et al [118]

formed TBAB semiclathrates with solutions of wTBAB=0.05 and wTBAB=0.20, but at

temperature ranges outside of those in the current study. In the case of wTBAB=0.05, Figure

2.8 shows that the data from the current study appears to be consistent with the data of Jin

et al [118].Due to the accuracy of the pressure transducer available for the current study

(±27 kPa), attempt was made to replicate the low-pressure data obtained by Jin et al [118]

Jin et al [118] also noted that TBAB semiclathrates formed from xenon undergo a

phase transition at roughly 287 K, depending on the concentration of TBAB. For the

current study, facilities were not available to conclusively establish the structure of the

TBAB semiclathrates formed in the presence of xenon. However, an examination of the

data for wTBAB=0.20 raises speculation that a phase change might be occurring since the

first three points and the last three points appear to be on different trajectories. Again,

though, with the available apparatus it was not possible to establish the structure of the

TBAB semiclathrates formed in the presence of xenon.

46

Table 2.4: Incipient Equilibrium Conditions for TBAB Semiclathrates Formed from Pure

xenon Gas


0.05 287.6 522

291.0 696

292.6 786

294.1 895

295.1 960

296.0 1069

297.0 1161

298.2 1285

300.8 1495

301.8 1602

302.7 1801

0.20 287.8 354

288.3 398

289.1 443

289.6 497

290.4 543

47


xenon in aqueous solutions of TBAB. ●,wTBAB=0.05(this work); ◊ wTBAB=0.05

[118]; ■,wTBAB=0.20(this work); ▲, wTBAB=0.20 [118]; +,xenon gas pure hydrates

[116]; ▬,xenon gas pure hydrates [117].

10

210

410

610

810

1010

1210

1410

1610

1810

2010

270 275 280 285 290 295 300 305

P/k

Pa

T/K

48

Table 2.5: Incipient Equilibrium Conditions for TBAC Semiclathrates Formed from Pure

xenon Gas


0.05 287.0 527

290.2 661

291.5 682

292.5 769

293.2 867

294.5 939

295.2 997

295.2 1009

297.1 1152

0.20 287.0 266

287.8 310

288.1 352

288.9 395

289.2 431

290.1 465

290.0 469

49


xenon in aqueous solutions of TBAC. ●,wTBAC=0.05; ■,wTBAC=0.20; +,xenon gas

pure hydrates [116]; ▬,xenon gas pure hydrates [117].

10

210

410

610

810

1010

1210

1410

1610

1810

285 287 289 291 293 295 297 299

P/k

Pa

T/K

50


formed from argon in aqueous solution of TBAB, and TBAC respectively.

■,wTBAB=0.05; ◊ wTBAB=0.2; ▲,wTBAC=0.05; ●, wTBAC=0.20

6.1

6.2

6.3

6.4

6.5

6.6

6.7

6.8

6.9

0.00342 0.00344 0.00346 0.00348 0.0035 0.00352 0.00354

ln(P

)

T-1 / K-1

51


formed from xenon in aqueous solution of TBAB, and TBAC respectively.

■,wTBAB=0.05; ◊ wTBAB=0.2; ▲,wTBAC=0.05; ●, wTBAC=0.20; dashed trend lines

represent TBAB and solid lines represent TBAC.

5.2

5.4

5.6

5.8

6

6.2

6.4

3.28 3.3 3.32 3.34 3.36 3.38 3.4 3.42 3.44 3.46 3.48 3.5

ln(P

)

(x103) T-1 / K-1

52

Figure 2.10 and Figure 2.11 are plots of ln(P) versus 1/T, which are constructed for the

sake of further examining the experimental data as well as for estimating the heat of

formation. Figure 2.10 shows that the data for TBAB and TBAC semiclathrates formed in

the presence of argon exhibits a very pronounced effect of salt concentration, whereas

Figure 2.11 shows that when formed in the presence of xenon the effect of the salt

concentration is relatively weak.

2.4.3 Heat of dissociation for TBAB and TBAC semiclathrates in the presence of argon

and xenon

The enthalpy of dissociation of a semiclathrate is the amount of heat required to

melt a crystal structure from a three phase system. One method for estimating the heat of

dissociation from experimental data is the Clapeyron approach [3, p. 234]. A Clapeyron-

type of equation is applied to the three-phase locus. The equation is given as follows:

∆𝐻𝑓 = −𝑧𝑅𝑑(𝑙𝑛𝑃)

𝑑(1/𝑇)

(2.1)

Where ∆𝐻𝑑𝑖𝑠 is the enthalpy of dissociation, z is the compressibility factor for the

gas phase, and R is the universal gas constant. In order to estimate the enthalpy of

dissociation, a plot of ln(p) versus 1/T is required to estimate the slope, and once this term

is calculated from Figure 2.10 and Figure 2.11, the enthalpy of fusion is calculated from

Equation (2.1).The heats of fusion are presented in Table 2.6 and Table 2.7.

53

Table 2.6: Estimated heat of dissociation ∆𝐻𝑑𝑖𝑠 for TBAB semiclathrates in the presence

of argon and xenon.

Gas wTBAB ∆𝐻𝑑𝑖𝑠/𝑘𝐽 ∙ 𝑚𝑜𝑙−1 ∙ 𝐾−1

Argon 0.05 132.5

0.20 169.4

Xenon 0.05 55.5

0.20 114.7

Table 2.7: Estimated heat of dissociation ∆𝐻𝑑𝑖𝑠 for TBAC semiclathrates in the presence

of argon and xenon.

Gas wTBAC ∆𝐻𝑑𝑖𝑠/𝑘𝐽 ∙ 𝑚𝑜𝑙−1 ∙ 𝐾−1

Argon 0.05 147.9

0.20 187.9

Xenon 0.05 57.2

0.20 126.6

As shown in Table 2.6 and Table 2.7, the energy to dissociate the semiclathrates formed

from TBAC is consistently higher than the energy required to dissociate semiclathrates

from TBAC. Values tend to be closer in the case of xenon (when the comparison is made

between the two promoters). However, this difference becomes more acute in the case of

argon, where TBAC heat of dissociation are 10% higher than those from TBAB

semiclathrates. The other difference that can be inferred from the calculated values for both

gases is that enthalpy of fusion increases with the concentration of the promoter.

54

Presentation of the thermodynamic model

In the present chapter, a thermodynamic model is proposed for representing the

phase equilibria of argon and xenon semiclathrates in the presence of two different

quaternary ammonium salts (QAS), TBAB and TBAC. Further predictions are also made

for TBAB and TBAC semiclathrates formed in the presence of CO2, CH4, N2, and H2

because experimental data is available in the open literature. The model is based on a

reaction equilibrium and it is solved by applying the Gibbs energy minimization. Initially

the model is developed for describing the liquid-solid equilibrium between the salt and

water, and subsequently it is extended to compute the dissociation temperature at given

pressure of a semiclathrate hydrate. Finally, the model developed for the two phase

equilibria is extended to the three phase equilibrium (liquid-solid-vapour) with the support

of the vdWP theory, in order to compute the dissociation temperature given the pressure

and global composition of the system.

3.1 Motivation for the new thermodynamic model

In the initial stages of this study, the objective was merely to use an existing model

to correlate the data obtained in Chapter 2. However, after examining the existing models,

it was felt that it was possible to improve upon what had been done previously. In the

literature survey presented in Chapter 1, it is noted that there have been at least seven

different attempts made to describe the phase behaviour of semiclathrates hydrates.

However, all attempts are based on two original works: the first one, was the one presented

by Paricaud [101] and the second, was developed by Eslamimanesh et al [94].

The model presented by Paricaud [101] requires only four adjustable parameters

therefore becoming the model with the least amount of fitting parameters among all the

models available, at the time of writing the present work. Paricaud’s approach relies on the

use of SAFT-VRE EoS to model the aqueous phase and the van der Waals and Platteuw

(vdWP) theory for describing the solid phase. However, the implementation of SAFT-VRE

EoS demands expertise in SAFT molecular theory and it is not familiar to the majority of

chemical engineers, the only other weakness observed in Paricaud’s model is the empirical

treatment of the Langmuir constants. In Paricaud’s particular case of treatment of the

55

Langmuir constant are presented as an empirical correlation proposed originally by Parrish

and Prausnitz [12]. In the present work, the Langmuir constants will be computed using a

rigorous methodology that involves the use of the Kihara potential.

The approach presented by Kwaterski et al [104] is based on the model of Paricaud

[101] and requires five adjustable parameters to correlate the phase equilibria, the extra

parameter comes from the use of a rigorous treatment of the Langmuir constant, in this

model, the authors use the Kihara potential for describing the interactions between the

guest molecule and the cage and this type of treatment requires three adjustable parameters.

Rather than performing a rigorous flash calculation, the model of Kwaterski et al

[104] uses Henry’s law for computing the solubility of the gases in the liquid phase.

Additionally, the vapour phase was assumed to be composed only by gas molecules and

the presence of water in the vapour phase is neglected. Therefore, it is felt that this

condition is in need of improvement. Because of this fact, the present work will use an EoS

in order to remove the simplification involved when using Henry’s law. Another

disadvantage seen in the Kwaterski’s model is the employment of the SRK EoS [105] for

describing the vapour phase because this equation of state is not very accurate when dealing

with polar compounds such as CO2 and H2O [119, p. 53]. In particular, it is inaccurate

when computing vapour pressure. This is one of the main reasons for selecting PSRK EoS

[111] in the current study. The PSRK EoS incorporates the Mathias-Copeman alpha

function [96] which improves significantly the vapour pressure calculation over the

original alpha function of the SRK EoS. Additionally, it is important to mention again that

in the publication of Kwaterski et al [104] the reported values of the Gibbs free energy at

the congruent point were wrongly reported and confirmed by the authors in a private

communication with the author of the present study (see Appendix E).

Models involving equality of fugacities typically involve more fitting parameters

such as the case of Eslamimanesh et al [94], in which 9 different adjustable parameters are

used to predict the three phase equilibria involving semiclathrates. The other disadvantage

of Eslamimanesh’s model [94] is the fact that it involves the computation of the vapour

pressure of the semiclathrate promoter that is present in the solution. This approach is not

very realistic because the promoter does not exist as a pure liquid in equilibrium with its

56

vapour. Eslamimanesh et al [94] used an empirical correlation but offered no description

of its origin. Another important disadvantage of modeling the equilibria by equality of

fugacities is the inability to represent the liquid-solid equilibria (in the absence of gases) of

promoter+water systems such as TBAB+H2O and TABC+H2O. Due to the aforementioned

reasons it is felt that the reaction equilibrium approach is better than the equality of

fugacities approach, and that is the reason for selecting it in the present work.

Thus, in the present work, a rigorous flash calculation will be performed for

describing the vapour-liquid equilibrium (VLE) of the electrolyte solution. The electrolyte

extension of the Predictive-Soave-Redlich-Kwong (PSRK) EoS has been selected for

modeling the vapour-liquid equilibria, in order to account for the non idealities due to

presence of electrolytes in the liquid phase. Another advantage of PSRK EoS, is the

improvement of the calculation of the vapour pressures of polar compounds such as water,

when compared to conventional EoS such as Peng-Robinson [95] or Soave-Redlich-

Kwong [105]. PSRK EoS also provides a single, continuous model for the vapour and

liquid phases.

In the first section of this chapter, the thermodynamic model for describing the

liquid solid equilibria is presented. In the same section, the model already developed for

the liquid-solid equilibria is extended to the solid semiclathrate phase. The second section

presents the parameter regression of the model. Parameters for computing the vapour liquid

equilibria are estimated. Parameters for the liquid-solid equilibria have also to be regressed

and finally, parameters for the solid-vapour-liquid equilibria are obtained. In the last

section of the chapter the results obtained from the model are presented.

3.2 Development of a new modelling approach for semiclathrates.

In this section, the theoretical model for describing the vapour-liquid-semiclathrate

hydrate (SVLE) phase equilibria is presented. The model is divided in two parts, in the first

part, a model for describing the liquid-solid equilibria is introduced and then, in the second

part, the liquid-solid modelling is combined with the vdWP theory in order to be extended

to the three phase equilibria. The liquid-solid equilibrium model is used to regress

parameters for the VLH model. The approach used in this model is based on reaction

57

equilibrium and it is solved by the minimization of the total Gibbs energy function, as

presented originally by Paricaud [101] and then modified by Kwaterski and Herri [104].

The current modelling approach will combine several “pieces”. Figure 3.1

illustrates the relationship between the various components in the current modelling

approach; each piece will be discussed in detail in the following sections. The first thing

that has to be completed is to determine the values of enthalpy and volume from the

regression of experimental values of liquid-solid equilibrium, for this stage the LIFAC

model is used to compute activity coefficient in the liquid phase. Once the parameters are

obtained the second step is to determine the missing interaction parameters in the PSRK

EoS which in this particular study are xenon+H2O and argon+H2O because these

parameters are not available in the PSRK tables [111]. When the parameters are calculated,

the following step is to compute the VLE in electrolyte solutions with the help of PSRK

and for doing this it is required to use the LIFAC model as it is shown in the flow diagram.

The next step is to regress the Kihara potential parameters and for doing this, it is required

to have the parameters from the liquid-solid equilibrium, the LIFAC model, and the PSRK

for electrolytes, and experimental data with the phase equilibrium between the solid

semiclathrate, the liquid and the vapour phase. At the end of the entire process, a total of

five parameters are obtained, two of these parameters come from the SLE and the three

remaining come from the SVLE. With these five parameters already determined, the

modelling can be done. Figure 3.1 is presented below:

58

Figure 3.1: Overall process diagram

3.2.1 Liquid-solid equilibrium

The dissociation of a semiclathrate hydrate, in a binary system constituted by a

single salt (CA) and water, can be seen as a combined chemical reaction and phase

equilibrium that is written as:

𝐶𝜈𝑐𝐴𝜈𝑎

∙ 𝜈𝑤𝐻2𝑂(𝐻) ⇌ 𝜈𝑤𝐻2𝑂(𝐿) + 𝜈𝑐𝐶𝑧𝑐+

(𝐿) + 𝜈𝑎𝐴|𝑧𝑎−|(𝐿)

(3.1)

Where 𝑧𝑐+ and 𝑧𝑎

− are the total charge of the cation and anion, respectively,𝜈𝑐 is the number

of C cations, 𝜈𝑎 is the number of A anions, and 𝜈𝑤 is the number of water molecules. The

equilibrium is described by the standard constant K(T) which is expressed in terms of the

composition as follows:

59

𝐾(𝑇) =(𝑎𝑤

𝐿 )𝜈𝑤(𝑎𝑐𝐿)𝜈𝑐(𝑎𝑎

𝐿)𝜈𝑎

𝑎ℎ𝐻 exp ( −

1

𝑅𝑇∫ (𝜈𝑤𝑉𝑤

𝐿 + 𝜈𝑐𝑉𝑐∞,𝐿 + 𝜈𝑎𝑉𝑎

∞,𝐿𝑃0

𝑃

− 𝑉ℎ𝐻) 𝑑𝑃)

(3.2)

Where the subscript h stands for the semiclathrate compound 𝐶𝜈𝑐𝐴𝜈𝑎

∙ 𝜈𝑤𝐻2𝑂, 𝑎ℎ𝐻 is the

activity of the hydrate in the solid phase. 𝑎𝑤𝐿 , 𝑎𝑐

𝐿 and 𝑎𝑎𝐿 denote the activity of the water,

cation, and anion in the liquid phase, respectively. 𝑉𝑤𝐿 and 𝑉ℎ

𝐻 are the molar volume of

pure liquid water and semiclathrate, respectively. 𝑉𝑐∞,𝐿

and 𝑉𝑎∞,𝐿

stand for the partial molar

volume of the cation and anion at infinite dilution of the salt, respectively. Due to the fact

that the semiclathrate phase is a pure phase, the activity value for the semiclathrate is unity:

𝑎ℎ𝐻 = 1

(3.3)

The activities of the compounds present in the liquid phase are expressed as the product of

the mole fraction and the activity coefficient. The activity of water is defined as:

𝑎𝑤𝐿 = 𝑥𝑤

𝐿 𝛾𝑤𝐿

(3.4)

The activities of the ionic compounds are defined as follows:

𝑎𝑐𝐿 = 𝑥𝑐

𝐿𝛾𝑐𝐿

(3.5)

𝑎𝑎𝐿 = 𝑥𝑎

𝐿𝛾𝑎𝐿

(3.6)

In the present work, the activity coefficients in the liquid phase (𝛾𝑤𝐿 , 𝛾𝑐

𝐿 , 𝛾𝑎𝐿) are evaluated

with the LIFAC model [120]. The reason for this choice is because LIFAC is the activity

60

coefficient model used in the extension of PSRK for dealing with electrolytes. LIFAC is

suitable for predictions of vapor liquid equilibria in electrolyte systems using the group

contribution concept. Equations (3.4), (3.5), and (3.6) combined with Equation (3.2), and

rearranged to give:

ln 𝐾(𝑇) = 𝜈𝑤 ln(𝑥𝑤𝐿 𝛾𝑤

𝐿) + 𝜈𝑐 ln(𝑥𝑐𝐿𝛾𝑐

𝐿) + 𝜈𝑎 ln(𝑥𝑎𝐿𝛾𝑎

𝐿)

−1



∞,𝐿 − 𝑉ℎ𝐻)

𝑃0

𝑃

𝑑𝑃

(3.7)

The chemical equilibrium constant can be calculated from the standard Gibbs energy of

dissociation:

∆𝑑𝑖𝑠𝐺0(𝑇) = −𝑅𝑇 ln 𝐾(𝑇) (3.8)

The total dissociation Gibbs energy of the system is given by the following expression

[119, p. 535]:

∆𝑑𝑖𝑠𝐺0(𝑇) = ∑ 𝜈𝑖𝜇𝑖0

𝑖

= 𝜈𝑤𝜇𝑤0,𝐿 + 𝜈𝑐𝜇𝑐

0,𝐿 + 𝜈𝑎𝜇𝑎0,𝐿 − 𝜇ℎ

0,𝐻 (3.9)

Where 𝜇ℎ0,𝐻

is the standard chemical potential of the semiclathrate in the hydrate

phase, 𝜇𝑤0,𝐿

, 𝜇𝑐0,𝐿

, and 𝜇𝑎0,𝐿

are the standard chemical potential of water, cation and anion in

the liquid phase, respectively. Combining Equations (3.7), (3.9) and Equation (3.8), leads

to the following expression:

61

−1

𝑅𝑇(𝜈𝑤𝜇𝑤

0,𝐿 + 𝜈𝑐𝜇𝑐0,𝐿 + 𝜈𝑎𝜇𝑎

0,𝐿 − 𝜇ℎ0,𝐻) = 𝜈𝑤 ln(𝑥𝑤

𝐿 𝛾𝑤𝐿) + 𝜈𝑐 ln(𝑥𝑐

𝐿𝛾𝑐𝐿)

+ 𝜈𝑎 ln(𝑥𝑎𝐿𝛾𝑎

𝐿)

−1




𝑃0

𝑃

𝑑𝑃

(3.10)

The standard chemical potential for the solid semiclathrate and water in the liquid phase

are given by Equation (3.11) and Equation (3.12), respectively:

𝜇ℎ0,𝐻(𝑇) = 𝜇ℎ

𝐻(𝑇, 𝑃) + ∫ 𝑉ℎ𝐻

𝑃0

𝑃

𝑑𝑃 (3.11)

𝜇𝑤0,𝐿(𝑇) = 𝜇𝑤

𝐿 (𝑇, 𝑃) + ∫ 𝑉𝑤𝐿

𝑃0

𝑃

𝑑𝑃 (3.12)

The standard chemical potential for cations, 𝜇𝑐0,𝐿(𝑇), and anions, 𝜇𝑎

0,𝐿(𝑇),are defined by

Equations (3.13) and (3.14), respectively:

𝜇𝑐0,𝐿(𝑇) = ∫ 𝑉𝑐

∞,𝐿𝑃0

𝑃

𝑑𝑃 + { lim𝑥𝑤→1

[𝜇𝑐𝐿(𝑇, 𝑃, 𝑥𝑤) − 𝑅𝑇 ln(𝑥𝑐)]}

(3.13)

𝜇𝑎0,𝐿(𝑇) = ∫ 𝑉𝑎

∞,𝐿𝑃0

𝑃

𝑑𝑃 + { lim𝑥𝑤→1

[𝜇𝑎𝐿(𝑇, 𝑃, 𝑥𝑤) − 𝑅𝑇 ln(𝑥𝑎)]}

(3.14)

The temperature dependence of the standard chemical potential of compound i can be

expressed in terms of enthalpy and isobaric heat capacities as follows:

𝜇𝑖0,𝐿(𝑇) =

𝜇𝑖0,𝐿(𝑇0)

𝑇0+ 𝐻𝑖

0,𝐿(𝑇0) (1

𝑇−

1

𝑇0) − ∫ (∫ 𝐶𝑝,𝑖

𝐿𝑇′

𝑇0

(𝑇′′)𝑑𝑇′′)𝑇

𝑇0

𝑑𝑇′

𝑇′2

(3.15)

62

𝐶𝑝,𝑖𝐿 is the isobaric heat capacity of compound i in the liquid phase, 𝐻𝑖

0,𝐿 is the standard

enthalpy of compound i in the liquid phase. Equation (3.15) has to be used for each

compound present in the liquid phase (cation, anion, and water), which leads to the

following expression:

1

𝑅𝑇0(𝜇ℎ

0,𝐻(𝑇0) − 𝜈𝑤𝜇𝑤0,𝐿(𝑇0) − 𝜈𝑐𝜇𝑐

0,𝐿(𝑇0) − 𝜈𝑎𝜇𝑎0,𝐿(𝑇0))

+1

𝑅(

1

𝑇−

1

𝑇0) (𝐻ℎ

0,𝐻(𝑇0) − 𝜈𝑤𝐻𝑤0,𝐿(𝑇0) − 𝜈𝑐𝐻𝑐

0,𝐿(𝑇0)

− 𝜈𝑎𝐻𝑎0,𝐿(𝑇0))

+1

𝑅∫ (∫ (𝜈𝑤𝐶𝑝,𝑤

𝐿 (𝑇′′) + 𝜈𝑐𝐶𝑝,𝑐𝐿 (𝑇′′) + 𝜈𝑎𝐶𝑝,𝑎

𝐿 (𝑇′′)𝑇′

𝑇0

𝑇

𝑇0

− 𝜈𝑤𝐶𝑝,ℎ𝐻 (𝑇′′)) 𝑑𝑇′′)

𝑑𝑇′

𝑇′2

= 𝜈𝑤 ln(𝑥𝑤𝐿 𝛾𝑤



𝐿)

−1




𝑃0

𝑃

𝑑𝑃

(3.16)

At this point, it is necessary to define the following molar

quantities: ∆𝑑𝑖𝑠𝑉0(𝑇), ∆𝑑𝑖𝑠𝐻0(𝑇), and ∆𝑑𝑖𝑠𝐶𝑝0(𝑇) that represent the change in the property

accompanying the dissociation reaction of the semiclathrate:

∆𝑑𝑖𝑠𝐻0(𝑇) = 𝜈𝑤𝐻𝑤0,𝐿(𝑇0) + 𝜈𝑐𝐻𝑐

0,𝐿(𝑇0) + 𝜈𝑎𝐻𝑎0,𝐿(𝑇0)−𝐻ℎ

0,𝐻(𝑇0) (3.17)

∆𝑑𝑖𝑠𝑉0(𝑇) = 𝜈𝑤𝑉𝑤𝐿 + 𝜈𝑐𝑉𝑐

∞,𝐿 + 𝜈𝑎𝑉𝑎∞,𝐿 − 𝑉ℎ

𝐻 (3.18)

∆𝑑𝑖𝑠𝐶𝑝0(𝑇) = 𝜈𝑤𝐶𝑝,𝑤

𝐿 (𝑇) + 𝜈𝑐𝐶𝑝,𝑐𝐿 (𝑇) + 𝜈𝑎𝐶𝑝,𝑎

𝐿 (𝑇) − 𝐶𝑝,ℎ𝐻 (𝑇)

(3.19)

63

By substituting Equations (3.9), (3.17), (3.18), and (3.19) into Equation (3.16), it leads to

the following expression:

∆𝑑𝑖𝑠𝐺0(𝑇)

𝑅𝑇0+

∆𝑑𝑖𝑠𝐻0(𝑇)

𝑅𝑇(1 −

𝑇

𝑇0) −

1

𝑅∫ (∫ (∆𝑑𝑖𝑠𝐶𝑝

0(𝑇))𝑇′

𝑇0

𝑑𝑇′′)𝑇

𝑇0

𝑑𝑇′

𝑇′2

= 𝜈𝑤 ln(𝑥𝑤𝐿 𝛾𝑤



𝐿)

−1

𝑅𝑇∫ (∆𝑑𝑖𝑠𝑉0(𝑇))

𝑃0

𝑃

𝑑𝑃

(3.20)

Equation (3.20) can be simplified by assuming that both ∆𝑑𝑖𝑠𝐶𝑝0(𝑇) and ∆𝑑𝑖𝑠𝑉0(𝑇) are

independent of temperature and pressure, respectively. With these assumptions, Equation

(3.20) can be written as follows:

∆𝑑𝑖𝑠𝐺0(𝑇0)

𝑅𝑇0+

∆𝑑𝑖𝑠𝐻0(𝑇0)

𝑅𝑇(1 −

𝑇

𝑇0) +

∆𝑑𝑖𝑠𝐶𝑝0(𝑇)

𝑅(1 + ln (

𝑇0

𝑇) −

𝑇0

𝑇)

+∆𝑑𝑖𝑠𝑉0(𝑇)

𝑅𝑇(𝑃 − 𝑃0) + 𝜈𝑤 ln(𝑥𝑤


𝐿𝛾𝑐𝐿)


𝐿) = 0

(3.21)

Equation (3.21) is the final equilibrium condition between the liquid electrolyte solution

and the solid that describes the semiclathrate. Note that Equation (3.21) is valid for the

situation in which the semiclathrate is formed in the absence of gases. However, certain

additional parameters in Equation (3.21) will be required when gases are present in the

mixture. The temperature at which the semiclathrate is dissociated under atmospheric

pressure (P0) and stoichiometric conditions, is called the congruent melting temperature,

whose compositions (molar fractions) at this point can be determined from the

stoichiometric composition of the semiclathrate phase, as follows:

For water:

64

𝑥𝑤(𝑠𝑡),𝐻 = 𝑥𝑤

𝐿 =𝜈𝑤

(𝜈𝑐 + 𝜈𝑎 + 𝜈𝑤) (3.22)

For cations:

𝑥𝑐(𝑠𝑡),𝐻 = 𝑥𝑐

𝐿 =𝜈𝑐

(𝜈𝑐 + 𝜈𝑎 + 𝜈𝑤) (3.23)

For anions:

𝑥𝑎(𝑠𝑡),𝐻 = 𝑥𝑎

𝐿 =𝜈𝑎

(𝜈𝑐 + 𝜈𝑎 + 𝜈𝑤) (3.24)

When Equation (3.21) is evaluated at the congruent melting point

(𝑇0 = 𝑇𝑐𝑔𝑟; 𝑃 = 𝑃0), the value of ∆𝑑𝑖𝑠𝐺0(𝑇0) can be obtained:


𝑅𝑇0= − (𝜈𝑤 ln(𝑥𝑤

(𝑠𝑡),𝐻𝛾𝑤𝐿) + 𝜈𝑐 ln(𝑥𝑐

(𝑠𝑡),𝐻𝛾𝑐𝐿)

+ 𝜈𝑎 ln(𝑥𝑎(𝑠𝑡),𝐻𝛾𝑎

𝐿))

(3.25)

In Equation (3.25), the activity coefficients are evaluated at the congruent melting point

composition (as calculated from Eq.(3.22),(3.23), and(3.24)) by means of the e-NRTL

activity coefficient model.

In order to calculate the values of ∆𝑑𝑖𝑠𝐻0(𝑇0) and ∆𝑑𝑖𝑠𝑉0(𝑇), it is necessary to regress the

experimental values from the solid-liquid phase equilibrium for the systems H2O+TBAB

and, H2O+TBAC.This is necessary because although these two parameters have already

been regressed and presented in the original publication of Paricaud [101], the values were

regressed using SAFT-VRE [102]. On the other hand, Kwaterski and Herri [104] used e-

NRTL [106]. In the present work LIFAC [120] is used for describing the liquid phase.

Thus, it is necessary to regress a new set of parameters which are presented in Section 3.3.4

of the present chapter.

65

3.2.2 Vapour–liquid-hydrate equilibrium

The results of the previous section were extended by Paricaud [101] by adding the model

of vdWP in order to describe the chemical potential of the semiclathrate that has been

formed in the presence of gases.

The chemical potential of the semiclathrate hydrates is derived by using the same

hypotheses as those made by van der Waals and Platteuw in their original work [101, 87].

The list of assumptions is presented below:

Position of host molecules (water and TBA+, Br-/Cl-) are fixed in a molecular lattice

(see Figure 1.9).

Cavities in the lattice are not distorted by gas molecules.

Cavities are assumed to be spherical.

Cavities only can trap one gas molecule.

Guest-guest interactions are neglected.

Quantum effects are neglected.

Paricaud [101] showed that in the case of semiclathrates, the chemical potential of the

semiclathrate, 𝜇ℎ𝐻,𝐹

,is given by the following expression:

𝜇ℎ𝐻,𝐹 = 𝜇ℎ

𝐻,𝛽+ ∑ 𝑛𝑖 ln (1 − ∑ 𝑌𝑖𝑗

𝑁𝑔𝑎𝑠

𝑗=1

)

𝑁𝑐𝑎𝑣

𝑖=1

(3.26)

Where 𝜇ℎ𝐻,𝛽

is the chemical potential per salt molecule in the empty metastable phase β, ni

is the number of cavities of type i per salt molecule, Yij is occupancy fraction of cavities

type i by the gas molecule of type j. 𝜇ℎ𝐻,𝛽

and Yij are defined by Equation (3.27), and Yij are

given by the following expressions:

𝜇ℎ𝐻,𝛽

= 𝜈𝑐𝜇𝑐𝐻,𝛽

+ 𝜈𝑎𝜇𝑎𝐻,𝛽

+ 𝜈𝑤𝜇𝑤𝐻,𝛽

(3.27)

66

𝑌𝑖𝑗 =𝐶𝑖𝑗𝑓�̂�

1 + ∑ 𝐶𝑖𝑗𝑓�̂�𝑁𝑔𝑎𝑠

𝑗=1

(3.28)

Where Cij is the Langmuir constant,𝑓𝑗̂ is the fugacity of molecule j in the mixture, which in

the present study is computed via PSRK EoS. If it is assumed that Langmuir constants are

only a function of temperature, and the cavities trapping the gas molecules are spherical,

Cij can be expressed as [121]:

𝐶𝑖𝑗 =4𝜋

𝑘𝑇∫ 𝑒−𝑤(𝑟)/𝑘𝑇𝑟2𝑑𝑟

𝑅𝑐𝑒𝑙𝑙−𝑎𝑖

0

(3.29)

Where k is the Boltzmann’s constant , 𝑅𝑐𝑒𝑙𝑙 is the radius of the cavity, 𝑎𝑖 is the radius of

spherical core of component i ,r is the distance of the guest molecule from the center of the

cavity, and 𝑤(𝑟) is the potential energy function for the interaction between the guest

molecule and the molecules constituting the cavity [122]. McKoy and Sinanogly [123]

suggested using the Kihara potential function [110, 107] to represent this interaction.

Typically, the Kihara potential is employed for the guest molecule-cavity interactions and

the expressions for its calculation are given by Equations (3.30) and (3.31):

𝑤(𝑟) = 2𝑧휀 [𝜎12

𝑅𝑐𝑒𝑙𝑙11 ∙ 𝑟

(𝛿10 +𝑎

𝑅𝑐𝑒𝑙𝑙𝛿11) −

𝜎6

𝑅𝑐𝑒𝑙𝑙5 ∙ 𝑟

(𝛿4 +𝑎

𝑅𝑐𝑒𝑙𝑙𝛿5) ]

(3.30)

𝛿𝑁 =

(1 −𝑟

𝑅𝑐𝑒𝑙𝑙−

𝑎𝑅𝑐𝑒𝑙𝑙

)−𝑁

− (1 +𝑟

𝑅𝑐𝑒𝑙𝑙−

𝑎𝑅𝑐𝑒𝑙𝑙

)−𝑁

𝑁

(3.31)

Where z is the coordination number of the cavity a, σ and ε are the radius of spherical

molecular core, collision diameter and minimum energy, respectively [121]. Finally, the

subscript N can take the values of 4,5,10 or 11. The Kihara potential parameters a, σ and ε

must be regressed using hydrate equilibrium data.

67

At equilibrium the Gibbs free energy is a minimum; or dG=0.Thus the equilibrium

condition for describing the chemical and phase equilibrium in the three phase system (S-

V-L) can be expressed in terms of the change in the Gibbs free energy during the

dissociation reaction as follows:

∆𝑑𝑖𝑠𝐺 = ∑ 𝜈𝑖𝜇𝑖

𝑖

= 𝜈𝑤𝜇𝑤𝐿 + 𝜈𝑐𝜇𝑐

𝐿 + 𝜈𝑎𝜇𝑎𝐿 − 𝜇ℎ

𝐻 (3.32)

Combining Equations (3.26), (3.12),(3.13),and (3.14) with Equation (3.32), leads to the

following equation:


𝑅𝑇=

1

𝑅𝑇(𝜈𝑤𝜇𝑤

0,𝐿(𝑇) + 𝜈𝑐𝜇𝑐0,𝐿(𝑇) + 𝜈𝑎𝜇𝑎

0,𝐿(𝑇) − 𝜇ℎ𝐻,𝛽(𝑇))

−1




𝑃0

𝑃

𝑑𝑃


𝐿) + 𝜈𝑤 ln(𝑥𝑤𝐿 𝛾𝑤


𝐿)

− ∑ 𝑛𝑖 ln (1 − ∑ 𝑌𝑖𝑗

𝑁𝑔𝑎𝑠

𝑗=1

)

𝑁𝑐𝑎𝑣

𝑖=1

= 0

(3.33)

Equation (3.15) can be evaluated for each of the chemical potential terms present in

Equation (3.33), and assuming that ∆𝑑𝑖𝑠𝐶𝑝0(𝑇) and ∆𝑑𝑖𝑠𝑉0(𝑇) are temperature and pressure

independent, respectively, the following expression for the equilibrium condition is

obtained [101]:

68


𝑅𝑇=


𝑅𝑇0+


𝑅𝑇(1 −

𝑇

𝑇0)

+ ∆𝑑𝑖𝑠𝑉0(𝑇0, 𝑃0)

𝑅𝑇(𝑃 − 𝑃0) + 𝜈𝑤 ln(𝑥𝑤


𝐿𝛾𝑐𝐿)


𝐿) − ∑ 𝑛𝑖 ln (1 − ∑ 𝑌𝑖𝑗

𝑁𝑔𝑎𝑠

𝑗=1

)

𝑁𝑐𝑎𝑣

𝑖=1

= 0

(3.34)

Equation (3.34) is the equation that needs to be solved to compute the three phase

equilibrium (semiclathrate-liquid-vapour). 𝑥𝑐𝐿, 𝛾𝑎

𝐿, and 𝛾𝑤𝐿 are computed in the present

work from LIFAC, which is the electrolyte activity coefficient model that is incorporated

into PSRK. For a given pressure, Equation (3.34) can be solved for temperature. In

Equation (3.34) the activity coefficient and Yij, are non-linear functions of temperature. The

value of ni for semiclathrates formed from TBAB and TBAC is 3 and 1, respectively.

Table 3.1: Summary of stoichiometric coefficients for TBAB and TBAC ion constituents

used in the present work.

Salt TBAB TBAC

Cation TBA+ TBA+

Anion Br- Cl-

𝜈𝑤 38 24

𝜈𝑐 1 1

𝜈𝑎 1 1

In order to solve this three phase equilibrium problem posed in the present work, it is

required to regress five different parameters: Two of these parameters are (∆𝑑𝑖𝑠𝐻0(𝑇0) and

∆𝑑𝑖𝑠𝑉0(𝑇)) which are calculated by solving a liquid-solid equilibrium along with

experimental data available; there is one set of parameters per promoter. The remaining

three parameters (a, σ and ε) are regressed from the actual three phase equilibria

experimental data and they will be calculated for each particular gas and promoter, for

69

example, CH4 has two different set of parameters depending on the type of promoter which

can be either TBAB or TBAC in the present work.

3.3 Parameter regression

In this section, the five parameters needed in the model presented in the previous

section are regressed and presented. Additionally, PSRK and LIFAC are validated to verify

their ability to correlate the solubility of gases in aqueous electrolyte and non-electrolyte

systems. Additionally, the osmotic coefficient and mean ionic activity coefficients will be

computed to further verify the capability of the activity coefficient model (LIFAC). Finally,

the liquid-solid equilibria and the solid-liquid-vapour equilibria will be computed in order

to determine the parameters that are needed in the model for describing the vapor-liquid-

hydrate equilibria.

3.3.1 Validation of PSRK for predicting the solubility of gases in water

The PSRK EoS (presented in full in Appendix B) is used in the present work to

describe the vapour and liquid phases when computing the VLE (vapour-liquid

equilibrium). The purpose of the vapour liquid equilibrium is to calculate the compositions

of all species in the mixture in both phases given the temperature and pressure of the

system. The previous approaches made simplifying assumptions such as neglecting the

presence of vapor in the vapour phase, assuming the validity of Henry’s law and neglecting

the presence the influence of electrolytes in the liquid phase over the phase equilibrium.

The methodology for finding the two phase equilibrium condition is the isothermal isobaric

flash, which is an algorithm that combines a mass balance along with the phase equilibria

relationships and it is presented in Appendix A. The vapour and liquid phases are

described, in the current work, using the PSRK EoS:

𝑃 =𝑅𝑇

𝑣 − 𝑏−

𝑎

𝑣(𝑣 + 𝑏)

(3.35)

70

Where P is pressure, T is temperature, v is molar volume, a is a term that relates attractive

energy, R is the universal gas constant, and b is the repulsive term. The term a and b are

given by the following expressions:

𝑎 = 𝑏𝑅𝑇 (

1

𝐴1

𝑔𝐸

𝑅𝑇+ ∑ 𝑥𝑖

𝑎𝑖

𝑏𝑖𝑅𝑇𝑖

+1

𝐴1∑ 𝑥𝑖𝑙𝑛 (

𝑏

𝑏𝑖)

𝑖

) (3.36)

𝑏 = ∑ 𝑥𝑖𝑏𝑖

𝑖

(3.37)

Where xi is the molar fraction of compound i in the mixture, A1 is a constant equal to -

0.64663, and gE is excess Gibbs free energy which in the PSRK EoS is computed with the

UNIFAC activity coefficient. ai, bi, and gE are defined as follows:

𝑎𝑖 = 0.42748(𝑅𝑇𝑐,𝑖)

2

𝑃𝑐,𝑖𝑓(𝑇)

(3.38)

𝑏𝑖 = 0.08664𝑅𝑇

𝑃𝑐,𝑖

(3.39)

𝑔𝐸

𝑅𝑇= ∑ 𝑥𝑖𝑙𝑛𝛾𝑖

𝑖

(3.40)

Where Tc,i, Pc,i, and ωi are the critical temperature, critical pressure and acentric factor of i

compound, respectively. f(T) is the alpha function which is given by the following

expression [124]:

𝑓(𝑇) = [1 + 𝑐1(1 − 𝑇𝑟,𝑖0.5) + 𝑐2(1 − 𝑇𝑟,𝑖

0.5)2

+ 𝑐3(1 − 𝑇𝑟,𝑖0.5)

3]

2

𝑇𝑟,𝑖 < 1 (3.41)

𝑓(𝑇) = [1 + 𝑐1(1 − 𝑇𝑟,𝑖0.5)]

2 𝑇𝑟,𝑖 > 1

(3.42)

71

Where c1, c2, and c3 are adjustable parameters regressed from vapour pressure experimental

data. Tr,i is the reduced temperature of compound i in the mixture. The temperature

dependent interaction parameters in UNIFAC are given by the following expression:

𝜏𝑚,𝑘 = 𝑒𝑥𝑝 [−𝑎𝑚,𝑘 + 𝑏𝑚,𝑘𝑇 + 𝑐𝑚,𝑘𝑇2

𝑇]

(3.43)

The parameters am,k, bm,k, and cm,k are the interaction parameters, which are regressed from

experimental data available in the open literature.

Several binary systems involving different type of gases and water that form

semiclathrates hydrates have been selected for the validation of PSRK in the computation

of the vapour-liquid equilibria. Given the temperature and pressure of the system, a flash

is solved in order to calculate the molar fraction of the gas in the liquid phase and in the

vapour phase. The binary systems selected are the following: CH4+H2O, CO2+H2O,

Ar+H2O, Xe+H2O, N2+H2O, H2+H2O.

The interaction parameters for the pairs CH4+H2O, CO2+H2O, N2+H2O, and

H2+H2O were taken from the original PSRK tables [111]. Interaction parameters for xenon

and argon in water were not available and thus, it was necessary to regress the parameters

from experimental solubility data for argon [125] and xenon [126] in water. The optimal

parameters were determined by minimization of the following objective function using the

Nelder-Mead method [127]:

𝐹 = ∑(𝑥𝑔𝑎𝑠𝑐𝑎𝑙𝑐 − 𝑥𝑔𝑎𝑠

𝑒𝑥𝑝)2

𝑁𝑃

𝑖

(3.44)

Where F is the objective function, NP stands for the number of experimental points used

in the optimization, and 𝑥𝑔𝑎𝑠𝑐𝑎𝑙𝑐 and 𝑥𝑔𝑎𝑠

𝑒𝑥𝑝 are the calculated and experimental molar fractions

of the gas in liquid phase, respectively. The Nelder-Mead method was implemented using

MATLAB and it was chosen because it does not requires analytical derivatives.

72

Results of the optimized parameters for the systems and their standard errors (described

Appendix E) are presented in Table 3.2:

Table 3.2: Newly optimized PSRK parameters for argon+H2O and xenon+H2O regressed

from experimental data [125], [126].The errors are at the 95% confidence uncertainties.

Parameters Argon Xenon

𝑎𝑚,𝑘 3435.11±10.15 4344.23±101.85

𝑎𝑘,𝑚 -1722.13±12.32 -2093.32±89.45

𝑏𝑚,𝑘 -9.83±0.052 -12.63±0.085

𝑏𝑘,𝑚 8.09±0.0035 10.10±0.079

The computed values obtained from PSRK are compared against experimental

solubility data and are presented in Table 3.3. From Figure 3.2,Figure 3.3, and Figure 3.4

along with the results seen in Table 3.3, it has been demonstrated the agreement between

experimental and predicted values therefore proving the reliability of PSRK EoS. The

correlating capacity of the PSRK EoS was also checked for CH4+H2O, CO2+H2O,

N2+H2O, and H2+H2O. In the case of CH4+H2O, Trebble [128, p. 93] noted that most EoS

could not even match the observed trends, qualitatively. Figure 3.4 shows that PSRK, on

the other hand, can accurately describe the solubility of CH4 in water.

73

Figure 3.2: Solubility of argon (1) in H2O (2) at various temperatures conditions,

and total P=0.1 MPa. Symbols stands for experimental data [125], and curve is the

prediction by PSRK.

270

275

280

285

290

295

300

305

310

315

320

325

1.5 2.0 2.5 3.0 3.5 4.0 4.5

T /

K

x1 (x105)

74

Figure 3.3: Solubility of argon (1) in water (2) at various temperatures conditions,

and total P=0.1 MPa. Symbols stands for experimental data [126], and curve is the

prediction by PSRK.

270

275

280

285

290

295

300

305

310

315

320

325

0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80

T /

K

x1 (x105)

75

Figure 3.4: Methane (1) solubility in water (2) at various temperature and pressure

conditions. Symbols stands for experimental data and curves are predictions by

PSRK.■,P=55.1 MPa;+,P=27.6 MPa; ▲, P=10.3 MPa; ●, P=4.1 MPa. Experimental

data from [129].

0

1

2

3

4

5

6

7

300 320 340 360 380 400 420 440 460

X1

(X1

05)

T / K

76


Ar+H2O, Xe+H2O, N2+H2O, H2+H2O, and CO+H2O.

system Number of

data/Source of data

Temp range/K Pressure range/MPa AARDa/ %

CH4+H2O 20/Culberson et al

[130]

310-444 4.1-55.1 5.2

CO2+H2O 18/Takenouchi et al

[131]

278-298 0.5-5.0 10.2

Ar+H2O 10/Clever [125] 273-318 0.1 0.1

Xe+H2O 10/Clever [126] 273-318 0.1 0.3

N2+H2O 11/ Gillespie et al

[132]

310-366 0.34-13.79 6.4

H2+H2O 17/ Gillespie et al

[132]

310-478 0.34-13.79 8.8

𝐴𝐴𝑅𝐷𝑎 =100

𝑁𝑃∑

|𝑥𝑔𝑎𝑠𝑐𝑎𝑙𝑐−𝑥𝑔𝑎𝑠

𝑒𝑥𝑝|


𝑁𝑃𝑖 Where NP is the number of experimental data points, xgas

is the molar fraction of the gas in the liquid phase, subscript calc is the predicted value,

and subscript exp stands for the predicted values.

77

Figure 3.5: Solubility of carbon dioxide (1) in water (2) at various temperature and

pressure conditions. Symbols stands for experimental data [131], and curves are

predictions by PSRK.□,T=298.25 K;○,T=288.26 K; x, T=278.22 K.

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0.000 0.005 0.010 0.015 0.020 0.025 0.030

P /

MP

a

x1 (x105)

78

Figure 3.6: Solubility of nitrogen (1) in water (2) at various temperature and


predictions by PSRK.□,T=298.25 K;○,T=288.26 K; x, T=278.22 K.

0

2

4

6

8

10

12

14

16

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014

P /

MP

a

x1(x101)

79

Figure 3.7: Solubility of hydrogen (1) in water (2) at various temperature and


predictions by PSRK.■,T=310.93 K;●,T=366.48 K; ▲, T=422.04 K.

0

2

4

6

8

10

12

14

16

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0

P /

MP

a

x1 (x104)

80

3.3.2 Verification of LIFAC model for computing the mean activity and osmotic

coefficient of dissolved salts

In order to incorporate the electrolyte LIFAC activity coefficient model (Equation

(3.45)) into the thermodynamic framework presented in the present work, it is necessary to

check the correlating capability of the interaction parameters available in the open

literature. The two systems involved in this validation process are TBAB+H2O and

TABC+H2O.The expression that defines the activity coefficient is presented below:

ln 𝛾𝑖 = ln 𝛾𝑖𝐿𝑅 + ln 𝛾𝑖

𝑀𝑅 + ln 𝛾𝑖𝑆𝑅

(3.45)

Where 𝛾𝑖 is the activity coefficient of compound i in the mixture, 𝛾𝑖𝐿𝑅 is the long-range

(LR) activity coefficient which is expressed in terms of the Debye-Hückel theory, 𝛾𝑖𝑀𝑅 is

the middle-range (MR) activity coefficient which represents the interactions caused by the

ion-dipole effect, and the short-range (SR) activity coefficient 𝛾𝑖𝑆𝑅 that is described using

the UNIFAC activity coefficient. The expressions for these terms are given in Appendix

C.

The results of the calculated and experimental mean activity coefficient for both

mixtures (TBAB+H2O and TBAC+H2O) is presented in Figure 3.8, in addition to the plot,

Table 3.4 presents the deviation of the calculated values from the experimental data. The

osmotic coefficients of water in the two liquid solutions (TBAB and TBAC) have also been

computed and results are plotted in Figure 3.9. The value for the errors between calculated

and experimental data are presented in Table 3.5.From Figure 3.11, it can be seen that the

predictions by the model are in agreement with the experimental data.

81

Figure 3.8: Mean activity coefficient of TBAB and TBA+H2O solutions at

T=298.15 K and atmospheric pressure. The symbols denote the experimental data

for different salts: □ , TBAB [133];Δ,TBAC [133].The solid lines are calculated

with the LIFAC activity coefficient model.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9

γ±

mTBAX / mol∙kg-1

82

Table 3.4: Summary of predictions of mean activity coefficient by means of electrolyte

NRTL for the systems TBAB+H2O and TBAC+H2O.

Salt Number of data/Source of

data

Molality range

mol/kg

AARDa/ %

TBAB 15/ Lindenbaum et al [133] 0.1-2.0 3.29

TABC 15/ Lindenbaum et al [133] 0.1-2.0 4.57


𝑁𝑃∑

|𝛾±𝑚𝑐𝑎𝑙𝑐−𝛾±𝑚

𝑒𝑥𝑝|

𝛾±𝑚𝑒𝑥𝑝

𝑁𝑃𝑖 Where NP is the number of experimental data points, 𝛾±𝑚

is mean activity coefficient of the salt, subscript calc is the predicted value, and subscript

exp stands for the predicted values.

83

Figure 3.9: Osmotic coefficient of water in TBAB and TBAC solutions at

T=298.15 K and atmospheric pressure. The symbols denote the experimental data

for the two salts: ○, TBAB [133]; □, TBAC [133].The solid lines are calculated

with the LIFAC activity coefficient model

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 1 2 3 4 5 6 7 8 9 10

φ

mTBAX / mol∙kg-1

84

Table 3.5: Summary of predictions of osmotic coefficient of water in TBAB and TBAC

solutions by means of electrolyte NRTL at T=298.15 K and atmospheric pressure.

Salt Number of data/Source of

data

Molality range

mol/kg

AARDa/ %

TBAB 27/ Lindenbaum et al [133] 0.1-10 4.14

TABC 27/ Lindenbaum et al [133] 0.1-10 4.36


𝑁𝑃∑

|𝜙𝑐𝑎𝑙𝑐−𝜙𝑒𝑥𝑝

|

𝜙𝑒𝑥𝑝

𝑁𝑃𝑖 Where NP is the number of experimental data points, 𝜙

is the osmotic coefficient of water in the solutions. Subscript calc is the predicted value,


3.3.3 Validation of the extension of PSRK for usage in aqueous electrolyte solutions

Due to the presence electrolytes in the liquid phase, it is necessary to verify that the

thermodynamic properties of the vapour-liquid mixtures are correctly described with

PSRK. In the same way it has been done with nonelectrolyte systems, computations of

compositions of vapour and liquid phases will be determined by performing a flash

calculation in a system in which the pressure and temperature are known.

Due to a lack of experimental solubility data of argon and xenon in electrolyte

solutions, CO2 and CH4 were chosen because they represent a wide range of solubilities in

water [128, p. 85]. Therefore, computations of solubilities require no additional parameters

and will be entirely predictive. These two gases can also form semiclathrates, and

experimental data is available in the open literature including solubility of CO2 in TBAB.

Results of the vapour-liquid calculations are summarized in Figure 3.7 through Figure 3.12.

In the case of the solubility of CH4 in NaCl, the average deviation between experimental

and predicted values (AARD) is 5.3%, and for the solubility of CO2 in NaCl, the deviation

is 1.7%. Figure 3.10 and Figure 3.11 correspond to the CH4+H2O+NaCl system, and

CO2+H2O+NaCl system, respectively. It can be seen that that the PSRK EoS is capable of

accurately computing gas solubilities in aqueous electrolyte solutions.

85

Figure 3.10: Solubility of CH4 (1) in H2O (2)+NaCl (3) at various total pressure

conditions and T=324.7 K.□,NaCl molality=1.0 mol/kg;+, NaCl molality =4.0

mol/kg. Symbols stands for experimental data [134] and curves are the predictions

by PSRK.

0

10

20

30

40

50

60

70

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

P /

MP

a

x1 (x103)

86

Figure 3.11: Solubility of CO2 (1) in H2O (2) + NaCl (3) at various pressure

conditions and fixed NaCl molality=4.0 mol/kg. □,T=313.15K;+,T

=333.15K.Symbols stands for experimental data [135] and curves are the

predictions by PSRK.

0

2

4

6

8

10

12

0.0 2.0 4.0 6.0 8.0 10.0 12.0

P /

MP

a

x1 (x103)

87

Figure 3.12: Solubility of CO2 (1) in H2O (2) +TBAB (3) at various pressure

conditions and T=283.15K;○,wTBAB=0.09 [56]; Curves are the predictions by

PSRK.


CH4+H20+NaCl, CO2+H2O+NaCl,and CO2+H2O+TBAB.

system Number of

data/Source of data

Temp

range/K

Pressure

range/MPa

AARDa/

%

CH4+H20+NaCl 33/O’Sullivan et al

[134]

324-375 10-61 5.3

CO2+H2O+NaCl 13/Takenouchi et al

[135]

313-333 0.47-9.65 1.7

CO2+H2O+TBAB 6/Lin et al [56] 283.15 0.392-1.678 2.1


𝑁𝑃∑

|𝑥𝑔𝑎𝑠𝑐𝑎𝑙𝑐−𝑥𝑔𝑎𝑠

𝑒𝑥𝑝|


𝑁𝑃𝑖 Where NP is the number of experimental data points, xgas

is the molar fraction of the gas in the liquid phase, subscript calc is the predicted value,


0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.00 0.50 1.00 1.50 2.00 2.50

x1

P / MPa

88

3.3.4 Regression of parameters for modelling solid-liquid equilibrium involving TBAB

and TBAC in the absence of gases.

The experimental values with the liquid-solid equilibria of the systems TBAB+H2O and

TBAC+H2O system were published in the work presented by Sato et al [136]. In the

aforementioned paper, data for temperatures of dissociation at different concentrations are

presented for both salts at atmospheric pressure. Since Sato et al [136] do not mention the

type of structure formed when conducting the experiments, the parameters (∆𝑑𝑖𝑠𝐻0(𝑇0)

and ∆𝑑𝑖𝑠𝑉0(𝑇)) will be fitted assuming each of the two possible structures than can be

present when forming semiclathrates of TBAB, and for each of the three possible structures

in the case of TBAC. To fit the parameters the following objective function is used:

𝐹(∆𝑑𝑖𝑠𝐻0(𝑇0), ∆𝑑𝑖𝑠𝑉0(𝑇)) = ∑(𝑇𝑐𝑎𝑙𝑐 − 𝑇𝑒𝑥𝑝)2

𝑁𝑃

𝑖

(3.46)

Where 𝐹(∆𝑑𝑖𝑠𝐻0(𝑇0), ∆𝑑𝑖𝑠𝑉0(𝑇)) is the objective function, NP stands for the number of

experimental points used in the optimization, and 𝑇𝑐𝑎𝑙𝑐 and 𝑇𝑒𝑥𝑝 are the calculated and

experimental dissociation temperatures of the mixture, respectively. The optimal

parameters were determined by minimization of the following objective function using the

Nelder-Mead method [127] and the results are presented in Table 3.7. From the error

calculation, it can be seen that the experimental data is best fitted assuming structure type

A in the case of TBAB, and structure type I in the case of TBAC. The temperature-

composition diagrams for both salts are depicted in Figure 3.13 (TBAB) and Figure 3.14

(TBAC), respectively.

89

Table 3.7: Semiclathrate parameters to compute solid-liquid equilibria of the systems

H2O+TBAB and, H2O+TBAC at atmospheric pressure

Salt Type ∆𝑑𝑖𝑠𝐻0(𝑇0)/

kJ/mol

∆𝑑𝑖𝑠𝑉0(𝑇)/

cm^3/mol AARD %

TBAB A 155.35±1.52 -5.36±0.045 0.09

TBAB B 162.56±1.36 -12.15±0.074 0.35

TBAC I 125.41±0.57 -9.27±0.0085 0.02

TBAC II 153.85±0.85 -7.58±0.0023 0.19

TBAC III 176.65±0.36 -6.53±0.0054 0.28

Figure 3.13: Temperature composition diagram of the H2O+TBAB mixture. The

liquid composition is expressed in terms of TBAB weight fraction (wTBAB).■ is the


structure type A, dashed line represents SLE curve assuming structure type B.

280

281

282

283

284

285

286

287

0 0.1 0.2 0.3 0.4 0.5 0.6

T /

K

wTBAB

90

Figure 3.14: Temperature composition diagram of the H2O+TBAC mixture. The

liquid composition is expressed in terms of TBAC weight fraction (wTBAC).■ is the


structure type I, dashed line represents SLE curve assuming structure type II. The

dash-dotted line represents predictions assuming structure III.

281

282

283

284

285

286

287

288

289

0 0.1 0.2 0.3 0.4 0.5 0.6

T /

K

wTBAC

91

3.3.5 Kihara Parameter regression and correlation of SVLE for systems containing

TBAB/TBAC semiclathrates and a gas

The three phase equilibrium is calculated with the following procedure: At the limit

of appearance of the semiclathrate phase, the mass of the semiclathrate phase is zero in

comparison to the quantities in the vapour and liquid phases because it is an incipient phase.

The conditions required in the vapour-liquid equilibrium are the equality of temperature,

pressure, and fugacities in all phases for all non-electrolyte compounds present in the

mixture. Aqueous ionic species are assumed to only be present in the liquid phase. The

equilibrium is solved by performing a flash which is explained in detail in Appendix A.

For a given temperature, pressure, and total composition of the mixture, the flash can be

solved in order to find the compositions in both liquid and vapour phases as outlined in

section 3.2.2. In order to find the dissociation conditions of the semiclathrate, the total

composition of the system, and either temperature or pressure needs to be specified. In the

present study, the pressure is the input, therefore the variable that needs to be computed is

the temperature. This can be done with Equation (3.34) because the only unknown in it is

the temperature value, and this can be solved by using the secant method which is a root-

finding algorithm [137, p. 10] so Equation (3.34) can be presented as an implicit function

(𝑔(𝑇)) of temperature:

𝑔(𝑇) =∆𝑑𝑖𝑠𝐺0(𝑇0)

𝑅𝑇0+


𝑅𝑇(1 −

𝑇

𝑇0) +

∆𝑑𝑖𝑠𝑉0(𝑇0, 𝑃0)

𝑅𝑇(𝑃 − 𝑃0)


𝐿) + 𝜈𝑤 ln(𝑥𝑤𝐿 𝛾𝑤


𝐿)

− ∑ 𝑛𝑖 ln (1 − ∑ 𝑌𝑖𝑗

𝑁𝑔𝑎𝑠

𝑗=1

)

𝑁𝑐𝑎𝑣

𝑖=1

= 0

(3.47)

In Equation (3.47) the activity coefficient terms depend on the temperature as well as the

occupancy factor Yij as well. Finally, the algorithm for solving the three phase equilibrium,

is presented in Figure 3.15 as follows:

92

Figure 3.15: Flow diagram for computing the dissociation pressure of

semiclathrates at a given temperature.

93

In order to perform the predictions of the three phase equilibria of the systems studied in

the present work, it was necessary to determine the optimal values of the Kihara potential

parameters because they were not available for semiclathrates. The Kihara potential is

implicit in the Yij term in Equation (3.47). The parameter optimization has been performed

using pressure-temperature equilibrium experimental data reported both in the present

work and the open literature. The optimized parameters for semiclathrates formed with

TBAB solutions are listed in Table 3.8, and the values obtained in the case of solutions

involving TBAC are presented in Table 3.9.The optimal parameters have been determined

by minimization of the following objective function using the Simplex-Nelder-Mead

algorithm [127].

𝐹 = ∑(𝑇𝑐𝑎𝑙𝑐 − 𝑇𝑒𝑥𝑝)2

𝑁𝑃

𝑖

(3.48)

Where F is the objective function, NP stands for the number of experimental points used

in the optimization, and Tcalc and Texp are the calculated and experimental dissociation

temperatures of the mixture, respectively.

Table 3.8: Regressed Kihara potential parameters for xenon, argon, CH4, CO2, H2, and N2,

in TBAB aqueous solutions.

Gas a (Å) σ (Å) ε/k (K)

xenon 0.25125±0.078 3.8589±7.5E-3 185.501±4.5E-3

argon 0.16523±0.065 2.6452±4.8E-3 190.523±2.90E-2

CH4 0.41521±0.023 2.95856±4.4E-4 304.25±6.3E-3

CO2 0.71086±0.015 3.05256±5.6E-3 242.156±4.8E-2

H2 0.29656±0.036 3.56415±6.5E-3 160.75±3.45E-3

N2 0.37856±0.047 3.9574±4.1E-2 174.236±7.45E-3

94

Table 3.9: Regressed Kihara potential parameters for xenon, argon, CH4, and CO2 in

TBAC aqueous solutions

Gas a (Å) σ (Å) ε/k (K)

xenon 0.23854±0.069 6.8521±7.8E-3 216.556±3.5E-2

argon 0.19421±0.037 4.3152±4.6E-3 194.562±4.5E-2

CH4 0.39605±0.036 4.5685±6.2E-4 211.554±1.2E-2

CO2 0.65388±0.054 3.9574±4.8E-3 180.504±6.1E-3

As previously noted, Paricaud’s model [101] did not used the rigorous Kihara potential.

This will be advantageous when the model is subsequently extended to describe

semiclathrate formation in the presence of gas mixtures.

3.4 Results and discussion

In the present work, several systems that form semiclathrates have been selected to

prove the accuracy of the thermodynamic model introduced in the previous section. Five

different type of gases such as CH4, CO2, H2 and the two gases object of the present study

(xenon and argon). The modelling is conducted using the procedure outlined in Figure 3.15

as well as the parameters regressed in the subsequent sections.

3.4.1 Xenon semiclathrates

Semiclathrates formed in the presence of xenon have been studied using the present

thermodynamic model in the presence of two different promoters (TBAB and TBAC) at

two different concentrations. The results from the simulation are in agreement with the

experimental data obtained in the present study. Two different levels of concentration

(wTBAX=0.05, 0.20) for the two promoters are used in the predictions. The value for AARD

in the case of the TBAB is 5.01% and 6.3% in the case of TBAC. No other thermodynamic

model available in the open literature has reported predictions involving semiclathrates of

xenon. At 20 wt%, there appears to be disagreement with the trend of the data at T>287 K.

As mentioned [34] it is suspect that there is a phase transition at that point. In the current

model Type A is assumed to always be the stable structure.

95


xenon+water/TBAB aqueous solution systems. Symbols stand for experimental

data and lines refer to the predicted values using the developed thermodynamic

model; ●, xenon+H2O system [117];◊, xenon+H2O system [116];□, xenon in the

presence of 5 wt. % TBAB [34];■, xenon in the presence of 5 wt. % TBAB [118];Δ,

xenon in the presence of 20 wt. % TBAB [34];▲, xenon in the presence of 20 wt.

% TBAB [118].

0

200

400

600

800

1000

1200

1400

1600

1800

2000

270 275 280 285 290 295 300 305

P /

kP

a

T / K

96


xenon+ water/TBAC aqueous solution systems. Symbols stand for experimental


model; ■, xenon+H2O system [117];●, xenon+H2O system [116];▲, xenon in the

presence of 5 wt. % TBAC [34];□, xenon in the presence of 5 wt. % TBAC [34].

0

200

400

600

800

1000

1200

1400

1600

1800

270 275 280 285 290 295 300

P /

kP

a

T / K

97

3.4.2 Argon semiclathrates

Results of predictions for semiclathrates formed from argon in aqueous solutions

of TBAB and TBAC are presented in Figure 3.18, and Figure 3.19, respectively. The results

show good agreement between the experimental measurements and model results. The

vertical axis for Figure 3.18 and Figure 3.19 are given in logarithmic scale due to the large

range of pressure especially in the case of pure hydrates. No other thermodynamic model

available in the open literature has reported predictions involving semiclathrates of argon.


argon+ water/TBAB aqueous solution systems. Symbols stand for experimental


model; ■, argon+H2O system [115];●, argon in the presence of 5 wt. % TBAB [34];

▲,argon in the presence of 20 wt. % TBAB [34].

1000

10000

100000

278 280 282 284 286 288 290 292 294

P /

KP

a

T / K

98


argon+ water/TBAC aqueous solution systems. Symbols stand for experimental


model; ■, argon+H2O system [115];●, argon in the presence of 5 wt. % TBAC [34];

▲,argon in the presence of 20 wt. % TBAC [34].

100

1000

10000

100000

278 280 282 284 286 288 290 292 294

P /

KP

a

T / K

99

3.4.3 Methane semiclathrates:

The vapour-liquid-solid equilibria for semiclathrates of methane in two different

promoters (TBAB and TBAC) have been correlated by solving the algorithm developed in

the present study. Results for TBAB and TABC are presented in Figure 3.20, and Figure

3.21, respectively. Three different concentrations of TBAB were selected for the

predictions: wTBAB=0.05, 0.10, 0.20 and 0.35. The value for AARD is 5.1% which is slightly

better than the predictions made by the model by Eslamimanesh et al [94] in whose work

the value for AARD is reported to be 5.6%, an also in the in the model presented by Shi et

al [100], the value of AARD is 11.2%. Again it should be noted that the current work

requires fewer parameters than Eslamimanesh model. In the case of semiclathrates formed

from TBAC, four different concentration of promoter have been selected for predictions:

wTBAC=0.05, 0.0997, 0.20, and 0.34. The present model predicts values which are also in

better agreement with the experimental values than the predicted values reported by Shi et

al [100];in this case the authors report a value of 4.9% whereas in the present work the

AARD value is 4.6%.The model by Shi et al [100] is based on Eslamimanesh’s approach

and this type of model relies on many assumptions that seem to be of questionable validity,

such as describing the vapour pressure of the promoter, the Langmuir constants, and the

activity coefficients in the liquid phase with empirical correlations. All of these

assumptions give the model of Eslamimanesh [94] 10 adjustable parameters. Results from

Paricaud’s are not available because the model was only trialed with carbon dioxide.

Kwaterski and Herri [104] reported a value for AARD equal to 8% for semiclathrates of

TBAB, the authors did not report any result for semiclathrates in the presence of TBAC.

In Figure 3.19 it can be seen that the prediction line for semiclathrates formed with 5 wt.

% TBAB solutions crosses the hydrate line at 294 K. This very same pattern has been

observed in the predictions made the model by Eslamimanesh. Experimental points also

replicate the same pattern.

100


methane +water/TBAB aqueous solution systems. Symbols stand for experimental


model;■,CH4+H2O system [138];▲,CH4 in the presence of 5 wt.% TBAB

[97];◊,CH4 in the presence of 5wt% TBAB [85]; ♦,CH4 in the presence of 5 wt.%

TBAB [85];-,CH4 in the presence of 10 wt.% TBAB;●,CH4 in the presence of 10

wt.% TBAB [57];□,CH4 in the presence of 10 wt.% TBAB [85];Δ,CH4 in the

presence of 20 wt.% TBAB [72];○, CH4 in the presence of 20 wt.% TBAB [85];x,

CH4 in the presence of 20 wt.% TBAB [97];+, CH4 in the presence of 20 wt.%

TBAB [22];*, CH4 in the presence of 35 wt.% TBAB [57].

0

5

10

15

20

25

30

35

40

45

50

280 285 290 295 300

P /

MP

a

T / K

101


methane +water/TBAC aqueous solution systems. Symbols stand for experimental


model; ■, CH4+H2O system [138]; ▲, CH4 in the presence of 5 wt. % TBAC

[139];♦, CH4 in the presence of 5 wt. % TBAC [73];●, CH4 in the presence of 9.97

wt. % TBAC [73];□, CH4 in the presence of 20 wt. % TBAC [73];Δ, CH4 in the

presence of 34 wt. % TBAC [140].

0

5

10

15

20

25

280 282 284 286 288 290 292

P/

MP

a

T/K

102

3.4.4 Carbon Dioxide semiclathrates

In the case of CO2 semiclathrates formed from TBAB, four different concentrations

have been selected for computing the predictions with the proposed model: wTBAB=0.05,

0.10, 0.19, and 0.40. The present model predicts a value of AARD equal to 2.3% which is

better than the predictions of models such as Paricaud’s (10.5%) [101] or the model of

Verrett et al [97] in which the AARD value is 4.7%. The improvement of the present model

over Paricaud’s model could be attributed to the extra adjustable parameter in the

computation of the Langmuir constant. Again it should be noted that with the present model

the computation of the Langmuir constant involves three parameters. Whereas Paricaud’s

Langmuir constant requires two adjustable parameters. Semiclathrates formed from TBAC

are also in good agreement with the experimental data. The value of AARD found in the

present work is 4.04%, whereas Shi et al [100] reports a value of 11.1%. It should be noted

that scatter in the CO2 experimental data, particularly at wTBAB=0.10, is greater than in

CH4.This may be due to the relatively high solubility of CO2 in H2O.

103


carbon dioxide +water/TBAB aqueous solution systems. Symbols stand for

experimental data and lines refer to the predicted values using the developed

thermodynamic model; ■, CO2+H2O system [141];▲, CO2 in the presence of 5 wt.

% TBAB [142];♦ CO2 in the presence of 5 wt. % TBAB [97];●, CO2 in the presence

of 10 wt. % TBAB [142];-, CO2 in the presence of 10 wt. % TBAB [97];Δ, CO2 in

the presence of 10 wt. % TBAB [22];x, CO2 in the presence of 19 wt. % TBAB

[142];○, CO2 in the presence of 40 wt. % TBAB [97].

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

273 278 283 288 293

P /

MP

a

T / K

104


carbon dioxide +water/TBAC aqueous solution systems. Symbols stand for


thermodynamic model; ■, CO2+H2O system [141];▲, CO2 in the presence of 4.337

wt. % TBAC [143];●, CO2 in the presence of 8.741 wt. % TBAC [143];Δ, CO2 in

the presence of 15 wt. % TBAC [100];○, CO2 in the presence of 34 wt. % TBAC

[140];-, CO2 in the presence of 34 wt. % TBAC [100].

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

278 280 282 284 286 288 290 292 294

P /

MP

a

T / K

105

3.4.5 Nitrogen semiclathrates.

Nitrogen semiclathrates in TBAB have also been used to test the proposed

thermodynamic model. The predictions are presented in Figure 3.24 in which the vertical

axis is log-scale due to the large pressure range; the calculated value of AARD is found to

be 4.71%, whereas Eslamimanesh et al [94] report a value of 11% when using their model.

Unlike the cases of methane and carbon dioxide, it was not possible to compute

semiclathrates of TBAC due to lack of experimental data. N2 semiclathrates form at much

lower pressure than N2 hydrates, which is shown by the new model.

106


nitrogen + water/TBAB aqueous solution systems. Symbols stand for experimental


model; ■, N2+H2O system [144];▲, N2 in the presence of 5 wt. % TBAB [84];◊,

N2 in the presence of 5 wt. % TBAB [57];●, N2 in the presence of 10 wt. % TBAB

[57];x, N2 in the presence of 10 wt. % TBAB [85];□, N2 in the presence of 20 wt.

% TBAB [84];○, N2 in the presence of 25 wt. % TBAB [57].

0.1

1

10

100

1000

280 282 284 286 288 290 292 294

P /

MP

a

T / K

107

3.4.6 Hydrogen semiclathrates.

Hydrogen semiclathrates in TBAB aqueous solutions are predicted with the

proposed model and it is found that predictions are in good match with the experimental

data. AARD from the present model is 5.8%. In the literature review it was observed that

no one had attempted to make predictions for H2 semiclathrates. Four different

concentrations of TBAB were used to perform the predictions, wTBAB= 0.05, 0.10, 0.25, and

0.43.The results are plotted in Figure 3.25 with pressures sketched in logarithmic scale. No

experimental data was available to perform predictions involving TBAC solutions. As with

N2, TBAB semiclathrates in the presence of H2 form at much lower pressure than H2

hydrates, which is also shown by the new model.

108


hydrogen + water/TBAB aqueous solution systems. Symbols stand for


thermodynamic model; ■, H2+H2O system [145];▲, H2 in the presence of 5 wt. %

TBAB [57];◊, H2 in the presence of 10 wt. % TBAB [57];+, H2 in the presence of

10 wt. % TBAB [85];-, H2 in the presence of 10 wt. % TBAB [77];Δ, H2 in the

presence of 25 wt. % TBAB [77];x, H2 in the presence of 43 wt. % TBAB [57];□,

H2 in the presence of 43 wt. % TBAB [77];○, H2 in the presence of 43 wt. % TBAB

[85].

0.1

1

10

100

1000

275 280 285 290 295

P /

MP

a

T / K

109

Results from all semiclathrates predictions are presented in Table 3.10, where range of

experimental data along with the error computed as AARD show that the new model

predictions are in good agreement with the experimental data. The new model is an

improvement to the model developed by Kwaterski and Herri [104] because it has been

extended to more gases such as xenon, argon, N2, and H2; additionally it has been used for

an additional promoter that is TBAC, whereas Kwaterski and Herri’s model was only tested

with CH4 and CO2 along with TBAB as promoter. The main reason for the improvement

in the new model over the predecessor is believed to be the rigorous treatment of the

vapour-liquid equilibrium, the present model besides guaranteeing the equilibria between

the liquid and vapour phase, considers the non idealities derived from the presence of

electrolytes in the liquid phase and do not neglect the polar nature of the compounds studied

in the present work such as water and carbon dioxide. Paricaud’s model remains as the

model with the least amount of adjustable parameters (one less than the present model) but

this can be justified by the fact of the treatment that is provided to the Langmuir constants

is not rigorous, it is simple empirical correlation without physical meaning.

Table 3.10: Model results for prediction of the equilibrium conditions of TBAB and

TBAC+H2O+gas (CH4, CO2, H2, N2, xenon, argon) systems.

System

number

of data

Temperature

range/K

Pressure range /

MPa

Salt mass

fraction range/

wt.

AARD

/ %

Xe+H2O+TBAC 16 287-297.1 0.266-1.152 0.05-0.20 6.3

Ar+H2O+TBAC 13 284-290.8 2.082-6.114 0.05-0.20 2.1

CH4+H2O+TBAB 51 283.2-298.15 0.708-41.369 0.05-0.20 5.1

CO2+H2O+TBAB 49 279.06-291.2 0.349-4.678 0.005-0.19 2.3

H2+H2O+TBAB 58 277.7-294.75 0.49-140.6 0.05-0.43 5.8

N2+H2O+TBAB 34 281.1-292.95 0.47-33.503 0.04337-0.34 4.71

Xe+H2O+TBAB 38 275.7-302.7 0.79-1.801 0.05-0.20 5.01

Ar+H2O+TBAB 13 283.5-291.6 1.569-4.126 0.05-0.20 4.1

CH4+H2O+TBAC 43 281.3-292.42 0.61-8.91 0.05-0.34 4.5

CO2+H2O+TBAC 30 280.1-293.33 0.36-4.42 0.04337-0.34 4.04

110

Conclusion and recommendations

4.1 Conclusions

Experiments were performed in a constant volume equilibrium cell to measure the

semiclathrate equilibrium formation conditions for TBAB and TBAC semiclathrates

formed from argon and xenon. For both TBAB and TBAC semiclathrates, in the presence

of both argon and xenon, the semiclathrates formed at a lower pressure than what is

required to form gas hydrates and the required pressure decreased as the salt concentration

increased. The effect of TBAB and TBAC salt concentration on the equilibrium pressure

was strong on the semiclathrates were formed in the presence of argon, but when formed

in the presence of xenon, the effect of the salt concentration the effect of salt concentration

was relatively weak. Also, the addition of TBAB and TBAC had a significant effect on the

pressure required for forming semiclathrates in the presence of argon, but only a

comparatively mild effect on the pressure required to form semiclathrates in the presence

of xenon. Additionally, the Clausius-Clapeyron equation was used to estimate the heat of

fusion for semiclathrate dissociation. For TBAB and TBAC semiclathrates formed in the

presence of argon, the heat of fusion varied from 132 to 188 kJ/mol K, whereas when the

semiclathrates were formed in the presence of xenon the estimated heat of fusion varied

from 56 to 127 kJ/mol K.

In the present work, a thermodynamic modelling approach correlates the

dissociation conditions of semiclathrates of xenon and argon in the presence of TBAB and

TBAC aqueous solutions. The proposed model was used to describe the liquid-solid

equilibrium of the TBAB+H2O and TBAC+H2O systems. The model has been developed

using the work done by Paricaud [101]. The model is based on a reaction equilibrium and

it is solved by applying the Gibbs free energy minimization. Another important feature of

the present thermodynamic model is the rigorous treatment of the Langmuir constants by

means of the Kihara potential. The PSRK EoS was used to describe the phase equilibrium

between the liquid and vapour phase, and a rigorous flash was performed to calculate

compositions of all species in the vapour and liquid phase. The presence of electrolytes in

111

the liquid phase was not neglected and it was included in the calculations of the vapour

liquid equilibrium. The optimal values of the model parameters have been regressed from

experimental data produced in the present work. Good agreement between the predictions

and the experimental data has been observed and when the results from this model were

compared to those from previous model the obtained values were slightly superior to its

predecessors due to the modifications previously listed. Although the model has been

applied to single gases, it can be easily extended to gas mixtures using the existing mixing

rules.

4.2 Recommendations

Based on the results of the current study, the following recommendations are made:

Conduct experimental measurement of solubilities of xenon and argon in

TBAB and TBAC at different pressure and temperature.

Conduct experimental measurement of osmotic coefficients and mean

activity coefficients of TBAB and TBAC aqueous solutions at different

temperatures in order to verify the capability of the LIFAC model at

temperatures different than 298.15 K.

Extend the thermodynamic model to gas mixtures, in the present study a

single gas was studied.

112

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135

APPENDIX A: ISOTHERMAL ISOBARIC FLASH COMPUTATION

Flash calculations refer to the process of calculating the liquid phase and vapour

phase mole fractions, xi and yi, respectively at the prevailing pressure and temperature when

given the overall composition of the mixture, zi , as shown in Figure A.1

Figure A.1: Scheme of vapor-liquid isothermal flash

The vapour and liquid phase mole fractions are constrained by the following equation,

where the summation of molar fraction in both liquid and vapour phase are equal to unity:

∑ 𝑦𝑖

𝑐

𝑖=1

= ∑ 𝑥𝑖 = 1

𝑐

𝑖=1

(A.1)

The equilibrium ratio or the K-value of component i is given by:

𝐾𝑖 = 𝑦𝑖

𝑥𝑖 =

𝜑𝑖𝐿

𝜑𝑖𝑉 → 𝑦𝑖 = 𝐾𝑖𝑥𝑖

(A.2)

Where 𝜑𝑖𝐿, 𝜑𝑖

𝑉 are the partial fugacity coefficients which in the present work are calculated

by means of the PSRK EoS and are presented in Appendix B. On the basis of one mole;

the sum of the vapour and liquid mole fractions is the unity (F = 1); i.e. L+V = 1. A material

balance on the i component results in:

𝑧𝑖 = 𝑦𝑖𝑉 + 𝑥𝑖𝐿 (A.3)

136

𝑧𝑖 = 𝑦𝑖𝑉 + 𝑥𝑖(1 − 𝑉) (A.4)

Substitution of yi from Equation (A.2) into Equation (A.4) yields:

𝑧𝑖 = 𝑦𝑖𝑉𝑥𝑖𝐾𝑖 + ( 1 − 𝑉)𝑥𝑖 (A.5)

Solving Equation (A.5) for xi

𝑥𝑖 = 𝑧𝑖

1 + 𝑉(𝐾𝑖 − 1) (A.6)

Solving Equation (A.4) for yi yields:

𝑦𝑖 = 𝐾𝑖𝑥𝑖 = 𝑧𝑖𝐾𝑖

1 + 𝑉(𝐾𝑖 − 1) (A.7)

Substitution of Equation (A.6) and Equation (A.7) into Equation (A.1) yields:

∑ 𝑦𝑖

𝑐

𝑖=1

= ∑ 𝑥𝑖 =

𝑐

𝑖=1

∑(𝑦𝑖 − 𝑥𝑖) = ∑ 𝑧𝑖(𝐾𝑖 − 1)

1 + 𝑉(𝐾𝑖 − 1) = 0

𝑐

𝑖=1

𝑐

𝑖=1

(A.8)

Therefore Equation (A.8) is the well-known Rachford-Rice equation [146] in which the

only unknown is V, which is an implicit function of V:

𝑓(𝑉) = ∑ 𝑧𝑖(𝐾𝑖 − 1)

1 + 𝑉(𝐾𝑖 − 1) = 0

𝑐

𝑖=1

(A.10)

The algorithm to solve Equation (A.10) is presented in Figure A.2, in which the pressure,

temperature and global composition are inputs in the algorithm and the program iterates by

substitution the molar fraction in both phases. The algorithm is composed by two loops:

the inner one computes the value of V and the outer one, computes the values of the mole

fractions.

137

Figure A.2: Algorithm for solving the isothermal flash (Reproduced from Elliot

and Lira [147, p. 617])

138

APPENDIX B: PSRK EQUATION OF STATE

The PSRK (Predictive-Soave-Redlich-Kwong) equation of state was developed by

Holderbaum and Gmehling [111] and it is based on the Soave-Redlich-Wong (SRK) [105]

equation of state. The PSRK equation of states is given by the following equation explicit

in pressure:

𝑃 =𝑅𝑇

𝑣 − 𝑏−

𝑎

𝑣(𝑣 + 𝑏)

(A.1)

Where P is pressure, T is temperature, v is molar volume, a is term that relates attractive

energy, and b is the repulsive term. Two modifications were done to make PSRK suitable

for vapour-liquid equilibria of polar as well as non polar mixtures. The first modification

relates the temperature dependence of the pure component parameter ai, which in the

original SRK EoS [105], is only function of the acentric factor ω, as presented in Equation

(B.1)

𝑎𝑖 = 0.42748(𝑅𝑇𝑐,𝑖)

2

𝑃𝑐,𝑖𝑓(𝑇)

(B.1)

𝑓(𝑇) = [1 + 𝑐1(1 − 𝑇𝑟,𝑖0.5)]

2

(B.2)

𝑐1 = 0.48 + 1.574𝜔𝑖 − 0.176𝜔𝑖2

(B.3)

𝑇𝑟,𝑖 =𝑇

𝑇𝑐,𝑖

(B.4)

Where R is the universal gas constant, Tc,i, Pc,i, and ωi are the critical temperature, critical

pressure and acentric factor of i compound, respectively. This temperature dependence

produces accurate vapour pressure values for nonpolar compounds, but when applied to

139

polar compounds, the results are not satisfactory. Thus, the expression proposed by Mathias

and Copeman [124]is employed in the PSRK equation:

𝑓(𝑇) = [1 + 𝑐1(1 − 𝑇𝑟,𝑖0.5) + 𝑐2(1 − 𝑇𝑟,𝑖

0.5)2

+ 𝑐3(1 − 𝑇𝑟,𝑖0.5)

3]

2

𝑇𝑟 < 1 (B.5)

𝑓(𝑇) = [1 + 𝑐1(1 − 𝑇𝑟,𝑖0.5)]

2 𝑇𝑟 > 1

(B.6)

Where c1, c2, and c3 are adjustable parameters regressed from vapour pressure experimental

data. The second modification introduced with PSRK EoS is the mixing rule for a.This

mixing rule for a is based on the work by Michelsen [148], which links the parameter a to

the excess Gibbs energy (𝑔0𝐸) at zero pressure. The general principle of all 𝑔𝐸 mixing rules

is that an activity coefficient model and an equation of state produce the same value for the

excess Gibbs energy:

𝑔𝐸,𝐸𝑜𝑆

𝑅𝑇=

𝑔𝐸,𝛾

𝑅𝑇 (𝑃 = 𝑃𝑟𝑒𝑓)

(B.7)

Where 𝑔𝐸,𝐸𝑜𝑆 and 𝑔𝐸,𝛾 are the Gibbs energy calculated from an EoS, and an activity

coefficient model, respectively. After applying Equation (B.7) to the original SRK EoS,

the following expression for a is obtained:

𝑎 = 𝑏𝑅𝑇 (1

𝐴1

𝑔𝐸


𝑎𝑖


+1

𝐴1∑ 𝑥𝑖𝑙𝑛 (

𝑏

𝑏𝑖)

𝑖

) (B.8)

𝑔𝐸

𝑅𝑇= ∑ 𝑥𝑖𝑙𝑛𝛾𝑖

𝑖

(B.9)

𝑏 = ∑ 𝑥𝑖𝑏𝑖

𝑖

(B.10)

𝑏𝑖 = 0.08664𝑅𝑇

𝑃𝑐,𝑖

(B.11)

140

Where A1 is constant whose value is -0.64663, xi is the molar fraction of the i compound,γi

is the activity coefficient of compound i, which is calculated with the UNIFAC activity

coefficient model.

Table B.1: Critical constants of argon, xenon, CH4, CO2, N2, H2, CO, and H2O required in

the PSRK model

Component Pc/ MPa Tc / K

Acentric

factor/

ω

c1 c2 c3 MW /

g∙mol-1

argon 4.898 150.86 0 0 0 0 39.948

xenon 5.842 289.733 0 0 0 0 131.243

CH4 4.5992 190.564 0.0114 0.4926 0 0 16.043

CO2 7.3773 304.128 0.2236 0.8252 0.2515 -1.7039 44.009

N2 3.3958 126.192 0.0372 0.5427 0 0 28.014

H2 1.315 33.19 -0.2187 0.1252 0 0 2.016

Water 22.064 647.096 0.3443 1.0783 -0.5832 0.5464 18.015

Unifac method

The UNIFAC method [149] was developed by Fredenslund et al to predict the

activity coefficients in nonelectrolyte mixtures. It is a method that combines the functional-

group concept with a model for activity coefficients based on an extension of the quasi

chemical theory of liquid mixtures (UNIQUAC) [150]. Due to the predictive nature of the

method, it is used only in those cases where there is no experimental data available.

Another issue to take into consideration when using UNIFAC (in its original

version) is that there are two parameters tables, one is for vapour-liquid equilibria and the

other is for liquid-liquid equilibria. It is important to choose the correct table because the

results may differ substantially. Also, it is required that the compounds present in the

mixture can be split into the functional groups of the UNIFAC tables. Otherwise, this

method cannot predict any values of the activity coefficients.

The first parameters to be found are the Group Volume (Rk) and Surface Area (Qk),

which are parameters characteristic of each group. Once these values are known, the

volume contribution of each molecule can be calculated using Equation (B.12).

𝑟𝑖 = ∑ 𝜐𝑘(𝑖)

𝑅𝑘

𝑘

(B.12)

141

Correspondingly, the surface fractions are determined by means of Equation (B.13).

𝑞𝑖 = ∑ 𝜐𝑘(𝑖)

𝑄𝑘

𝑘

(B.13)

Table B.2: Molecular and group parameters for UNIFAC and LIFAC

Component group 𝜐𝑘(𝑖)

𝑅𝑘 𝑄𝑘

argon Ar 1 1.177 1.116

xenon Xe 1 1.13 1.13

CH4 CH4 1 1.129 1.124

CO2 CO2 1 1.3 0.982

N2 N2 1 0.856 0.93

H2 H2 1 0.416 0.571

Water H2O 1 0.92 1.4

TBAB

CH3 4 0.9011 0.848

CH2 12 0.6744 0.54

N+ 1 3.0 3.0

Br- 1 1.2331 1.1510

TBAC

CH3 4 0.9011 0.848

CH2 12 0.6744 0.54

N+ 1 3.0 3.0

Cl- 1 0.9861 0.9917

The variable νk(i) is the number of times that the group k is present in the molecule i. These

two properties, ri and qi, are molecular properties, and they do not depend on the

composition. The surface fraction of each molecule and each group (Qi, qi) are used to

determine the matrix ek,i, which contains the surface fraction of each group in each

molecule as shown in Equation (B.14).

𝑒𝑘,𝑖 = 𝜐𝑘

(𝑖)𝑄𝑘

𝑞𝑖

(B.14)

The following step is to determine the surface area fraction for each group in the mixture,

and it is carried out based on Equation (B.15).

142

𝜃𝑘 = ∑ 𝑥𝑖𝑞𝑖𝑒𝑘,𝑖𝑖

∑ 𝑥𝑗𝑞𝑗𝑗 (B.15)

The interaction parameters between groups (𝑎𝑚,𝑘, 𝑏𝑚,𝑘, 𝑐𝑚,𝑘), which are taken from the

UNIFAC tables [149], along with the temperature, are used to calculate the variable τm,k as

given by Equation (B.16).

𝜏𝑚,𝑘 = 𝑒𝑥𝑝 [−𝑎𝑚,𝑘 + 𝑏𝑚,𝑘𝑇 + 𝑐𝑚,𝑘𝑇2

𝑇]

(B.16)

This variable is used to compute βi,k (Equation (B.17)) and sk (Equation (B.18)).

𝛽𝑖,𝑘 = ∑ 𝑒𝑚𝑖𝜏𝑚,𝑘

𝑚

(B.17)

𝑠𝑘 = ∑ 𝜃𝑚

𝑚

𝜏𝑚,𝑘 (B.18)

The last two steps before calculating the activity coefficients are to calculate the volume

and surface area contributions for the molecules, Ji and Li (Equation (B.19) and Equation

(B.20) as follows:

𝐽𝑖 = 𝑟𝑖

∑ 𝑥𝑗𝑟𝑗𝑗 (B.19)

𝐿𝑖 = 𝑞𝑖

∑ 𝑥𝑗𝑞𝑗𝑗 (B.20)

Finally, with all these parameters, the activity coefficients can be computed. Usually, this

is performed separating the combinatorial and residual part of the activity coefficient, as

143

presented in Equation (B.21) and Equation (B.22), respectively. The sum of both parts

provides the value of the activity coefficient as shown in Equation (B.23).

𝑙𝑛(𝛾𝑖𝐶) = 1 − 𝐽𝑖 + 𝑙𝑛(𝐽𝑖) − 5𝑞𝑖 [1 −

𝐽𝑖

𝐿𝑖 + 𝑙𝑛 (

𝐽𝑖

𝐿𝑖)]

(B.21)

𝑙𝑛(𝛾𝑖𝑅) = 𝑞𝑖 [1 − ∑ (𝜃𝑘

𝛽𝑖,𝑘

𝑠𝑘 − 𝑒𝑘,𝑖𝑙𝑛 (

𝛽𝑖,𝑘

𝑠𝑘))

𝑘

] (B.22)

𝑙𝑛(𝛾𝑖) = 𝑙𝑛(𝛾𝑖𝑅) + 𝑙𝑛(𝛾𝑖

𝐶) (B.23)

Extension of PSRK to electrolytes

For systems dealing with electrolytes PSRK have been successfully used for

prediction of gas solubilities [151, 152, 153].Basically, the PSRK can be used without any

further modification in the vapour phase by assuming that electrolytes are not present in

the vapour phase. For the liquid phase, Equation (B.8), Equation (B.9), and Equation (B.10)

are changed to:

𝑎 = 𝑏𝑅𝑇 (1

𝐴1

𝑔𝐸


′𝑎𝑖


+1

𝐴1∑ 𝑥𝑖

′𝑙𝑛 (𝑏

𝑏𝑖)

𝑖

) (B.25)

𝑏 = ∑ 𝑥𝑖′𝑏𝑖

𝑖

(B.26)

𝑔𝐸

𝑅𝑇= ∑ 𝑥𝑖

′𝑙𝑛𝛾𝑖

𝑖

(B.27)

Where 𝑥𝑖′ is the salt-free mole fraction of component i (gas or solvent) in the liquid phase,

which allows the computation of a and b of the mixture parameters in the liquid phase

144

despite the fact that critical properties such as Pc and Tc of the ions are not available.

Another modification to the liquid phase is the calculation of the activity coefficients;

therefore, in Equation (B.27), 𝛾𝑖 is calculated with the LIFAC model [120]. The LIFAC

model is also based on the group contribution concept, and can be used to predict vapour-

liquid equilibria, osmotic coefficients and mean activity coefficients for electrolyte

systems.

LIFAC model

When a salt, such as TBAB or TBAC, is dissociated into ions in a liquid solvent,

three possible types of interaction can occur: solvent-solvent, ion-solvent, and ion-ion. The

aforementioned interactions are responsible for the thermodynamic properties of the

system. In the LIFAC model [120], several types of interactions are considered for

computing the excess Gibbs energy, as shown in Equation:

𝐺𝐸 = 𝐺𝐿𝑅𝐸 + 𝐺𝑀𝑅

𝐸 + 𝐺𝑆𝑅𝐸

(B.29)

Where 𝐺𝐿𝑅𝐸 term represents the long range (LR) interaction parameter due to Coulomb

electrostatic forces between the charged species (ions). 𝐺𝑀𝑅𝐸 represents the contribution of

the charge-dipole interactions and the charge-induced dipole interactions. 𝐺𝑆𝑅𝐸 expresses

the short range (SR) interactions and it is described using the UNIFAC method [149].

Finally, 𝐺𝐸 represents the excess Gibbs energy of the system. The activity coefficient of

the long range contribution is given by the following expression in the case of the solvent:

ln 𝛾𝑠𝑜𝑙𝑣,𝑠𝐿𝑅 =

2𝐴𝑀𝑠𝑑

𝑏3𝑑𝑠[1 + 𝑏𝐼0.5 − (1 + 𝑏𝐼

12)

−1

− 2 ln(1 + 𝑏𝐼0.5)] (B.30)

The activity coefficient for ion j is given by:

ln 𝛾𝑖𝑜𝑛,𝑗𝐿𝑅 =

2𝐴𝑀𝑠𝑑

𝑏3𝑑[1 + 𝑏𝐼0.5 − (1 + 𝑏𝐼0.5)−1 − 2 ln(1 + 𝑏𝐼0.5)]

(B.31)

145

Where Ms is the molecular weight of the solvent, I is the ionic strength, ds is the molar

density of the solvent s calculated from DIPPR tables, d is the density of the solvent

mixture, and A and b are both the Debye-Huckel parameters, which are calculated with the

following expressions

𝐴 = 1.327757 ∙ 10−5𝑑0.5

(𝐷𝑇)1.5

(B.32)

𝑏 = 6.359696𝑑0.5

(𝐷𝑇)0.5

(B.33)

The density of the solvent mixture is described with the following formula:

𝑑 = ∑ 𝜐𝑠𝑑𝑠

𝑠

(B.34)

Where 𝜐𝑠 is the salt-free volume fraction of the solvent is in the liquid phase and expressed

as follows:

𝜐𝑠 =𝑥′𝑠𝑑𝑠

∑ 𝑥′𝑖𝑑𝑖𝑠

(B.35)

Where 𝑥′𝑖 is the liquid phase mole fraction of the solvent i in a salt-free basis, T is the

temperature, and D is the dielectric constant of the solvent mixture, which in the case of a

binary solvent, is given by the following expression:

𝐷 = 𝐷1 + [(𝐷2 − 1)(2𝐷2 + 1)

2𝐷2− (𝐷1 − 1)] 𝜐2

(B.36)

D1 and D2 are the pure component dielectric constant of the solvents. The activity

coefficient for the middle range contribution is presented for both solvents and ions as

follows:

146

ln 𝛾𝑠𝑜𝑙𝑣,𝑘𝑀𝑅 = ∑ 𝐵𝑘,𝑖𝑜𝑛𝑚𝑖𝑜𝑛

𝑖𝑜𝑛

−𝑀𝑘 ∑ ∑ 𝜈𝑘

(𝑖)𝑥′

𝑖𝑖𝑘

𝑀∑ ∑ [𝐵𝑘,𝑖𝑜𝑛 + 𝐼𝐵′

𝑘,𝑖𝑜𝑛] 𝑥′𝑘𝑚𝑘

𝑖𝑜𝑛𝑘

− ∑ ∑ [𝐵𝑐,𝑎 + 𝐼𝐵′𝑐,𝑎] 𝑚𝑐𝑚𝑘

𝑎𝑐

(B.38)

ln 𝛾𝑠𝑜𝑙𝑣,𝑀𝑅 = ∑ 𝜈𝑘

(𝑖)ln 𝛾𝑘

𝑀𝑅

𝑘

(B.39)

ln 𝛾𝑖𝑜𝑛,𝑗𝑀𝑅 =

1

𝑀∑ 𝐵𝑘,𝑖𝑜𝑛𝑥′𝑘

𝑖𝑜𝑛

−𝑧𝑗

2

2𝑀∑ ∑ 𝐵′

𝑘,𝑖𝑜𝑛𝑥′𝑘

𝑗𝑘

+ ∑ 𝐵𝑐,𝑎𝑚𝑐

𝑘

−𝑧𝑎

2

2∑ ∑ 𝐵′𝑐,𝑎𝑚𝑐𝑚𝑎

𝑎𝑐

(B.40)

Where 𝑥′𝑘 is the salt-free mole fraction of solvent group k, 𝜈𝑘(𝑖)

Mk is the molecular weight

of solvent group k, M is the molecular weight of mixed-solvent. In the LIFAC model, Bi,j

is a second virial coefficient that represents the interaction between the species i and j. The

expressions for the ion-ion interaction parameter Bc,a and ion-solvent group interaction

parameter Bk,ion are presented as follows:

𝐵𝑐,𝑎 = 𝑏𝑐,𝑎 + 𝑐𝑐,𝑎𝑒𝑥𝑝(−𝐼0.5 + 0.13𝐼) (B.41)

𝐵𝑘,𝑖𝑜𝑛 = 𝑏𝑘,𝑖𝑜𝑛 + 𝑐𝑘,𝑖𝑜𝑛𝑒𝑥𝑝(−1.2𝐼0.5 + 0.13𝐼) (B.42)

Where bi,j and ci,j are the middle range parameters between compound i and j, which can

be either ion or solvents groups; bi,j and ci,j are product of regression of experimental data,

and its values are found in the LIFAC tables [120].

147

The complete expression for the activity coefficient of solvent s is given by the following

expression:

𝑙𝑛𝛾𝑠 = 𝑙𝑛𝛾𝑠𝐿𝑅 + 𝑙𝑛𝛾𝑠

𝑀𝑅 + 𝑙𝑛𝛾𝑠𝑆𝑅

(B.43)

The activity coefficient for ion j, the following equation is presented:

𝑙𝑛𝛾𝑗 = 𝑙𝑛𝛾𝑗𝐿𝑅 + 𝑙𝑛𝛾𝑗

𝑀𝑅 + 𝑙𝑛𝛾𝑗𝑆𝑅 − 𝑙𝑛 (

𝑀𝑠

𝑀+ 𝑀𝑠 ∑ 𝑚𝑖

𝑖𝑜𝑛

𝑖

) (B.44)

Where mi is the molality of the i ion.Finally, the partial fugacity coefficient in PSRK EoS

for compound i in the mixture is calculated with the following equations which are valid

for both liquid and vapour phases:

𝑙𝑛𝜑𝑖 =𝑏𝑖

𝑏(𝑧 − 1) − 𝑙𝑛 (

𝑃(𝑣 − 𝑏)

𝑅𝑇) − �̅�𝑖𝑙𝑛 (

𝑣 + 𝑏

𝑣)

(B.46)

�̅�𝑖 =1

𝐴1(∑ 𝑥𝑖𝑙𝑛𝛾𝑖

𝑖

+ ∑ 𝑥𝑖𝑙𝑛 (𝑏

𝑏𝑖)

𝑖

) + ∑ 𝑥𝑖

𝑎𝑖


(B.46)

Where z is the compressibility factor which is calculated with the analytical method of

Cardano and it is explained in detail in Appendix C, and v is the molar volume of the

mixture.

148

APPENDIX C: ANALYTICAL SOLUTION TO CUBIC EQUATIONS

A cubic equation of state can be solved by trial and error or analytically. In the

present work the roots of the equation of state were calculated analytically. Given an

expression for any cubic equation of state in terms of the compressibility factor, z: [119, p.

653]:

𝑧3 + 𝑈𝑧2 + 𝑆𝑧 + 𝑇 = 0 (C.1)

With the following abbreviations:

𝑃 =3𝑆 − 𝑈2

2

(C.2)

𝑄 =2𝑈3

27−

𝑈𝑆

3+ 𝑇

(C.3)

The discriminant can be determined to be

𝐷 = (𝑃

3)

3

+ (𝑄

2)

2

(C.4)

For D>0, the equation has only one real root:

𝑧 = (√𝐷 −

𝑄

2)

1/3

−𝑃

(3 (√𝐷 −𝑄2)

13

)

−𝑈

3 (C.5)

For D<0, there are three real roots. With the following abbreviations:

Θ = √−𝑃3

27 𝑎𝑛𝑑 Φ = arc cos (

−𝑄

2Θ)

(C.6)

The roots are given by the following expressions:

𝑧1 = 2Θ1/3𝑐𝑜𝑠 (Φ

3) −

𝑈

3

(C.7)

149

𝑧2 = 2Θ1/3𝑐𝑜𝑠 (Φ

3+

2𝜋

3) −

𝑈

3

(C.8)

𝑧3 = 2Θ1/3𝑐𝑜𝑠 (Φ

3+

4𝜋

3) −

𝑈

3

(C.9)

The largest and the smallest of the three values correspond to the vapour and to the liquid,

respectively. The middle root one has no physical meaning.

150

APPENDIX D: ERROR CALCULATION IN THE PARAMETER ESTIMATION

Once the parameter estimation is performed, the standard error of the parameters

can be obtained as the square root of the corresponding diagonal element of the inverse of

the matrix A* multiplied by the variance [121].

�̂�𝑘𝑖= �̂�𝜀√{[𝐴∗]}𝑖𝑖

(D.1)

Where:

�̂�𝜀 =𝐹(𝑎𝑛,𝑚, 𝑏𝑛,𝑚)

𝑑. 𝑓.

(D.2)

𝐴 = ∑ [𝜕𝒆𝑇

𝜕𝒌]

𝑁𝑃

𝑖=1

∙ [𝜕𝒆𝑇

𝜕𝒌]

𝑇

(D.3)

𝑒𝑖 = 𝑥𝑔𝑎𝑠𝑐𝑎𝑙𝑐 − 𝑥𝑔𝑎𝑠

𝑒𝑥𝑝 (D.4)

Where k is the vector of estimated parameters, �̂�𝑘𝑖 is the standard error of parameter

ki d.f. is the degrees of freedom, given by the number of experimental points minus the

number of parameters.

151

APPENDIX E: PRIVATE COMMUNICATION WITH KWATERSKI:

Dear Matthew

Please find the answer of Matthias Kwaterski

Best regards,

Well, I have checked the results of the co-worker of Mr Clarke and they

are basically correct except from a minor error: the gamma values as

well as the absolute values of the results for delta_diss_G_0 are

correct, however, they have to be multiplied by -1. In other words, the

numerical values of delta_diss_G_0 have to be positive. In the formula

for delta_diss_G_0 of the 38-semi-clathrate as presented in his MS Word

document, he erroneously wrote 285,15 K. Of course, this temperature

value should be replaced by 283,5 K, but I’ve checked the results

provided in his table and they are correctly calculated. Thus, when

using the isothermal parameter values for the eNRTL parameters at

298,15 K, I ended up at the numerical results 24438,6 J mol-1 and

25378,5 J mol-1 for delta_diss_G_0 of the 26 and the 38 semi-clathrate

hydrate, respectively. With except from the sign, these values do match

with the values I have calculated. Therefore, also the calculated gamma

values (table two in the MS Word document) as well as the values

presented in table 1 do match with my results. That further means that

the values cited in the poster shown in China were unfortunately not

presented correctly.

Le 19/01/2015 18:18, Matthew Clarke a écrit : Dear Jean-Michel,

How are you? I have a student who is currently working on computing

equilibrium in semi-clathrates, using your approach, and he is having

difficulty reproducing your values for the Gibbs Free Energy Change of

Dissociation. I have asked him to type up a detailed sample calculation

to show the numbers that he is using to compute the Gibbs Free Energy

Change and I have attached this document. In this document he not only

shows his calculation of ΔG but he also shows that his eNRTL model

reproduces the results of Chen. Can you kindly take a moment to compare

his numbers with yours? I've been working with my student for over a

month and I'm not able to see any glaringly obvious mistakes in his

numbers. Perhaps we have misinterpreted something that was written?

Thank you for your time Jean-Michel

Matthew Clarke

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Measuring and Modelling of the Thermodynamic Equilibrium