UNIVERSITY OF CALGARY
Thermodynamic Formation Conditions for Propane Hydrates in Equilibrium with Liquid Water
by
Kayode Israel Adeniyi
A THESIS
SUBMITTED TO THE FACULTY OF GRADUATE STUDIES
IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE
DEGREE OF MASTER OF SCIENCE
GRADUATE PROGRAM IN CHEMISTRY
CALGARY, ALBERTA
AUGUST, 2016
© Kayode Israel Adeniyi 2016
Abstract
Accurate knowledge for hydrate formation conditions of pure propane in the presence of liquid
water is important to avoid flow assurance issues in processing, storage and transportation of
liquefied petroleum gas, as well as modeling hydrate based separation processes involving type
II hydrates. Experimental dissociation conditions were measured using the phase boundary
dissociation method, which has the advantages of reduced experimental time and generation of
more equilibrium dissociation data when compared to other hydrate measurement techniques. In
this study, phase equilibra measurements are reported for 99.5 mol % and 99.999 mol % propane
at the phase boundary for the liquid water (Lw)-hydrate (H)-propane vapour (C3H8 (g)) and Lw H-liquid propane (C3H8 (l)) loci. The results were modeled using van der Waal and Platteeuw
model for the hydrate phase and reduced Helmholtz energy equation for the fluid phases. Results
are compared with highly variant literature data, where large deviations observed for the Lw-HC3H8 (l) phase boundary can be mainly attributed to purity and experimental techniques used in
the literature.
ii
Acknowledgements
I am deeply grateful to my Supervisor, Dr. Robert Marriott for the opportunity to work on this
project, patience, support, understanding and supervision of this thesis. Much of my
understanding on phase behavior of hydrate formers is credited to his mentorship.
Thank you to Dr. Viola Birss and Dr. Peter Kusalik for agreeing to serve on my committee and
their support.
I acknowledge the financial support of University of Calgary, Dr Marriott’s NSERC ASRL
Industrial Research Chair in Applied Sulfur Chemistry, Department of Chemistry Graduate
Student Award (2014, 2015 and 2016) and Faculty of Graduate Studies Travel Grant (2015 and
2016).
My sincere appreciation goes to Alberta Sulphur Research Ltd. (ASRL) staff for providing a
friendly working atmosphere. I am grateful to Connor Deering for his help with modeling,
experimental expertise, reviewing my writing and more importantly, his friendship. Thanks to
Zachary Ward for always answering my unending questions about the experimental setup. I also
wish to thank the postdoctoral researchers, Dr. Payman Pirzadeh and Dr. Fadi Alkhateeb for their
words of encouragement and constructive criticism of my writings.
iii
My appreciations also go to the graduate coordinator, Janice Crawford, Patricia Alegre of ASRL
and other administrative staff of the department for their advice and guidance on university
regulations. I would also like to thank my colleagues in the research group for their support.
Omowumi, thank you for your understanding, support, sacrifices and giving me the most
valuable gift of life that I cherished. To my beloved parents, I say thank you for your support and
guidance through the years.
iv
Dedication
This thesis is dedicated to God Almighty, the GREAT architect of life and all positive
inspiration.
v
Table of Contents
Abstract ........................................................................................................................................ ii
Acknowledgements ..................................................................................................................... iii
Dedication.......................................................................................................................................v
Table of Contents ........................................................................................................................vi
List of Tables ................................................................................................................................ix
List of Figures ............................................................................................................................. xi
List of Symbols, Abbreviations and Nomenclature................................................................xiii
Chapter One: Introduction………………………………………………………………….…..1
1.1 Outline ………………………………………………………………………………….……1
1.2 Motivation for study…………………………………………………………………….…...1
1.3 History of gas clathrate hydrates………………………………………………………….....3
1.4 Structure and formation of clathrate gas hydrate…………………………………………....4
1.4.1 Structure I…………………………………………………………………….…….....7
1.4.2 Structure II ………………………………………………………………………........7
1.4.3 Structure H…………………………………………………………..…………….…..7
1.5 Applications of gas hydrate………………………………………………………………….8
1.5.1 Hydrogen (H2) storage………………………………………..…………………….....8
1.5.2 Separation processes…………………………………………………………………..9
1.5.3 Desalination of seawater……………………………………………………………..10
1.5.4 Potential source of energy……………………………………………………………10
1.5.5 Natural gas hydrate in flow assurance………………………………………………..11
1.5.5.1 Gas hydrate prevention and control……………………………..…………...…..12
1.6
Importance of propane hydrate formation conditions studies………………………..........12
1.7
Phase behavior and avoiding hydrate formation………….……………………………….13
vi
1.7.1 Phase behavior of a hydrocarbon hydrate former………….…………..……….......14
1.7.2 Phase behavior of C3H8 + H2O system…………………………………..……........15
1.7.3
Semi empirical model for hydrate dissociation correlation……….……….............17
1.8 Experimental dissociation data for C3H8 hydrate……………………………………..…...17
1.9 Review of gas hydrate thermodynamic models………………………………………...….20
Chapter two: Review of Literature Techniques, Experimental Procedure
and Calibration……………………………………………………………………………...…24
2.1 Outline………………………………………………………………………………..........24
2.2 Methods of studying gas hydrate equilibra…………………………………...………...….24
2.2.1 Dynamic method………………………………………...………………...……......24
2.2.2 Static method…………………………………………………………..……......….25
2.2.2.1 Isothermal method…………………………………………..………...…..25
2.2.2.2 Isobaric method……………………………………………..………...…..26
2.2.2.3 Isochoric method………………………………………………...…....…..26
2.2.2.4 Phase boundary dissociation method………………………………...…..27
2.3 Review of selected experimental apparatus for hydrate dissociation studies……………...29
2.4 Apparatus used for this study………………………………………………………….......32
2.4.1 Interfacing experimental setup with Laboratory Virtual Instruments
Engineering Workbench (LABVIEW) for data acquisition…………………..…....34
2.5 Materials…………………………………………………………………………………...36
2.6 Experimental procedure……………………………………………………………….......36
Chapter three: Experimental Results and Modelling…………………………………….....38
3.1 Outline……………………………………………………………………………………..38
3.2 Thermodynamic modeling………………………………………………………………....38
3.2.1 Fluid phase modeling……………………………………………………………….....39
3.2.2 Description of the hydrate phase……………………………………………….….…..46
3.2.2.1 The Vander Waal and Platteeuw hydrate model…………………………….…....46
3.2.2.2 Calculation of hydrate phase fugacity……………………………………….…...48
vii
3.2.2.3 Hydrate cage occupancy……………………………………………………….......51
3.2.2.3.1 Optimization of Kihara potential paramaters……………………………........53
3.3 Algorithm for calculating equilibrium hydrate formation condition………………………..54
3.4 Experimental results and discussion…………………………………………………...........56
3.4.1 Model comparisons to experimental and literature data along the Lw-H-C3H8(g)…......61
3.4.2 Model comparisons to experimental and literature data along the Lw-H-C3H8(l)….......67
3.4.3 Comparisons of upper quadruple points of this study and literature………………........72
Chapter four: Conclusion, recommendation and future work……………………………....74
4.1 Conclusion……………………………………………………………………………..…....74
4.2 Recommendations……………………………………………………………………….….75
4.3 Future work………………………………………………………………………………....75
Appendix A: Calibrations and results…………………..………………………………………77
A.1.1 Pressure calibration………………………………………………………………………………77
A.1.1.1 Primary transducer calibration through the use of Deadweight Testers……………..77
A.1.1.2 Secondary transducer calibration………………………………………………….……….78
A.1.1.3 Results and discussions……………………………………………………………………….79
A.1.2 Temperature calibration………………………………………………………………………….81
A.1.2.1 The International Temperature Scale……………………………………………………….81
A.1.2.2 Calibration procedure………………………………………………………………….........82
A.1.2.3 Result and discussion……………………………………………………………………….…83
A.1.3 Volume calibration…………………………………………………………………….…85
Appendix B: Pressure versus temperature plots of the experimental run for the dissociation
points along Lw-H-C3H8(g) and Lw-H-C3H8(l) phase boundaries reported in this study………86
Appendix C: Parameters and coefficients used in the reduced energy Helmholtz EOS for
calculation of thermodynamic properties of C3H8 in equation 3.13……………………….…....91
Appendix D: First derivative of  and the reducing function  r and Tr with respect to n i …...92
Appendix E: Copyright Permissions……………………………………………………………93
viii
References……………………………………………………………………………………....95
ix
List of Tables
Table 1.1. Comparison of structure I, structure II and structure H hydrates…………………….6
Table 1.2. Summary of experimental dissociation conditions along the Lw–H–C3H8 (g) phase
boundary…………………………………………………………………………………………18
Table 1.3. Summary of the experimental dissociation conditions along the Lw–H–C3H8 (l) phase
boundary…………………………………………………………………………………………19
Table 1.4. Summary of C3H8 hydrate upper quadruple points reported in literature…………...20
Table 2.1. Methods used for the study of C3H8 hydrate dissociation conditions….……………32
Table 2.2. Measured gas impurities (in moles) in C3H8 used for this work……….……………36
Table 3.1. Binary parameters of the reducing functions for density and temperature used in
equations 3.18 and 3.19………………………………………………………………………….44
Table 3.2. Thermodynamic reference properties for structure II used in this study…………….50
Table 3.3. Optimised Kihara potential paramaters used for this study………………………….54
Table 3.4. Experimental dissociation conditions for C3H8 hydrates along the Lw–H–C3H8(g)
phase boundary…………………………………………………………………………………..58
Table 3.5. Experimental dissociation conditions for 99.999 mol % C3H8 hydrates along the Lw–
H–C3H8(l) phase boundary………………………………………………………….…………...59
Table 3.6. Model comparison to the experimental data along the Lw–H–C3H8(g) locus……....61
Table 3.7. Summary of literature data along the Lw-H-C3H8(g) phase boundary compared.…..63
Table 3.8. Model comparison to the experimental data along the Lw–H–C3H8(l) phase
boundary…………………………………………………………………………………………68
Table 3.9. Summary of literature data and corresponding purities along the Lw–H–C3H8(l) phase
boundary…………………………………………………………………………………………70
Table 3.10. Quadruple points conditions from this study and literature….…………………...73
Table A.1. Comparison of the pressures measured by the calibrated primary Paroscientific
Transducer, p cal , and the uncalibrated Paroscientific Pressure Transducer, p meas ………….…80
Table A.2. Comparison of the pressures measured by the calibrated primary Paroscientific
Transducer, pcal, and the uncalibrated Keller Pressure Transducer, p meas ……………………...80
Table A.3. The experimentally measured melting points of H2O with the corresponding
deviations………………………………………………………………………………………..84
x
Table A.4. Comparison of the measured temperatures from the calibrated, Tcal, and uncalibrated
PRT, Tmeas (used inside the water bath)……………………………………...…………….…….84
xi
List of Figures
Figure 1.1. Classification of clathrate gas hydrates……………………………………………..5
Figure 1.2. Typical pressure − temperature diagram for a hydrocarbon hydrate former………15
Figure 1.3. Pressure − temperature diagram of propane + water system……………………….16
Figure 2.1. A Typical pressure-temperature curve of a gas hydrate formation and dissociation
using the isochoric method………………………………………………………………………27
Figure 2.2. Experimental schematic of Deaton and Frost apparatus for phase equilibra studies of
gas hydrates……………………………………………………………………………………...30
Figure 2.3. Diagram of the dynamic apparatus designed and constructed by Hammerschdmit for
studying hydrate formation conditions………………………………………………………….31
Figure 2.4. Schematic diagram of the setup used for the measurement of hydrate dissociation
points…………………………………………………………………………………………….34
Figure 2.5. LabVIEW front panel for the experimental propane hydrate dissociation setup used
in this study……………………………………………………………………………………...35
Figure 3.1. Correlation plot comparing the experimental mole fraction of water in propane to
those calculated using REPFROP 9.1……………………………………………………..……..46
Figure 3.2. Simplified flowchart for the calculation of dissociation temperature used for the
thermodynamic modeling in this study…………………………………………………………..55
Figure 3.3. Pressure versus temperature plot of this study experimental, literature data and this
thermodynamic model along the Lw–H–C3H8 (g) and Lw–H–C3H8 (l) phase boundaries………60
Figure 3.4. Pressure versus temperature plot of this study experimental result, model, empirical
correlations and literature data along the Lw–H–C3H8 (g) locus………………………………...64
Figure 3.5. Temperature deviations between model and this study experimental data, literature
data and correlations along the Lw-H-C3H8(g) locus……………………………………………65
Figure 3.6. Relationship between propane purities and variance of the literature data along the
Lw-H-C3H8(l) locus to the model presented in this study…………………………………….....67
Figure 3.7. The pressures versus temperatures plot of this study dissociation conditions, model
and literature data along the Lw–H–C3H8 (l) locus……………………………………………....69
Figure 3.8. Temperatures deviation of the model presented in this study to the literature data
along the Lw-H-C3H8 (l) locus…………………………………………………………………..71
xii
Figure 3.9. A graphical representation of upper quadruple point determination from the point of
intersection of the Lw-H-C3H8 (g) locus and Lw-H-C3H8 (l) loci……………………………….72
Figure A.1. A representative temperature-time plot showing the water freezing points for the
PRT probe calibration……………………………………………………………………………83
Figure B.1. Pressure versus temperature profile for 99.999 mol % C3H8 + H2O showing the
cooling, hydrate formation and heating stages along the Lw-H-C3H8(g) locus…………………86
Figure B.2. Pressure versus temperature profile for 99.5 mol % C3H8 + H2O showing the
cooling, hydrate formation and heating stages along the Lw-H-C3H8(g) locus…………………86
Figure B.3. Pressure versus temperature profiles for C3H8 + H2O showing the cooling, hydrate
formation and heating stages along the Lw-H-C3H8(l) locus……………………………………87
xiii
List of Symbols, Abbreviations and Nomenclature
Abbreviations and Symbols
Definition
A
Helmholtz energy
AD
Average deviations
a
Spherical molecular core radius
Acorr
Cross sectional area
Aid .mix
Ideal gas mixture contribution to the
Helmholtz energy
AE
Departure function or excess contribution to
the Helmholtz energy
Ar
Pure fluid residual Helmholtz energy
Aο
Ideal gas Helmholtz energy
Amj and Bmj
Langmuir constant fitting parameter related to
guest molecule j in cavity m
Ap
Buoyancy corrected factor
b
Temperature correction coefficient
CCS
Carbon capture and sequestration
C mj
Langmuir constant of gas j in cavity type m
c οp
Ideal gas heat capacity
c οpw
Reference standard difference in heat capacity
between ice and liquid water
EOS
fi
Equation-of-state
Fugacity of gas component i in a mixture
xiv
f H2O ,C3H8
Fugacity of water in propane phase
f C3 H 8 , H 2 O
Fugacity of propane in water phase
f wH
Fugacity of water in hydrate
f wL
Fugacity of pure water
f w
Fugacity of water in empty hydrate lattice
F
Degree of freedom in a thermodynamic system
Fij
Binary interaction parameter for mixture i and j
ITS
The International Temperature Scale
g
Gravitational constant
H
Enthalpy
HBGS
Hydrate based gas separation
hο
Ideal gas enthalpy
hοο
Ideal gas enthalpy at arbitrary reference state
I
Ice water
KB
Boltzmann constant
Lw
Liquid water
LHC
Liquid hydrocarbon
LPG
Liquefied petroleum gas
mi
Number of face of type i hydrate former
N
Number of moles
ni
Number of edges in the hydrate cage of type i
hydrate former
xv
N
NP
P
PBD
p
Number of components in a thermodynamic
system
Number of experimental data points
Number of phase in a thermodynamic system
Phase boundary dissociation method
Pressure
PT
Pressure transducer
POI
Polynomial term in the dimensionless residual
Helmholtz energy equation
p   T
PRT
Pressure-density-temperature
Platinum resistance temperature
p meas
Pressure measured from uncalibrated
transducer
p cal
Pressure measured from a calibrated transducer
R
Ideal gas constant
sI
Structure I
sII
Structure II
sH
Structure H
sο
Ideal gas entropy
s οο
Ideal gas entropy at arbitrary reference state
Tt
Triple point
T
Temperature
Tο
Arbitrary reference temperature
Tc
Critical temperature
xvi
Tr
Reduced temperature
TtTc
Vapour pressure of a hydrocarbon
Tcal
Temperature measured from a calibrated probe
Tcalc
Calculated temperature
Texp
Experimentally measured temperature
R
Distance between encaged gas molecule from
the center of the cavity
r
Cavity radius
u&v
Coefficient for speed of sound derived from
Einstein equation
VA1
Inlet feed valve
VA2
Outlet valve
w(r)
Cell potential function for the interaction
between guest molecules
V
Vapour phase
V
Volume
VBA
Microscoft visual basic for applications
vdWP
Van der Waal and Platteeuw model
VLE
Vapour liquid equilibrium
vm
Number of type m cavities per water molecule
Q
Quadruple point
Q1
Lower quadruple point
Q2
Upper quadruple point
X
Structure I or II hydrate former
xvii
xi
Mole fraction of component i
Y
Structure H hydrate former
y H 2 O , C3 H 8
z
Mole fraction of saturated water in propane
Coordination number
h wο
Reference enthalpy difference between the
empty hydrate lattice and ice phase.
hw
Enthalpy change between empty hydrate lattice
and liquid water
 w  L
Difference in chemical potential between
empty hydrate cage and pure water phase
 w  H
Difference in chemical potential between
empty hydrate cage and filled hydrate cage
 wο
Reference chemical potential difference
between empty hydrate lattice and pure water
∆s
Specific entropy change
∆V
Specific volume change
v m
Reference volume difference between empty
hydrate lattice and pure ice water phase

Fugacity coefficient
w
Activity coefficient of water
(r )
Potential energy for interaction between
molecule within the cavities

Characteristic minimum energy

Collision diameter
 jm
Fractional cage occupancy of gas molecule i
within the hydrate cavity m.

Chemical potential of empty hydrate lattice
xviii
L
Chemical potential of liquid water
H
Chemical potential of filled hydrate lattice

Dimensionless Helmholtz energy
ο
Dimensionless ideal gas Helmholtz energy
r
Dimensionless residual Helmholtz energy

Density
ο
Ideal gas density
c
Critical density
 fluid
Density of deadweight hydraulic fluid
r
Reduced density

Reduced density

Reduced temperature
ο
Reduced temperature at reference state
512
Pentagonal dodecahedron
51262
Tetrakaidecahedron
51264
Hexakaidodecahedron
435663
Irregular dodecahedron
51268
Icosahedron
xix
CHAPTER ONE: Introduction
1.1
Outline
The aim of this study was to measure the dissociation conditions for propane hydrates and to
develop a thermodynamic model for accurate calculation for hydrate formation conditions. At
equilibrium, formation conditions are equivalent to the dissociation conditions. In this chapter,
the history, fundamental background and formation conditions of clathrate hydrate is presented
in order to provide the motivation for this work. The different structures of gas hydrates as well
as their applications are discussed. Because propane is the hydrate of interest in this study,
threats posed by its occurrence in flow assurance and application as a potential replacement for
the energy intensive amine processes for removal of acid gas in sour natural gas production are
discussed. A literature review of available formation conditions also is presented. Finally, a
review of the thermodynamic models used for predicting clathrate hydrate formation conditions
is briefly discussed in the final section of this chapter.
1.2
Motivation for study
Propane is the most prevalent liquefied petroleum gas (LPG),1-2 where LPG consumption is
steadily increasing because of its applications as low carbon energy for transportation, farming,
power generation, cooking and heating purposes.1–4 LPG is primarily propane (C3H8) and butane
(C4H10). C3H8 is formed naturally and is found in association with reserves of oil and natural
gas.5 LPG is normally separated from crude oil or natural gas as a by-product, where natural gas
purification produces approximately 55 % of all LPG while crude oil refining accounts for the
remaining 45 %.1,6 Depending on the source of the LPG and production history of the reservoir,
1
non-hydrocarbon impurities also may be present such as water (H2O), hydrogen sulfide (H2S)
and carbon dioxide (CO2) that must be removed before the LPG can be transported in pipelines
and trucks as a salable product.1,4,6
Gas clathrate hydrates are crystalline solid compounds formed from water and suitably small
molecules at appropriate conditions, typically at high-pressure and low-temperature. These
molecules can be hydrocarbons such as methane (CH4), ethane (C2H6), ethene (C2H4) and C3H8
or non-hydrocarbons, such as like H2S and CO2.7-8 Water naturally coexists with oil and gas
inside all subsurface reservoirs. During production, operating conditions inside producing wells,
subsea transfer lines, risers and pipelines can fall within the conditions that favour hydrate
formation.7,9- 10 Gas hydrates can also form in a single phase fluid with dissolved water such as
in transportation pipelines, transport trucks and LPGs storage facilities.11 Both hydrate
formations can lead to serious safety problems and operational shutdowns which can result in
large economic losses. Because of the potential threats presented by hydrate formation in those
aforementioned areas and C3H8 being a main component of LPG, accurate calculation of C3H8
hydrate formation conditions is important, so as to avoid and control its occurrence during LPG
production.
Although gas hydrates are undesirable during production in the oil and gas industry, they have
been considered for the separation of gas impurities from sour natural gas (i.e., natural gas that
contains an appreciable quantity of H2S), coal and CO2 streams for carbon capture and
sequestration (CCS) technologies.9,12–15 C3H8 hydrates have small unoccupied cages; its hydrate
can be formed so that the small sized gas impurities like H2S and CO2 are captured inside the
2
unoccupied hydrate cages and then dissociated to release the captured impurities. This process
offers an alternative to the energy intensive amine treatment that is currently used for removing
CO2 and H2S from sour natural gas. The design of such a process, otherwise known as hydrate
based gas separation (HBGS), requires accurate knowledge for the formation conditions of C3H8
hydrate and mixed hydrates with other impurities.
1.3
History of gas clathrate hydrates
The discovery of gas clathrate hydrates is often attributed to Sir Humphrey Davy in 1810.7–8,16
However in 1778, John Priestley first observed that sulphur dioxide (SO2) would impregnate
water and cause it to repeatedly freeze, whereas hydrochloric acid (HCl) and silicon tetrafluoride
(SiF4) would not induce this effect when he left the window of his laboratory open overnight in
winter.7,17 Unlike Davy’s experiments, Priestley’s temperature (265 K) of the gas mixture was
well below the ice point and it was not clear whether the structure formed was a hydrate.7-8 Davy
reported the hydrate of chlorine, in which he noted that, the ice-like solid formed at temperatures
greater than the freezing point of water, and the solid was composed of more than just water,
thus, a compound structure must have been formed.16
Villard (1888) first discovered and reported the natural gas hydrates of CH4, C2H6, C2H4 and
C3H8.7,18–19 Clathrate hydrates were referred to as a scientific curiosity up until 1934 when
Hammerschmidt discovered that they were the main culprit for plugging oil and gas pipelines in
Canada.20 Natural gas has suitably sized components such as CH4, C2H6, and C3H8 that are
capable of forming clathrate hydrates in pipelines or other facilities under favourable
3
thermodynamics conditions.7,20 Hammerschmidt discovered the presence of natural gas hydrates
in pipelines at relatively high-pressures and low-temperatures, where it was then believed that it
was impossible for ice to exist.20 However, he noted that removal of the liquid water phase
completely eliminated the possibility of hydrate formation in a pipeline, as the hydrate could not
form until the liquid water dew point was reached. 9,20 The understanding has been revised with
recent work, where it is now well understood that hydrates can form without dense-phase water.8
1.4
Structure and formation of clathrate gas hydrate
Gas clathrate hydrates are structures in which suitably sized small molecules are enclosed or
enclathrated in cages formed by water molecules.7-9 The small molecules are commonly referred
to as a “guest” or “former” while the water forming the hydrate cages are called “host”
molecules.8 Water makes up ca. 85 % of the composition of a hydrate lattice while the guest
molecules constitute ca. 15 %.7,9 Within the cavity, small guest molecules can freely rotate and
vibrate, but have limited translational motion.7,13 The cages are composed of hydrogen-bonded
water molecules mainly in the form of five and six–membered rings.7-8
Von Stackelberg and coworkers (1949) studied and identified the crystal structure of hydrates
using X-ray diffraction techniques.21 They classified the structure into cubic structure I (sI) and
cubic structure II (sII) based on the type of cages found in the crystal and guest molecule
sizes.7,21 Hexagonal structure H (sH) was later discovered by Ripmeester et al. in 1987 through
the use of solid state nuclear magnetic resonance and X-ray diffraction techniques.22 The general
classifications and nomenclature for clathrate gas hydrates are shown in Figure 1.1.
4
m
Jeffrey et al.(1984) proposed a nomenclature in the form ni i for hydrate structures where ni
represent the number of edges in a face of type i and superscript mi is the number of faces.23 The
structure of these rings defines the types of hydrates formed by a guest molecule.9 The three
structures discussed above are composed of small dodecahedral cages (512) as a building block.
The 512 cage is composed of twelve pentagonal faces, formed by water molecules that are
bonded to each other by hydrogen bonding, with the oxygen atoms at each vertex.7–9 The 51262
cage is formed from twelve pentagonal and two hexagonal faces because 512 cages alone will
experience strain on the hydrogen bonds.9,21 The 51264 cage is made up of 512 cages and four
hexagonal faces that further relieve the hydrogen bond strain on the 512 cages when they are
connected to each other through the faces.7,9,22 The irregular dodecahedron (435663) cage consists
of six pentagonal, three square and three hexagonal faces that have a considerable amount of
bond strain.9,22,24 The 435663 cage is slightly larger and less spherical than the 512 cage, but both
cages can accommodate small guest molecules like CH4 with the 51268 cage being slightly larger
in size.24 Table 1.1 compares the hydrate structures at different level of cage occupancy.
Figure 1.1. Classification of clathrate gas hydrates. 512, 51262, 51264, 435663 and 51268 represent
pentagonal dodecahedron, tetrakaidecahedron, hexakaidecahedron, irregular dodecahedron and
icosahedron cages, respectively.25(reproduced with permission)
5
Table 1.1. Comparison of structure I, structure II and structure H hydrates.7-8,13
Structure I
Structure II
Structure H
Crystal system
Cubic
Cubic
Hexagonal
Lattice description
Primitive
Face centered
Hexagonal
Water Molecules per unit cell
46
136
34
512
20
20
20
51262
24
-
-
51264
-
28
-
51268
-
-
36
435663
-
-
20
All cages filled
X+5 ¾ H2O
X+5 ⅔H2O
5X+Y+34 H2O
Mole fraction hydrate former
0.1481
0.1500
0.1500
Structure I
Structure II
Structure H
Only large cages filled
X+7 ⅔ H2O
X+17 H2O
-
Mole fraction hydrate former
0.1154
0.056
-
Volume of unit cell (m³)
1.728 × 10-27
5.178 × 10-27
-
Coordination number of
cages in different structure*
Theoretical Formula†
† Where X is the structure I and II hydrate formers while Y is a structure H former.
*Number of oxygens at the periphery of each cavity.
6
1.4.1 Structure I
In a sI hydrate, each unit cell consists of 46 water molecules that form two small dodecahedral
(512) and six large tetradecahedral (51262) cages.7,21 Both cages in the sI gas hydrate can
accommodate only small guest gas molecules with molecular diameters up to 6.0 Å such as CH4,
C2H6, CO2, and H2S.7-8,21 The 512 cages have a internal free diameter of ca.5.1 Å, thus, they can
accommodate the smaller guest molecules less than the size of their diameter. Guest molecules
like CH4 (4.36 Å diameter) can effectively occupy the small cages while larger molecules, such
as C2H6 (5.5 Å diameter), occupy the 51262 cages which have a diameter of 5.86 Å.7-9
1.4.2 Structure II
The unit cell of sII gas hydrate consists of sixteen small dodecahedral (512) cages and eight large
hexakaidecahedral (51264) cages formed by 136 water molecules. The 51264 cages have a internal
free diameter of 6.66 Å that allows them to accommodate gases with molecular diameters in the
range of 5.9 to 7.0 Å, such as C3H8 and C4H10.7,9 As is the case for sI, unoccupied 512 cages in sII
can potentially accommodate small molecules like CO2, H2S and CH4.9 This is the principle used
in hydrate based gas separation processes (HBGS) such as flue gas removal and CCS
technology.26–32 sII hydrates are the most common form of gas hydrates encountered in natural
gas production.9
1.4.3 Structure H
The unit cells of structure H (sH) are each made up of three small dodecahedral (512) cages, two
medium irregular dodecahedral (435663) cages, and one large icosahedral cage (51268) formed
from 34 water molecules.7-8,22 Examples of sH hydrate formers are 2,2-dimethylbutane, 2,3dimethylbutane, cyclopentane (C5H10). sH hydrates are always double hydrates, meaning small
7
guest molecules, commonly referred to as “help gas” such as CH4 are required to occupy and
stabilize the small (512) and medium (435663) cages of the structure, while large molecules with
sizes ranging from 7.5 to 8.6 Å, such as the ones listed above, occupy the large 51268 cages.7-8,24
The presence of guest molecules inside the hydrate cages causes a stabilization, so when the
majority of the cages are unoccupied, the hydrate structure collapses.7-9 This stabilization is
postulated to be due to van der Waal forces because there are usually no other bonds available
between the host water cages and a guest molecules.7-9,13 Aside from presence of a stabilizing
molecule, the formation of gas hydrate also depends on two other main conditions: (a) the right
combination of pressure and temperature (i.e., typically high-pressure and low-temperature), and
(b) a sufficient amount of water.7-8,16 Gas clathrate also can be non-stoichiometric, i.e., their cage
occupancy is a function of the pressure and temperature conditions and not the number of cages
available.8 Hydrate formation is commonly favoured in locations such as gas valves where
narrowing within the valves causes Joule-Thompson temperature reduction.8,20 Factors that can
enhance kinetic hydrate formation are nucleation sites such as imperfections in pipelines, weld
spots, or pipeline fittings and high-velocity turbulence.
1.5
Applications of gas hydrate
1.5.1 Hydrogen (H2) storage
Dyadin discovered the clathrate hydrate of hydrogen (H2) in 1991 which has since been
extensively studied for hydrogen storage.33–38 With a diameter of 2.72 Å, H2 typically forms a
type II structure at a pressure of 300 MPa with an H2 to H2O ratio as high as 1:2 due to the 512
8
and the 51264 cages holding up to 2 and 4 molecules of H2 respectively.33,35-36 Because of the
relatively high pressures required for pure H2 hydrates, binary hydrates (i.e., hydrate cages
containing two different guest molecules) are easier to form and study, to reduce the pressure for
H2 storage in the clathrate hydrate cages.38 Binary hydrates have been reported for H2 and
tetrahydrofuran (THF) at relatively lower pressures.34,36,38 Although, THF reduces the hydrate
formation pressure and increases the dissociation temperature, it also occupies some of the
hydrate cages that can potentially be used for hydrogen storage.26,39
C5H10 has been reported as an alternative to THF because it also reduces H2 hydrate formation
pressures but it also occupies the hydrate cages.39-40 Skiba et al. (2009) reported that at pressures
between p = 200 – 250 MPa, the double hydrate of the H2–C3H8–H2O system decomposes at a
temperature 20 K higher than pure C3H8 hydrate and H2 occupancy inside the hydrate cages also
increases.41 This makes C3H8 a better potential alternative to both C5H10 and THF for hydrogen
storage.
1.5.2 Separation processes
The World’s proven conventional natural gas reserves are abundant with a large number of the
reserves considered to be sour.7,13,42 Separation of CO2 and H2S from natural gas in the vapour
phases has been done on an industrial scale for at least the last 70 years and has been typically
achieved using aqueous amines.43-44 HGBS processes are being developed for separating CO2
and H2S from sour natural gases as an alternative to this energy intensive amine processes
currently used.26,39,44–45 The captured gas is released at high-pressure and low-temperature,
reducing costs for liquefaction of the CO2 and H2S products.26,39 Another advantage of an HBGS
9
process are that they operate near room temperature during the dissociation process, thereby
reducing heating costs.39 Most HGBS processes are found to be kinetically limited but
thermodynamically feasible.
1.5.3 Desalination of seawater
Seawater is an important source of potable water in many countries with shortages of fresh
water. Desalination technologies such as multi-stage distillation, reverse osmosis and electrodialysis are energy intensitve.46-48 Hydrate based separation processes provides a viable
alternative that, again, are estimated to consume less energy.39 The separation of salts and other
gaseous impurities can be achieved by sequential clathrate hydrate formation and dissociation.
When the hydrates form, the cages omit the salts so that when the structure dissociates, only the
gas and pure water is released.48-49
1.5.4 Potential source of energy
At T = 273 K and 1 atmosphere, a cubic meter (m3) of completely filled natural gas hydrate
contains ca. 96 kg of CH4, giving it the potential to be a future natural gas source.13 Large
quantities of natural gas exist as hydrates in the arctic and permafrost regions of the earth, as well
as in the ocean bottoms.7,50–52 The available energy from gas hydrate sources alone is estimated
to be twice that of all other fossil fuels combined.50,53 Although more studies need to be done to
prevent uncontrollable sand production and rapid depressurization of the hydrate bed during
exploration.54
10
1.5.5 Natural gas hydrate in flow assurance
Despite the beneficial uses of hydrate already discussed above, their occurrence during natural
gas production is undesirable because they can plug pipelines and block process facilities during
transportation. Hydrate plugs can form when conditions are favourable such as (i) during a
production start-up, (ii) during a restart following an emergency shut-down and operational shutin due to temperature gradient and because reservoir heat is not available, (iii) having uninhibited
water of condensation, or (iv) when local cooling occurs due to flow across a valve or
restriction.7– 9 If not appropriately predicted and avoided, normal operating conditions inside
producing wells, flow lines, valves and pipelines can fall within the hydrate stability zones where
hydrates can lead to the complete shutdown of operations and results in loss of billions of dollars
in revenue.7,9,13
Beside the economic impacts, there can be serious safety hazards associated with gas hydrates.
During the dissociation of hydrate plugs in the pipeline, there can be a substantial pressure drop
across a plug and before the plug detaches from the pipe wall. Upon releasing from the pipeline
wall, the pressure difference can cause the hydrate to reach speeds greater than 300 km hr-1
within the pipeline.7 Not only can the plug itself break through the pipeline, but this phenomenon
also facilitates the compression of the downstream gas which can result in pipelines blowouts or
ruptures possibly leading to the injury or death of any nearby workers.7,13,55 In 2013, two oil
workers working at a ConocoPhillips site in Alberta were struck by a hydrate plug resulting in
the death of one and serious injuries to the other.56
11
1.5.5.1
Gas hydrate prevention and control
Gas hydrate formation in flow assurance can be prevented through one of the following
processes: (a) maintaining the system temperature above the hydrate formation temperature
through the use of heat or insulation, (b) dehydrating the hydrocarbon fluid to below a specified
level, (c) operating at lower pressure than the hydrate formation pressure, or (d) injection of a
chemical inhibitors to prevent or mitigate hydrate formation: (i) thermodynamic inhibitors such
as methanol or glycol to decrease the hydrate formation temperature and prevent crystal
formation, (ii) kinetic inhibitors such as poly N-vinyl pyrrolidone to decrease the rate of
formation and growth of hydrate crystals and (iii) anti-agglomerates, such as quaternary
ammonium salts, allow hydrates to form but to a controlled crystal size so that they can still be
transported through pipelines.7-8,55,57-58 Regardless of the hydrate prevention methods, accurate
calculation of formation conditions are very important in order to determine the best prevention
strategy.
1.6
Importance of C3H8 hydrate formation conditions studies
C3H8 with a diameter of 6.3Å typically forms a sII clathrate hydrate in the presence of water;
however, C3H8 is too large to occupy the 512 cages; therefore, it occupies the large cages of 51264
(6.66 Å) leaving the 512 cages empty.7,9,11,59-60 The 512 cages in the sII hydrate can potentially
accommodate the molecules with small diameters such as H2S, CO2 and CH4 at appropriate
temperature and pressure conditions. In fact, these smaller formers often stabilize the sII hydrates
even more than just the primary former such as C3H8. The components of gas mixtures are
12
partitioned through forming hydrates according to their relative difference in occupation
thermodynamics and kinetics within hydrate cavities.9,13,61
As discussed earlier, hydrates can affect the processing, storage and transportation of LPGs. If
the conditions are accurately known or predictable, operators are able to properly design schemes
to deal with hydrate issues in flow assurance and develop HBGS processes. Furthermore, C3H8 is
the reference material for which most sII calculations are based on, i.e., it is thought to be better
studied than other sII hydrates.
Thus, the specific aims and objectives of this study are to:
1. Measure the hydrate formation conditions (i.e., pressure and temperature) of C3H8 in the
presences of liquid water by using high purity C3H8 (99.999 mol %).
2. Develop a semi–empirical model based on the Clausius–Clapeyron relation for the rapid
estimation of hydrate formation condition of C3H8.
3. Develop a robust thermodynamic model based on reduced Helmholtz equation–of–state
(EOS) and the van der Waals and Platteuw (vdWP) model for the accurate prediction of
C3H8 hydrate formation conditions.
1.7
Phase behaviour and avoiding hydrate formation
Fundamental thermodynamic properties of natural gas components and mixtures are required for
calculating fluid behaviour over natural gas production conditions.62-64 When considering the
phase behaviour of a hydrate (mixture), the Gibbs Phase Rule is a useful tool in recognizing
changes in the temperature-pressure behavior when changing thermodynamic degrees of
13
freedom. Thermodynamic degrees of freedom (F) or variance of a chemical system is given
as7,65-68
F = C + 2 – P.
(1.1)
Where C and P represent the number of components and phases respectively in a chemical
system. For example, a system with pure hydrate former in the gas phase, liquid water and
hydrate phases (C = 2 and P = 3) is univariant at any given temperature, i.e., either pressure or
temperature can be changed independently without changing the state of the system while the
same system with four different phases is invariant (neither pressure nor temperature can be
changed).
1.7.1 Phase behaviour of a hydrocarbon hydrate former
The typical phase diagram for a hydrocarbon former in the presence of water is shown in Figure
1.2. The broken blue line “TtTc” represents the vapour pressure of a pure hydrocarbon hydrate
former. Tt and Tc represent the triple and critical points of the hydrate former, respectively. The
triple point is the point of coexistence of solid, vapour (V) and liquid hydrocarbon (LHC) phases
of the hydrocarbon former while at or beyond the critical point there are no phase boundaries.
For most hydrocarbon formers, four different phases of vapour, hydrate (H), liquid water (Lw)
and ice water (I) can exist together in equilibrium at a relatively lower pressure and temperature
referred to as the lower quadruple point, Q1. LHC, V, Lw and H phases also coexist at a relatively
higher pressure and temperature at the upper quadruple point Q2. The quadruple points (Q) are
invariant for any particular hydrocarbon / hydrate former according to equation 1.1.7-8,66 The
curve 1–1´– 1´´ represents the limit of the hydrate stability region where hydrates formed at
pressures above the curve are thermodynamically stable.66 At higher temperatures, on the right
14
side of line Q2Tc, hydrate phase cannot be formed, but at lower temperature hydrate is formed as
shown at the left hand side of the line Q1Q2. At even lower-temperature and pressure conditions,
an ice phase coexisting with hydrocarbon vapour prevails over hydrate phase formation. 7-8,66
Three different phases of I-H-V form along the line labelled 1, usually below T = 273.15 K and
relatively low pressure, while Lw-H-V phases exists in equilibrium with each other at the line
labelled 1´ greater than T = 273.15 K. At relatively high temperatures and pressures, LHC-H-Lw
phases also coexist at the line labelled 1´´.
1
LHC-Lw
LHC–H
I–LHC
ln p
Tc
Q2
Tt
V–LW
1
Q1
I–V
273.15 K
T
Figure 1.2. Typical p−T diagram for a hydrocarbon hydrate former. Q1and Q2 represents the
lower and upper quadruple points respectively. Line TtTc represents the vapour pressure line of
hydrocarbon while 1–1´–1´´ represents the hydrate stability region consisting the I–H–V, Lw–
H–V and LHC–H–Lw phase boundaries.
1.7.2. Phase behaviour of C3H8 + H2O system
C3H8 hydrates can exist at temperatures above the normal melting point of either
ice or
C3H8.11,59- 60 The phase diagram of the C3H8 + H2O system is shown in Figure 1.3. The red line
and blue curve represents the vapour pressure and hydrate stability region of C 3H8 respectively.
15
C3H8 has two quadruple points in the presence of liquid water: Q1 and Q2.7-8,11,59-60 Beyond Q2,
above temperature T = 279 K, C3H8 hydrate will not form at any pressure but at temperatures
lower than T ≈ 279 K and pressures greater than p ≈ 0.16 MPa, hydrate phase can coexist with
other phases present. However, at temperatures and pressure lower than T = 273.15 K and
p ≈ 0.16 MPa respectively, up to Q1, hydrate phase can coexists with other phases, notably ice
water and C3H8 (g) as shown in the Figure 1.3. Between Q1 and Q2 three different phases of
C3H8 (g), Lw and H coexist in equilibrium while the steep blue line from Q2 represents the
condition for coexistence of C3H8(l)–H–Lw phases typically at relatively high pressure
p ≈ 0.4 MPa and temperatures between T = 273.16 - 279 K.7-8 C3H8 hydrate will begin to
dissociate when the pressure is decreased or temperature is increased while along the hydrate
stability curve.7-8,11,60
1.80
1.60
1.40
Lw + H
I+ H
1.20
p / MPa
1.00
C3H8(l) + Lw
0.80
0.60
Q2
0.40
0.20
C3H8(g) + Lw
Q1
0.00
265
267
269
271
273
275
277
T/ K
279
281
283
285
Figure 1.3. p – T diagram of C3H8 + H2O system. Q1 is the lower quadruple point and Q2
represent upper quadruple point of C3H8 hydrate. Blue and red lines represent the hydrate
stability region and vapour pressure of C3H8 respectively.
16
Different values of Q1 and Q2 have been reported by various authors. In addition, there is a large
variance in reported dissociation pressure and temperature for the hydrate stability zones. The
literature review for the dissociation conditions and the quadruple points will be discussed in the
next section.
1.7.3 Semi – empirical model for hydrate dissociation correlation
The Clausis-Clapeyron equation provides a relation for estimating the phase transition between
two different phases of a pure component. The slope of tangent (dp/dT) of the line separating two
phases is mathematically expressed as7-8,67-68
dp
s
H
,


dT V TV
(1.2)
where ∆H, T, ∆V and ∆s represent the enthalpy change, temperature, specific volume change and
the specific entropy change of the phase transition, respectively. Experimental data at the
Lw-H-V and Lw–H–LHC phase boundaries can be correlated using this relation for the rapid
estimation of dissociation pressures and temperatures. This relation also enables the estimation
of H which is difficult to measure experimentally.67-68
1.8
Experimental dissociation data for C3H8 hydrate
Hydrate formation conditions for C3H8 are more commonly reported in the literature along the
Lw–H–C3H8(g) phase boundary because it is easier (experimentally, empirically and
theoretically) to work with C3H8 in the gas phase versus the liquid phase.69-84 This can be
attributed to difficulties of measurement in the liquid phase, where the effect of impurities such
as nitrogen and small leaks in the experimental setup can be more pronounced. Also, there are no
17
empirical correlations for the Lw–H–C3H8(l) phase boundary, because the available data is
variable in this region. Thus, the locus above Q2 is estimated at constant temperature. Table 1.2
shows the summary of the experimental data covering the Lw–H–C3H8(g) equilibrium
conditions. Carroll and Kamath fit C3H8 hydrate dissociation data on the Lw–H–C3H8(g) locus
from T = 273.15 - 278.75 K with a Clausius – Clapeyron type relation.8,69
Carrol gives the relation p (MPa) = exp (–259.5822 + 0.58 * T + 27150.7 / T) while Kamath
correlations gives the relation p (KPa) = exp (67.13 + (−16921.84) / T), where T is in Kelvin in
both correlations.
Table 1.2. Summary of experimental dissociation conditions along the Lw–H–C3H8(g) phase
boundary.
Source
No. of data
% Purity
T range / K
p range / MPa
Miller and Strong82
8
-
273.20 - 277.13
7.844 - 18.682
Reamer et al.70
6
> 99.00
274.3 - 277.2
0.2401 - 0.4140
Tumba et al.71
3
99.50
274.6 - 278.1
0.250 - 0.540
Engelos and Ngan73
6
99.50
274.2 - 278.3
0.2296 - 0.5353
Robinson and Mehta74
5
99.50
274.20 - 278.87
0.2068 - 0.5516
Patil75
5
99.50
273.60 - 278.00
0.207 - 0.248
Verma et al.72
9
> 99.5
273.9 - 278.4
0.188 - 0.562
Kubota et al.76
15
> 99.5
273.25 - 278.45
0.712 - 0.552
Deaton and Frost77
5
99.80
273.70 - 277.04
0.1827 - 0.3861
Thakore and Holder78
5
99.90
274.00 - 278.15
0.2170 - 0.5099
18
Source
No. of data
% Purity
T range / K
p range / MPa
Den Huevel et al.79
11
99.95
276.77 - 278.55
0.368 - 0.547
Nixdorff80
10
>99.995
273.55 - 278.52
0.18824 - 0.5490
Maekawa84
10
99.999
274.2 - 278.1
0.211 - 0.509
In contrast to the vapour phase, there are relatively few experimental data along the Lw–H–
C3H8(l) phase boundary due to the difficulties previously mentioned. Table 1.3 shows the
literature summary of the dissociation conditions in the Lw–H–C3H8(l) phase boundary.
Table 1.3. Summary of the experimental dissociation conditions along the Lw–H–C3H8(l) phase
boundary.
Source
No. of data
% Purity
T range / K
p range / MPa
Wilcox et al.81
7
-
278.6 - 279.2
0.0807 - 0.6115
Makogon83
9
99.95
278.05 - 278.28
0.555 - 34.999
Den Heuvel et al.79
17
99.95
278.75 - 278.86
0.893 - 9.893
Verma et al.72
4
> 99.50
278.4 - 278.6
0.562 - 11.2999
Reamer et al.70
3
> 99.00
278.6 - 278.8
0.0684 - 0.2046
Reamer et al.1952 reported dissociation data using the lowest C3H8 purity of < 99.5 mol % while
Nixdorff, 2007 used the highest purity of 99.995 mol % C3H8.70,80 There are discrepancies in the
experimental data reported for C3H8 hydrate in the vapour and liquid regions, potentially caused
by the various purities and or the buoyancy of the hydrate used which will be discussed in detail
19
in chapter three. Note that unspecified impurities can increase or decrease the relative stability of
the hydrate phase with respect to the fluid phases.
The reported quadruple points for C3H8 in the presence of liquid water are also sparse. Carroll
(2003) reported T = 273.05 K and p = 0.172 MPa while Harmen and Sloan (2009) measured a
similar pressure to Carroll but a slightly higher temperature of T = 273.10 K for Q1.7,60 Q2 are
more often reported than Q1; a summary of Q2 values reported in the literature are shown in
Table 1.4.
Table 1.4. Summary of C3H8 hydrate upper quadruple points reported in literature.
Source
% Purity
p / MPa
T/K
Makogon83
99.95
0.555
278.3
Robinson and Mehta74
99.5
0.5516
278.872
Den Heuvel et al.79
99.95
0.6a
278.62
Carroll8
-
0.556
278.75
Verma et al.72
99.5
0.562
278.4
0.566
278.588
Average a
(Standard deviation)
0.004
0.2378
79
Average pressure does not include the value of Den Heuvel et al. due to lower
reported precision.
a
1.9
Review of gas hydrate thermodynamic models
In 1959 Van der Waals and Platteuw proposed the solid state and fluid solution theory for
modeling the hydrate phase based on the equality of water chemical potential between the
20
hydrate phase and the co-existing water phases, i.e., ice, liquid or vapour water.85 The model was
developed based on the assumption that hydrate formation is similar to gas adsorption with the
following conditions: (i) each hydrate cavity is a spherical cage which can hold one gas molecule
at a time, (ii) there is no interaction between the guest molecules and London forces are the only
force present between the guest-host interactions; all other polar forces are assumed to be
integrated in the hydrogen-bonded hydrate lattice, (iii) the host molecule’s contribution to the
free energy is independent of the mode of occupancy by the guest molecules, so the guest
molecule inside the hydrate cage does not distort the hydrate cage, (iv) the enclathrated guest
molecule can only undergo rotational and vibrational motion within the cavities but no
translational motion and, (v) classical statistic mechanics is valid under all conditions. 7,85-86
The Langmuir constant C(T) is an important parameter used to define the interaction behavior
between the gas and water molecules within the cavities and it can be calculated from either the
Lennard-Jones 6-12 potential, Lennard–Jones–Devonshire model or the Kihara potential
model.85 McKoy and Sinanoglu (1963) suggested that the Kihara potential model with a
spherical core was more suitable for estimating the gas-water interactions within the cavities.86-87
In 1972, Parrish and Prausnitz presented a modification to the vdWP model for calculating gas
hydrate equilibra in multi-component systems by introducing the Kihara potential model for
estimating the interaction between the guest and host molecules.7,86 They also presented a
detailed algorithm for calculating dissociation pressure and temperature in the I-H-V and Lw-HV loci as well as reference hydrates for different lattice structures.86
21
Holder et al. in 1980 introduced reference thermodynamic properties for gases and hydrates to
replace the reference hydrates initially used for the different cages of hydrate structures in
Parrish and Praunnitz model.88 Their lattice distortion and guest-guest interactions depend on the
properties of the guest and are not accounted for in previous models.89 Chen and Guo (1996)
introduced the concept of equality of fugacities in coexisting phases at equilibrium in gas-water
mixture to hydrate modeling and used the Lennard-Jones 6-12 potential for calculating the gaswater interactions.90 Klauda and Sandler’s (2000) fugacity model attempted to correct some of
the assumptions made by van der Waal and Platteeuw, primarily by taking into account different
degrees of lattice distortion caused by each guest and therefore subsequent changes in the empty
lattice fugacity or Gibbs free energy. Their proposed model removes the reference parameters
widely used in the previous vdWP type models for hydrate structure and instead introduced some
guest specific parameters.85,91 Klauda and Sandler (2000) also included the energy contributions
from the surrounding second and third shells for calculating the Kihara potential parameter.91
Later in 2003, Klauda and Sandler accounted for the guest-guest interactions and dual occupancy
of guest molecules in the cavities in their model.92 Unlike Klauda and Sandler, Ballard (2004)
accounted for the hydrate lattice distortion due to presence of guest molecules by suggesting
empty hydrates lattice of CH4, C3H8 and CH4 + neohexane as the reference standard hydrates for
sI, sII and sH respectively. Perturbation from the standard states is then accounted for by using
the activity coefficient in his model.7,89 Thus, C3H8 hydrate condition are important, because the
C3H8 hydrate is considered the reference material for type II hydrates in general.
Cubic EOSs such as the Peng-Robinson, Soave-Redlich-Kwong, Valderrama-Patel-Teja
equations and the statistical associating fluid theory are commonly used to model the fluid
22
phases in hydrate dissociation modeling. These are used for computational speed, whereas, for
reference quality results, more accurate EOS such as the reduced Helmholtz energy equation can
be used. Further descriptions of the vdWP model and modeling will be discussed in chapter
three.
23
CHAPTER TWO: Review of Literature Techniques, Experimental Procedure and
Calibration
2.1
Outline
The description of the methods and selected earlier apparatus used for studying the dissociation
conditions of gas hydrate are presented in this chapter. The experimental setup and procedures
employed in this study along the Lw–H–C3H8(g) and Lw–H–C3H8(l) phase boundaries also are
presented. The calibration procedures and results for the pressure transducers, platinum
resistance thermometer and autoclave volume are discussed in Appendix A.
2.2
Methods of studying gas hydrate phase equilibra
There are two primary methods for studying gas hydrate phase equilibra in the laboratory:
dynamic and static methods.93
2.2.1 Dynamic method
In this method, a gas is continuously flowed through a chamber or loop, which is maintained at
condition that favours hydrate formation, usually at low temperature before adding water to the
gas.20,93 This method is more suited for the studies of hydrates formation kinetics, or the effect of
electrolytes and inhibitors on hydrate formation, but it can be used to measure dissociation
conditions.93 In this case, continuous flow of gas through the loop or chamber can be stopped to
allow the system to equilibrate at the desired pressure for hydrates to form. Once hydrates are
formed, the temperature or pressure of the system can then be increased and decreased,
24
respectively, until the hydrates begins to melt so that the dissociation conditions can be measured
by a change in effluent composition.20,93
2.2.2 Static method
This method involves the growth of hydrate crystals in a static high-pressure vessel or autoclave
vessel followed by the subsequent dissociation of the formed hydrate while measuring the
temperature and pressure (with or without a visual window).7,93 For phase equilibra studies of
gas hydrate, this method is preferred because of the ease by which intensive properties such as
the dissociation pressure and temperature can be measured.93 Generally, this method can be
subdivided into either: isothermal, isobaric, or isochoric techniques. 93-94
2.2.2.1 Isothermal Method
In this method, the pressure of a gas-water system is increased above an estimated hydrate
formation pressure at a constant temperature until hydrate begins to form.7,73,77,84,93 The pressure
is usually controlled by withdrawal or addition of gas or aqueous liquid. Hydrate formation
creates a temporary increase in temperature and rapid reduction in the pressure until the hydrate
crystal is formed completely. The temperature increases temporarily because the formation is
exothermic and the pressure reduction is due to the enclathration of the gas molecules inside the
hydrate cages.7,77 Provided there is excess H2O, this gas molecule enclathration causes a pressure
reduction to either Lw–H–V or Lw–H–LHC phase boundary condition.7,70,79 After complete
hydrate formation, the system temperature remains constant and the equilibrium dissociation
pressure is determined by decreasing the pressure and taken as the point of dissolution of the
hydrate crystal phase.7,77,93-94
25
2.2.2.2 Isobaric method
In an isobaric system, the pressure of a gas-water system is maintained constant by suitable
sources such as a positive displacement pump or gas exchange through an external reservoir.7,94
The temperature is then lowered and the initiation of hydrate formation is indicated by a
significant reduction in volume of the gas from the source. The formed hydrates crystals are
heated continuously or stepwise to measure the equilibrium hydrate dissociation temperature by
(i) visually observing through a sight glass, the temperature of complete disappearance of the
hydrate phase, or (ii) point of intersection between the cooling and heating pressure curves or
(iii) heat released using a calorimeter.7,74,94
2.2.2.3 Isochoric method
This method is similar to the isobaric method, but in the isochoric method the system is
examined in a constant volume vessel or cylinder where the gas-liquid water system is cooled to
form hydrates after which the temperature is increased to dissociate the hydrates.7,84,94 Figure 2.1
shows the typical p–T curve of a gas–water system for the cooling, formation and dissociation
processes of hydrates in an isochoric system. The system is first allowed to equilibrate at
conditions above the hydrate formation condition at point “A” before cooling it down to a lower
temperature at point “B” where the hydrate crystals begins to form. The hydrate formation is
characterised by a drastic reduction in pressure, as shown in the curve from “B” to “C”. The
formed hydrate is slowly heated from “C” to “A”. The equilibrium point “D” (intersection
between the cooling and hydrate heating curves) represents the condition where hydrates are
completely melted. At the inflection point, point “D”, the thermodynamic degrees of freedom
increased from 1 to 2 upon heating from “C” to “A”. Increasing the temperature outside the
26
hydrate stabilization regions beyond “D” will results in smaller pressure changes that are
associated with vapour-liquid-equilibra (VLE).
A
C3H8 + H2O loaded
into autoclave
Hydrate begin
to form
D
Equilibrium
T and p
Hydrate formation
Pressure
B
C
Complete hydrate
formation
Temperature
Figure 2.1. A Typical p-T curve of a gas hydrate formation and dissociation using the isochoric
method. A-B; gas cooling, B-C; hydrate formation, C-D; hydrate dissociation.
The isochoric method can provide more pressure and temperature information near a hydrate
forming system in less time when compared to the isobaric and isothermal systems because the
procedure can be automated to be carried out continuously. Also, because the method is a nonvisual technique in most cases, it can be less subjective than the other methods described above
because there is no human error associated with visual observation of hydrate dissociation. 80,94
2.2.2.4 Phase boundary dissociation method
In 2012, Loh et al., presented the phase boundary dissociation (PBD) method for measuring
dissociation conditions of methane hydrates in fresh and sea water in a porous media for
27
pressures ranging from p = 2.30 - 17.00 MPa and varying water compositions.95 This approach is
a modification to the isochoric method in which the hydrates are formed by reducing the
temperature of gas-water system inside a constant volume cylinder or autoclave followed by a
controlled dissociation of the formed hydrates along a phase boundary (e.g Lw–H–V).95-96
According to the Gibbs phase rule (F = C – P + 2), for a pure hydrate former and liquid water
system, i.e., C = 2, P = 3 and F = 1, either the temperature or pressure can be changed without
affecting the number of phases along the phase boundaries/loci. As a result, the system would
dissociate along the triple loci for as long as the three phases coexist and the point of intersection
between the cooling and heating curves is taken as the equilibrium hydrate condition similar to
the isochoric method.94-96 The PBD method has the advantage of generating more equilibrium
data in a short period of time compared to all other previously discussed methods. Ward et al.
2015, reported ca. 4.8 hours per data point when using the PBD method compared to 40 – 45
hours per data point for the isochoric method for the hydrate dissociation condition of H2S along
the Lw-H-V phase boundary.96
Irrespective of the method used for studying gas hydrate phase equilibra, agitation is important
for the design of any apparatus.7-8,20,93 In 1896, Villard first observed that an increase in agitation
caused a decrease in the quantity of liquid water phase and or increase in hydrates formation.
Likewise, Hammerschmidt (1934) also noted that some form of agitation such as gas bubbling
through water, increased velocity (turbulence) or flow fluctuations, initiated and accelerated the
hydrate formation.18,20 The apparatus developed by Deaton and Frost (1946) for studying hydrate
dissociation conditions also was oscillated about a horizontal axis to create a form of agitation
and facilitate good hydrate formation rates.77 It is well known that any type of mild agitation can
28
enhance nucleation in supernatant fluids.97-98 Studies also show that higher stirring rate increase
the nucleation rate and surface area of gas hydrates.97-101 Generally, in a gas-liquid system,
agitation helps to facilitate the mass transfer from one phase to another thereby promoting the
rate of hydrate formation.7-8,100-101
2.3
Review of selected experimental apparatus for hydrate dissociation studies
John Cailletet developed an apparatus in 1887 commonly referred to as the Cailletet apparatus
for the original purpose of studying the liquefaction of oxygen, but it was used later on to study
the dissociation conditions of some mixed hydrates such as CO2 + PH3.79,102-104
The apparatus mainly consists of a thick-walled pyrex capillary tube about 50 mm in length, with
an internal and external diameter of 3 and 10 mm respectively which enabled the visual
observation of phase transition. One end of the tube is closed while the other end is open with a
conical thickening to allow an autoclave vessel made of stainless steel to be mounted on it. The
capillary tube is filled with gas-water mixture constrained over mercury which serves as a
pressure transmitting fluid and prevents the mixture from being contaminated with the silicon oil
used in an adjacent hydraulic device for generating pressure. A glass coated iron rod on top of
the vessel enables mechanical stirring of the mixture and the glass tube is kept at the desired
temperature by a thermostat with circulating oil.102-103 The temperature and pressure were
measured by using a platinum resistance thermometer and a deadweight pressure gauge,
respectively.102
29
Later, in 1937 Deaton and Frost constructed a static hydrate equilibrium apparatus which was
used as a model for many other apparatuses that are use today.7,77 Figure 2.2 shows the schematic
diagram of Deaton and Frost experimental setup main parts. It consists of a high-pressure cell
stainless steel cell containing a quartz or sapphire window. The cell is thermally regulated
through a cooler or heater and is connected to a rocking motor to agitate the system. Gases are
flowed above the liquid water through the valve into the cell and are allowed to exit through an
outlet valve connected to a vacuum or pressure gauge. The isothermal method was used for their
study of hydrate dissociation; the pressure was reduced by letting out some gases inside the cell
to cause hydrate dissociation which was visually observed for the estimation of equilibrium
hydrate pressure.7,77
Rocking
motor
Rocking cell
Heater
Gas + water
Cooler
Water bath
Figure 2.2. Experimental schematic of Deaton and Frost’s apparatus for phase equilibra studies
of gas hydrates.
Hammerschmidt (1946) designed and used the first dynamic apparatus to investigate natural gas
hydrate kinetic and thermodynamic formation conditions (see Figure 2.3).20,80 The setup consists
of a temperature controlled water bath marked “6” and Pyrex glass tube, “5” which contains the
30
compressed gas. The gas is flowed through the inlet, “1”, and follows through the loop, “2”,
made of copper tubing and immersed inside the water bath. Water is added from the reservoir
marked “3” by gravity flow to the gas when passing through the internal copper tube, “4”.
Temperature was measured with an iron–constantan thermocouple, “7”, and the gas exit the
setup at “11” through a gas meter, “12” which enables the measurement of the amount of gas
exiting the apparatus. The pressure was measured with a Bourdon tube gauge, “10”, which was
calibrated by a piston gauge. The precooled water bath “6” allows the gas to pass through the
apparatus at various velocities, pressures and temperatures. This apparatus was mainly used to
investigate different kinetic factors that facilitate hydrate formation.20
Figure 2.3. Diagram of the dynamic apparatus designed and constructed by Hammerschdmit for
studying hydrate formation conditions. 1 – gas inlet, 2 – copper precooling coil, 3 – water supply
reservoir, 4 – copper tube, 5 – pyrex glass, 6 – constant temperature bath, 7 – thermocouple
junction , 8 – millivolt meter, 9 – drip, 10 – pressure gauge, 11 – pressure reducing valve, 12 –
gas meter. 20 (reproduced with permission)
31
Different methods are used in the literature for the studies of C3H8 hydrate dissociation
conditions. These methods are summarized in Table 2.1, where the isothermal and isobaric
methods are more common.
Table 2.1. Methods used for the study of C3H8 hydrate dissociation conditions.
Study
Method
Reamer et al.70
Isothermal and Isobaric
Tumba et al.71
Isochoric
Verma et al.72
Isothermal
Engelos and Ngan73
Isothermal
Robinson and Mehta74
Isobaric
Patil75
Isobaric
Kubota et al.76
Isobaric
Deaton and Frost77
Isothermal
Thakore and Holder78
Isothermal
Den Heuvel, et al.79
2.4
Isothermal (Cailletet apparatus)
Nixdorff80
Isochoric
Wilcox et al.81
Isothermal
Miller and Strong82
Isothermal
Makogon83
Static and dynamic
Maekawa84
Isothermal and Isochoric
Apparatus used for this study.
The setup used in this study was initially assembled by a previous graduate student, Zachary
Ward, for the study of dissociation conditions for mixed sour gas hydrates; a description of the
32
commissioning can be found in Zach’s thesis.105 Repairs were made to the setup for the study of
dissociation conditions of C3H8 hydrate i.e., the PRT and pressure transducer were replaced
because of leaks and electronic malfunction, respectively. Figure 2.4 shows a schematic of the
experimental setup. The autoclave vessel is constructed of Hastelloy–C276 coupled with
magnetic stirrer to enhance the mass transfer in the C3H8-H2O mixture. The autoclave has a
volume of about 45.00 cm3 and has a maximum working pressure p = 20.68 MPa (3000 psia) and
working temperature of T = 263.15 - 308.15 K which are within this study experimental
conditions. The apparatus was originally commissioned with a Paroscientific Inc. Digiquartz
410KR-HT-101 Pressure Transducer and a four-wire 100 Ω platinum resistance thermometer
with a PT-104 temperature data logger (Pico Technologies) which have measurement precisions
of δp =  3.45 104 MPa and δT = ± 0.001K, respectively. The Paroscientific Inc. Digiquartz
410KR-HT-101 Pressure Transducer was later replaced with a Keller Druckmesstechnik PA-33X
transducer with a precision of δp = ± 0.001 MPa for measurements in the Lw-H-C3H8(l) locus.
The calibration procedure and results for the PRTs and the pressure transducers are discussed in
Appendix A.106-111 The autoclave vessel was placed inside a PolyScience PP07R-40 refrigerated
circulating bath controlling the temperature to within ± 0.004 K. The stirring assembly was
controlled by an in-house assembled voltage regulation controller and a Hall Effect speed sensor.
C3H8 fluid was injected into the autoclave cell through an inlet high pressure valve, VA1. For
data acquisition, the setup was interfaced with Laboratory Virtual Instrument Engineering
Workbench (LabVIEW) which records the pressure and temperature of the system continuously
and averages every 30 seconds. The PRT and pressure transducer were calibrated by the supplier
and checked by comparison to the ice melting point and comparisons to a previously calibrated
33
transducer at different pressures as well as under a vacuum of 2.5 107 MPa at different
temperatures.
Data logging
computer
Vent
VA2
VA1
Propane tank
PT
Platinum resistance
thermometer
Impeller
Glycol + water bath
Autoclave
Propane + degassed water
PolyScience water chiller bath
Figure 2.4. Schematic diagram of the setup used for the measurement of C3H8 hydrate
dissociation points. VA1, VA2 and PT represent the inlet feed valve, outlet valve and pressure
transducer, respectively.
2.4.1 Interfacing experimental setup with LabVIEW for data acquisition.
The LabVIEW interface was used to merge and communicate with all electronic components of
the setup on a single operating window and thus enabled the automation of experiments while
controlling the temperature and pressure of the system for hydrate formation or dissociation. The
front panel of the experimental setup is shown in Figure 2.5. The indicators labeled “A” and “B”
are used to monitor the pressure and temperature plots versus time inside the autoclave and
throughout the experimental run. The button labelled “C” is use to execute the code for shutting
34
down the system automatically. The elapse interval for saving the time averaged data from the
experimental run is entered into “D” (normally set to 30 s) while the indicator “E” shows the
current temperature set-point during the experiment. The PolyScience circulating water bath is
restarted or shut down with the control knob “G” for temperature regulation of the experiment.
Each desired temperature set-point is entered alongside the duration of time required for each
temperature step. The pressure transducer, PRT and the circulator water bath are switch on or off
by the code executed by the control in box “I”. The temperature indicator or readout “J”
represents temperatures measured by the PRTs, i.e., temperatures measured at room condition
(not shown in Figure 2.4), inside the autoclave cell and water bath (shown in Figure 2.4). This
LabVIEW code allows the user to precisely repeat temperature and pressure conditions with ease
for checking reproducibility.
E
F
D
J
G
C
H
I
B
A
Figure 2.5. LabVIEW front panel for the experimental C3H8 hydrate dissociation setup used in
this study: A & B, graphic indicator showing the current temperature and pressure measured by
the PRT and pressure transducer respectively inside the autoclave cell; C, control to stop the
experiment run; D, time interval to record the averaged pressures and temperatures for each
experimental run; E, indicator showing the current set temperature of the experimental run; F,
experimental runs name; G, control knob used for stopping or restarting the chiller on the
PolyScience circulator water bath; H, automated temperature set-points program control; I,
control knobs for stopping or restarting the pressure transducer, PRT and the circulating water
bath; J, temperature readout indicator that measures the autoclave cell, room and circulating
water bath temperatures.
35
2.5
Materials.
C3H8 with listed purities of 99.999% and 99.5% was supplied by Linde Canada Ltd. and Praxair
Inc., respectively. The purity and compositions of C3H8 gases were analyzed with a Bruker 450gas chromatograph (GC) equipped with a thermal conductivity detector (TCD) and a flame
ionization detector (FID).
Table 2.2. Measured gas impurities (mol %) in C3H8 used for this work.
Supplied by
N2
CO2
CH4
C3H8
i-C4H10
Praxiar Inc
0.4094
0.002
0.006818
99.425
0.1573
Linde Ltd.
0.0000248
ND*
0.00138
99.999
ND*
ND* refers to as not detectable
All water used was taken from EMD Millipore model Milli-Q Type 1 water purification system
polished to a resistivity of 18 MΩ·cm and degassed under vacuum for at least 12 hours.
2.6
Experimental procedure
The PBD method was used to measure the hydrate dissociation conditions of C3H8. The
autoclave was place under a vacuum of 2.5 × 10-7 MPa for a period of 24 hours before each
experiment to flush out impurities from the system. Prior to an experiment, the apparatus was
leak tested by pressurizing the autoclave cell with C3H8 and waiting for 6 hours for pressure
stabilization. C3H8 gas was then flowed through the feed valve, VA1, as shown in Figure 2.4,
36
into the autoclave cell to purge any impurities that may still be trapped in the feed lines or cell
before loading with C3H8 to the desired pressure above the hydrate stability region. For the study
in the Lw–H–C3H8(g) phase boundary, ca. 10 cm3 of polished and degassed water was injected
into the evacuated autoclave by suction. The amount of water corresponds to a mole ratio of 77:1
for water to C3H8 so that liquid water phase was always in excess throughout the experiment. For
measurements in the Lw–H–C3H8(l) region, varying quantity of water was delivered to the
autoclave through a syringe pump after loading the autoclave to a desired pressure.
The C3H8-H2O mixture was then mixed for 8 hours until pressure was stable to within ± 0.005
MPa. Once the system had reached equilibrium, it was cooled and held at 273.35 K for 18 hours
to form hydrates. Figure 2.1, previously shown in sub-section 2.2.2.3, shows a typical curve for
gas cooling, hydrate formation and dissociation stages. A large pressure drop in the autoclave
vessel signifies the hydrate formation.7,94 The formed hydrate was then heated slowly in steps of
0.2 K along the Lw–H–C3H8(g) and 0.05 K along the Lw–H–C3H8(l) phase boundaries,
respectively, to obtain the dissociation points. The system was allowed to equilibrate for
approximately 4 hours at every step before recording the pressure and temperature. Typically,
increasing the temperature in the hydrate stability region causes a sharp increase of pressure due
to the release of enclathrated gas molecules into the gas phase, however, increasing the
temperature outside the hydrate stability region results in a smaller increase in pressure which is
as a result of gas expansion in the autoclave.7,77,94 The pressure versus temperature profile of the
experimental run for each of the data points along the Lw–H–C3H8(g) and Lw–H–C3H8(l) loci
reported in chapter three is shown in Appendix B.
37
CHAPTER THREE: Experimental Results and Modeling
3.1
Outline
C3H8 hydrate dissociation conditions were studied by using the phase boundary dissociation
method described in chapter two. Two purities (99.5 and 99.999 mol %) of C3H8 were used for
the study along the Lw–H–C3H8(g) phase boundary in order to investigate the effect of
impurities on dissociation pressures and temperatures. C3H8 with a listed purity of 99.999 mol %
was used for measurements in the Lw–H–C3H8(l) region. The results obtained from the
experiments were used to fit a Clausius–Clapeyron semi–empirical correlation and to calibrate
more rigorous equations used for thermodynamic modeling of dissociation pressure and
temperature. A mathematical description of the van der Waal and Platteuw model and the
reduced Helmholtz energy EOS used in this study for modeling the hydrate and the fluid phases
respectively are discussed. The algorithm for calculating the dissociation temperature is
presented as well. Finally, the model results were compared to the available literature data and
the deviations are discussed.
3.2
Thermodynamic modeling
In order to define the equilibrium between coexisting species, the partial molar free energy
(chemical potential or fugacity) of each individual species in each phase needs to be well defined
at relevant temperatures, pressures and molar compositions. By definition, equilibrium is
obtained when the free energy of each species in each phase is equal. For example, the
equilibrium condition of three different phases of L, V and H co-existing with each other can be
represented by equation 3.1,64
38
 i , L   i ,V   i , H
or
f i , L  f i ,V  f i , H
,
(3.1)
where µ and f are the chemical potential and fugacity of component i, respectively. By solving
equation 3.1 iteratively, one can mathematically find the equilibrium conditions, i.e., pressure
and temperature, between H2O in a C3H8 gas and H2O which has been incorporated into a C3H8
gas hydrate. The hydrocarbon fluid (vapour and / or liquid C3H8) and hydrate phase fugacities for
this study were calculated by using the reduced Helmholtz energy EOS and the modified van der
Waal and Platteeuw model proposed by Chen and Guo (1996) because of the sound physical
background and high accuracy of these equations.85,90,112-114
3.2.1 Fluid phase modeling
The reduced Helmholtz energy EOS of Lemmon et al., (2009) was used for the calculation of
pure C3H8 thermodynamic parameters such as pressure, density, fugacity and saturation
properties in the fluid phase by using the Reference Fluid Thermodynamic and Transport
Properties (REFPROP 9.1) software.112,115 The equation can be applied from the triple point
temperature of C3H8, T = 85.525 - 650 K and for pressures up to 1000 MPa.112 The reduced
Helmholtz energy equations are composed of ideal and real gas contributions of the fluid. The
ideal gas terms consist of the ideal gas equation and a relation which is used to account for the
isobaric heat capacity at zero pressure, while the real gas contribution describes the residual
behaviour of the fluid.112,116-117 The equation formulated with the Helmholtz energy expressed as
a fundamental properties of density and temperature can be expressed as 112-114,116-117
A , T  A ο  , T   A r  , T 
  ( ,  )   ο ( ,  )   r .

RT
RT
(3.2)
39
Where A, Aο , Ar,  ο and αr represent the Helmholtz energy, ideal gas Helmholtz energy, pure
fluid residual Helmholtz energy, dimensionless form of ideal and real (residual Helmholtz
energy) gas contributions to the Helmholtz energy, respectively. The reduced temperature and
density are defined as ( ) 

Tc
and ( ) 
. The ideal gas contribution to the Helmholtz energy
c
T
is given as
Aο  h ο  RT  Ts ο ,
(3.3)
where s ο is the ideal gas entropy and h ο represents the ideal gas enthalpy which is expressed as
T
h ο  hοο   c οp dT .
(3.4)
Tο
ο
ο
In equation 3.4, hο and c p denotes the ideal gas enthalpy and heat capacity at an arbitrary
reference temperature ( T ο = 273.15 K), respectively. The ideal gas entropy is given as 112
T
s s 
ο
ο
ο
T0
where
c οp
 T 
 ,
dT  R1n
T
  οTο 
(3.5)
 ο represents the ideal gas density at an reference arbitrary pressure pο  0.001 MPa
and temperature T ο = 273.15 K:
ο 
pο
.
Tο R
(3.6)
The ideal gas contribution to the Helmholtz energy can be expressed as equation 3.7 by
40
substituting equations 3.4 and 3.5 into equation 3.3
ο
T

cp
 T
A  h   c dT  RT  T  sο  
dT  R1n
T

  οTο
Tο
Tο

T
ο
ο
ο
p

 .

(3.7)
Alternatively, equation 3.7 can be rewritten in a dimensionless and simplified form:
ο
 ο
hοο sοο
 ο   c p
1 cp
 
  1  1n

d   d .
RTc R
 ο R ο  2
R ο 
ο
(3.8)
ο
For C3H8, the correlation used for calculating c p in equation 3.8 was developed by fitting the
heat capacity experimental data of Trusler and Zarari (1996) which gives the relationship112,118
c οp
R
6
 4   vk
k 3
u k2 exp(u k )
expu k   12
,
(3.9)
where u and v are coefficients derived from Einstein’s vibrational frequencies equation, which
are given as v3 = 3.043, v4 = 5.874, v5 = 9.337, v6 = 7.922, u3 = 393 K / T, u4 = 1237 K / T, u5 =
1984 K / T, u6 = 4351 K / T and R = 8.3144 J / mol / K.
The functional form of the ideal gas Helmholtz energy can be obtained by substituting equation
3.9 into equation 3.8, 112,114,117
6
 ο  1n  31n  a1  a 2   vi 1n1  exp  bk  ,
(3.10)
i 3
where
bk 
uk
.
Tc
(3.11)
The general real gas contribution (residual Helmholtz energy) to the Helmholtz energy is
41
described with an empirical model which is expressed as a sum of the polynomial and
exponential terms: 112,116-117
I POI
  ,     N k 
r
tk
k 1
dk

I POI  I Exp
N 
k  I POI 1
k
tk
 d exp( l ) .
k
k
(3.12)
r
The   ,   term for C3H8 contains an additional Gaussian term which helps to improve the
prediction of properties in the critical region and is expressed in equation 3.13 as 112
5
11
18
 r  ,     N k t  d   N k t  d exp(  l )   N k  d  t exp   k (   k 2   k (   k ) 2 ) .
k
k 1
k
k
k
k 6
k
k
k
12
(3.13)
The values of the parameters and coefficients N k , t k , l k ,  k , and
 k were obtained from the
nonlinear regression of the available experimental data for C3H8 vapour liquid equilibrium
(VLE) and pressure-density-temperature (p-ρ-T) by National Institute of Science and Technology
(NIST) which are shown in Appendix C.112
The thermodynamic properties for C3H8 + H2O mixtures are calculated by accounting for the
mixing of the two components which uses a generic mixing equation based on the corresponding
state principle. The Helmholtz energy of the mixture, A, is the sum of the ideal gas, real gas and
mixing or excess contributions which is expressed in the form113,116-117
A  Aid .mix  A E ,
(3.14)
42
id .mix
where A
and A E denote the ideal mixture and mixing contributions to the Helmholtz energy
respectively. A
id .mix
ο
is the sum of the ideal gas Helmholtz energy ( A i ) and pure fluid residual
r
Helmholtz energy ( Ai ) of the component i, in the mixture which can be expressed in the form:
n


A id .mix (  , T , x)   xi Aiο  , T   Air  ,   RT 1nxi ,
(3.15)
i 1
where n and x i represent total number of components and mole fractions of component, i, in the
mixture at temperature T and density ρ, respectively. The functionalized form of the ideal gas,
 ο , and residual energy of pure fluid,  r , contributions to the Helmholtz energy are expressed
by equations 3.16 and 3.17 as
n
 A ο  , T 

 ο (ρ, T, x )   xi  i
 1nxi  ,
i 1
 RT

(3.16)
and
n
 r  , , x    xi ir  ,    E  , , x  .
(3.17)
i 1
Where  i is the residual term of the reduced Helmholtz free energy of component i which can
r
be calculated from equation 3.13 for C3H8 and the Wagner and Pruß (2002) equation for H2O.114
The excess contribution to the Helmholtz energy or departure function
C3H8 + H2O system. The reduced density for a mixture,  =
a mixture,  =

 r (x)
AE
is not required for the
, and reduced temperature of
Tr ( x)
, require mixing function by corresponding states:114,116-117
T
43
3
n
n
xi  x j 1  1
1
1 

,
  xi x j  v ,ij  v ,ij 2

 r ( x) i 1 j 1
 v,ij xi  x j 8   c1,/i3  c1,/ j3 
(3.18)
and
n
n
Tr ( x)   xi x j  T ,ij  T ,ij
i 1 j 1
xi  x j
 T2,ij xi  x j
Tc ,i Tc , j .
(3.19)
In equations 3.18 and 3.19, c,i is the critical pressure for component i and  c, j represents the
critical pressure of component j while Tc ,i and Tc, j represent critical temperatures of component
i and j respectively. The binary parameters used in the equations 3.18 and 3.19 for C3H8 are
shown in Table 3.1.
Table 3.1. Binary parameters of the reducing functions for density and temperature used in
equations 3.18 and 3.19. 116
Mixture i-j
 vij
 vij
 T ,ij
 T ,ij
C3H8−H2O
1.0
1.011759763
1.0
0.600340961
Different derivative functions are formulated from the reduced Helmholtz energy equations for
calculating thermodynamic properties such as pressure, compressibility factor, speed of sound,
isochoric heat capacity, isobaric heat capacity, Gibb energy, internal energy, enthalpy and
entropy using differentiation with respect to density or temperature.114,116 The results obtained by
using the Helmholtz energy equations for pure C3H8 are accurate to within 0.01 to 0.03 % for
densities from T = 85.525 - 350 K, 0.5 % for heat capacities from T = 85.525 - 650 K.112
44
Accurate modeling of the hydrate phase conditions depend on the correct fugacities of the fluid
phases. If the fluid phases are not modeled correctly, this can lead to errors in the modeling of
the hydrate phase because the optimized hydrate parameters would now be based on the
erroneous fluid fugacities.89 The fugacity of component i, in a binary mixture can be calculated
from the expression: 64,113
f i (T , p, n)  xi pi (T , p, n) .
(3.20)
Where  (T , p, n) denotes the fugacity coefficient of component i in the mixture which can be
calculated from the relationship between molar derivate of  r :113,116
  ( n r ) 

i (T , p, n) = RT exp 
.
 ni T ,v ,n j
(3.21)
Equation 3.21 can be substituted into equation 3.20 to give
  (n r ) 

f i (T , p, n)  xi RT exp 
 ni T ,v ,n j
(3.22)
where n is the number of moles in the mixture in component i and j. The explicit function of this
derivative and other derivatives are shown in Appendix D. Because fugacities for each
component in a gas mixture cannot be directly measured experimentally, experimentally
determined mole fractions of saturated water in C3H8 reported in the literature were compared to
calculated saturated mole fractions at the same conditions for temperature T = 235.55 - 399.89 K
and pressure p = 0.7720 - 67.3962 MPa in order to verify the accuracy of the mixing rules used
for calculating the fugacities of the components in the fluid phase.119-124 The mole fractions of
the saturated component are iteratively calculated from REFPROP 9.1 by using the solver
45
routine within Microsoft Excel. A correlation plot of the calculated mole fraction of saturated
water in C3H8 versus experimental literature data is shown in Figure 3.1. The results obtained by
using this equation are accurate to within an AAD of < 0.2 % with a coefficient of determination
(R2) value of 0.988.
0.10
0.09
yH2O, C3H8 (calc)
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.00
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
yH2O , C3H8(exp)
Figure 3.1. Correlation plot comparing the literature experimental mole fraction of saturated
H2O in C3H8, y H 2O , C3H8 (exp) to those calculated using REFROP 9.1, y H 2O , C3 H 8 (calc).115 □, Song
and Kobayashi (1994);119 ∆, Song et al.(2004); 120 ○, Kobayashi and Katz (1953); 121 +, Sloan et
al. 1986; 122 *, Bukacek (1955). 123
3.2.2 Description of the hydrate phase
3.2.2.1 The Van der waal and Platteeuw hydrate model
Van der Waal and Platteeuw (vdWP) in 1959 proposed the first hydrate model based on the
statistical thermodynamic and Langmuir adsorption theory for calculating the chemical potential
of a hydrate phase based on some assumption described in the last section of chapter one.85
46
The presence of guest molecules inside the hydrate cavity provides the cavity stability, so when
the majority of the cavities are unoccupied, the hydrate cavities dissociate and collapse. The
stability of the hydrate phase is measure as the chemical potential (µ) of water forming the
cavities.7,24 To develop a hydrate model, the hydrate formation process can be viewed as taking
place in two steps. The first step is to form a hypothetical empty hydrate cage from pure water
and the second stage is to fill the hydrate lattice with the former: 8

pure water (  w ) → empty hydrate lattice (  w ) → filled hydrate lattice(  w ).
L
H
The change in chemical potential accompanying this process is given as
 wH  L (  wH   wL ) = (  w   wH ) + (  wL   w ),

(3.23)
where  w ,  w and  w represents the chemical potential of water in the filled hydrate lattice,
H
L
empty hydrate lattice, and pure liquid water respectively.
At equilibrium conditions, the chemical potential of water in the hydrate phase is equal to any


other coexisting phases present;  w =  w or  w (ice).7-8,24,86 By introducing  w to this
H
L
equilibrium condition, equation 3.24 is obtained:
 wH   wL or  w ,

(3.24)




where  w   w   w ,  w   w   w and  w   w   w .
H
H
L
L
Van der waal and Platteeuw derived the change in chemical potential of water in a hydrate phase
and the hypothetical empty hydrate cage as 84
47
 wH  RT  vm ln(1   jm ) ,
m
(3.25)
j
where v m represents the number of cavities of type m per water molecule and  jm is the
fractional occupancy of the guests molecules j within the hydrate cavities m.
3.2.2.2 Calculation of hydrate phase fugacity
Many attempts have been made to improve the accuracy of the vdWP model over the years. The
breakthrough work of Prausnitz and Parrish in 1972 simplified the use of the vdWP model and is
employed within several commercial computer programs.86 Other modifications to the vdWP
model have been discussed in section 1.5. The modified vdWP model by Chen and Guo (1996)
was used for modeling the hydrate phase in this stability here.90 The model was based on the
concept of equality of fugacities of water for all the phases present at equilibrium as shown in
equation 3.26 rather than chemical potential because of the relative ease of calculating fugacities
for gas mixture components:
f wH = f wL or f w ,
H
L
(3.26)

where f w , f w and f w represents the fugacity of water in hydrate, liquid water and ice phase
respectively.  is related to f by 67
   ο  RT 1n
f
,
fο
(3.27)
where the subscript ○ denotes a reference state which is taken as the ideal solution. It follows
that f w can be expressed in term of w
H
 H
from equation 3.27, 90
48
f
H
w
   w   H
 f exp
 RT

w

,


(3.28)

where f w is the reference fugacity of the empty hydrate cavity which is expressed as
   w   L 
.
f  f exp

RT



w
L
w
(3.29)
f wL was calculated from the Wagner and Pruß (2002) reduced Helmholtz energy EOS.114 Classical
thermodynamics can be used to derive the expression for  w
of Holder et al. (1980) was used for calculating  w
 L
 L
, where the simplified method
by directly integrating over pressure and
temperature while using hexagonal ice (ice Ih) water as a reference point from the relationship88
 L
 w
RT
 wο T hw
v w


dT

 RT dp  ln  w xw ,
RTο Tο RT 2
pο
p
(3.30)
where  w is the experimentally determined reference chemical potential difference between
ο
water in the empty hydrate lattice and pure water (L) or the ice (α) phases, at an arbitrary
reference temperature T○ (T○ = 273.15 K) and absolute zero pressure p ο . v w represents the
reference volume difference between the empty hydrate cage and pure ice water phase. The
volume of the hydrate lattice does not change at low pressures, p < 20 MPa, and ice Ih can be
used as a reference lattice at this condition for estimating v w .86,89 The last term, ln  w x w , in
equation 3.30, is use to account for the deviation in the chemical potential of a pure liquid or ice
water relative to a water rich mixture.88,125 The activity coefficient,  w , is normally assumed to be
unity unless an inhibitor or a highly soluble gas is present, xw denotes the mole fraction of water
in the liquid water phase, while hw represents the molar enthalpy difference between the empty
49
hydrate lattice and liquid water phase. The molar enthalpy difference ( hw ) between the empty
hydrate lattice and liquid water is temperature dependant and is expressed as88
T
hw  hwο   c pw dT ,
(3.31)
Tο
where h w is the enthalpy difference between the empty hydrate lattice and ice, at T = 273.15 K
ο
and zero pressure. The change in heat capacity ( c pw ) between the empty hydrate and pure
water phases also depends on temperature:
c pw  c οpw Tο   bT  Tο  ,
(3.32)
ο
where c pw is the reference standard difference in heat capacity between ice and liquid water at
temperatures above 273 K and b represent the coefficient of temperature correction. Table 3.2
shows the values of constants in equation 3.30, 3.31 and 3.32 used in this study.
Table 3.2. Thermodynamic reference properties for structure II used in this study.
Reference
Parameter
Structure II
Source
hwο
1025 J mol-1
Dharmawardhana et al.126
 wο
883.8 J mol-1
Sloan7
c οpw
-38.13 J mol-1 K-1
Holder et al.88
v m
b
3.4 cm3 mol-1
0.141
Parrish and Prausnitz86
Holder et al.88
50
The equality of coexisting phase fugacities in equation 3.26 is used to solve for the equilibrium
H
pressure and temperature in this study. f w and 
L
are calculated from equation 3.28 and
the relationship developed by Holder et al. in equation 3.30 respectively.85,88
3.2.2.3 Hydrate cage occupancy
The fractional occupancy of the gas molecule within the cavities is calculated by using the
Langmuir adsorption theory which is expressed as 85-86,88-92
 jm 
C jm f j
1   j C jm f j
,
(3.33)
where f j is the fugacity of hydrate former j in cavity m, which was calculated from the reduced
Helmholtz energy EOS described in section 3.3.1. The Langmuir constant ( C jm ) is used to
measure the attraction between the enclathrated gas and water molecules in the cavity and is
given as
C jm 
4
TK B


0
  w(r )  2
r dr ,
exp
 TK B 
(3.34)
where KB, w(r) and r represents the Boltzmann constant, cell potential function (average
resulting field of the enclathrated gas molecules in all position within the cavity), and the
distance between the centre of the encaged gas and water molecules respectively. Van der waal
and Platteuw calculated the contribution to the potential energy due to the interaction of the guest
molecules within the cavity by using the Lennard-Jones 6–12 potential.85 However, McKoy and
Sinanoglu (1963) suggested that the Kihara potential with a spherical hard core provides a better
estimate for the gas-water interactions within the cavity.87 By first considering a gas-water
51
molecule interaction and assuming that the core diameter of water is zero. The potential energy
for the interaction (r ) is given by
(r )  ,
for
r  2a
(3.35)
and
  12    6 
=
4

(r )
 
 

r

2
a


 r  2a   ,

for
r ˃ 2a,
where   2a is the collision diameter, i.e., the distance at which (r ) = 0.
(3.36)

and a represents
the characteristic energy and spherical molecular core radius respectively. McKoy and Sinanoglu
summed the interaction between the gas and water molecules within the cavities and give the
relationship86-87,90
  12  10
a  6  4
a 
w(r )  2 z  11     11   5     5 
R  R r
R 
R r 
(3.37)
where
N 
1
N
 r a   N  r a   N 
1     1    
 R R  
 R R 
(3.38)
and N can be 4, 5, 10 or 11, z is the coordination number (number of oxygen atoms at the
periphery of the cavity), R is the cavity radius and r represents the distance between the gas
molecule from the center of the cavity. The Kihara cell potential parameters (a,  and  ) can be
determined in two ways:127 (i) from gas viscosity and second virial coefficient data for pure
substances (ii) by correlating gas hydrates experimental dissociation data to the Kihara potential
parameters. For gas molecules such as C3H8 that only occupy the large cages of structure II, the
Langmuir constant can also be estimated from the relationship developed by Bazant and Trout
52
(2001) which relates the calculated fluid phase fugacity and the experimentally determined
change in chemical potentials via the relationship7,128-129
 17  w  H

exp
 1
K
T
B
1


C jm (T) =
fj
 H
where  w
(3.39)
and f j represent the chemical potential difference between a hydrate phase and
empty hydrate cage and the fugacity of the hydrate former respectively.
Based on recommendations made by Mckoy and Sinanoglu, Parrish and Prausnitz (1972)
reported better estimates for dissociation pressures and temperatures that were close to
experimental dissociation conditions by using the Kihara potential for calculating gas-water
molecule interactions for the hydrate modeling of multi-component gases.86-87 They proposed an
equation for calculating the Langmuir constant by correlating the experimental dissociation data
for pure hydrate formers and gave the relationship86
C jm (T ) 
Amj
T
exp
Bmj
T
,
(3.40)
where Amj and Bmj are the fitting parameters related to guest type j in type m cavity at
temperature T. Their parameters were re-optimised with the experimental data measured here to
give a better correlation of experimental conditions.
3.2.2.3.1 Optimization of Kihara potential parameters
6
The parameters of Parrish and Prausnitz ( Ajm  999.99210 K / MPa and B jm  3794 .48 K)
given by Karakatsani and Kontogeorgis (2013) were adjusted by minimizing the difference
53
between the fugacities of H2O in the C3H8 phase and hydrate phase.130 The optimized parameters
were determined by minimization of the following objective function using the least square
regression method expressed as
NP

Obj.F   f H 2O, C H  f
i
3
8

2
H
w
.
(3.41)
The optimized values for A jm and B jm are shown in Table 3.3 and can be used to iteratively
solve for the formation temperature at any pressure.
Table 3.3. Optimised Kihara potential paramaters used for this study.
3.3.
Phase boundary
Ajm 106 (K / MPa)
B jm (K)
Lw–H–C3H8(g)
999.992
10719.292
Lw–H–C3H8(l)
999.992
23266.346
Algorithm for calculating equilibrium hydrate formation temperature
To obtain the temperature that gives equilibrium between the hydrate phase and liquid water
phases at a given pressure, one solves equations 3.28 by iteration until the equality of fugacities
in equation 3.26 is satisfied. The flowchart of the steps followed in Microscoft Excel Visual
Basic Application for the estimation of the equilibrium hydrate formation temperature is shown
in Figure 3.3.
54
Input p
Is hydrate formation
possible?
Yes
No
Return error message
Guess T
Calculate: yH2O, C3H8 , yC3H8, H2O ,
fH2O , C3H8 & fC3H8, H2O at p and guessed T
Calculation of Cjm
H
Calculate f w
Solve Abs | fH2O,C3H8 – f wH | ≤ 10-10
No
Assign new T
Yes
Return T & p
Figure 3.2. Simplified flowchart for the calculation of dissociation temperature used for the
thermodynamic modeling in this study.
Pressure p is first input and an initial temperature T is guessed (this is currently done using an
empirical fit as an ancillary equation). The program proceeds to execute the next four steps by
first calculating the mole fractions of the two fluid phases (saturated H2O in C3H8, y H 2O ,C 3 H 8 and
saturated C3H8 in H2O, y C 3 H 8, H 2O ), along with their respective fugacities i.e., f H 2O , C3 H 8 and
f C3 H 8 , H 2O at the guessed temperature and specified pressure from REFPROP 9.1. The Langmuir
H
constant, C jm , and fugacity of water in the hydrate phase, f w , are calculated by using equations
55
3.40 (with the optimized A jm and B jm in Table 3.4) and 3.28 respectively. Temperature is
optimized using the solver routine within Excel.
3.4
Experimental results and discussion
C3H8 with listed purities of (99.5 and 99.999) mol % were used for the study in the Lw–H–
C3H8(g) locus for temperatures T = 273.63 - 278.63 K and pressure p = 0.1887 - 0.5774 MPa
while 99.999 mol % C3H8 was used for measurements along the Lw–H–C3H8(l) phase boundary
for pressures p = 0.5717 - 18.2622 MPa and temperature T = 278.64 - 278.75 K. Previous
studies conducted using the experimental setup used for this study have demonstrated the
accurate determination of equilibrium conditions of CH4 and H2S gas hydrates.96 The
experimental dissociation conditions of C3H8 hydrates for the two purities (99.5 and 99.999 mol
%) along the Lw–H–C3H8(g) phase boundary measurements are shown in Table 3.4. The
dissociation pressure of 99.5 mol % C3H8 is, on average, 0.015 MPa larger than that of the
99.999 mol % C3H8 for the same range of temperature T = 273.63 - 278.63 K and this can be
attributed to the presences of impurities like N2, CO2 and CH4 (assessed by GC TCD/FID as
shown in repeated Table 2.6 below). These molecules can occupy both the small (512) and large
(512 64) cages of a sII hydrate but have a higher propensity for occupying the small cages. C3H8,
on the other hand, only occupies the large cages (512 64) in type II hydrate leaving the small cages
(512) empty. While a larger fraction of cage occupancy by some impurities can result in further
stabilization of the hydrate crystal structure (lower dissociation pressure), impurities in the fluid
phase also lead to destablitization (higher pressure stabilization of the fluid phase).7-9 In this case,
the freezing-point depression is small but apparent; whereas, if the impurities were species such
56
as H2S, one would expect the opposite effect. The experimental dissociation conditions for
99.999 mol % C3H8 in the Lw–H–C3H8(l) phase boundary are shown in Table 3.5 and Figure 3.3
presents the summary of the measured conditions for this study with literature data and
calculated values predicted by the model along the Lw–H–C3H8(g) and Lw– H– C3H8(l) phase
boundaries.70-84
Table 2.6. Measured gas impurities (mol %) in C3H8 used for this work. (Repeated)
Supplied by
N2
CO2
CH4
C3H8
i-C4H10
Praxiar Inc
0.4094
0.002
0.006818
99.425
0.1573
Linde Ltd.
0.0000248
ND*
0.00138
99.999
ND*
ND* refers to as not detectable
57
Table 3.4. Experimental dissociation conditions for C3H8 hydrates along the Lw–H–C3H8(g)
phase boundary.
99.999 mol % C3H8
99.5 mol % C3H8
T / Ka
p / MPab
T / Ka
p / MPab
273.63
0.1887
273.63
0.2052
273.83
0.1957
273.83
0.2130
274.03
0.2038
274.03
0.2212
274.23
0.2131
274.23
0.2298
274.42
0.2223
274.43
0.2398
274.62
0.2318
274.63
0.2489
274.83
0.2420
274.83
0.2589
275.02
0.2524
275.03
0.2698
275.22
0.2637
275.23
0.2799
275.43
0.2758
275.43
0.2929
275.63
0.2882
275.63
0.3048
275.83
0.3005
275.83
0.3175
276.03
0.3135
276.03
0.3305
276.22
0.3280
276.23
0.3441
276.42
0.3434
276.43
0.3582
276.62
0.3594
276.63
0.3731
276.82
0.3754
276.84
0.3887
277.03
0.3927
277.04
0.4062
277.22
0.4109
277.23
0.4226
277.42
0.4302
277.44
0.4404
277.63
0.4501
277.63
0.4637
277.83
0.4709
277.83
0.4840
278.03
0.4939
278.03
0.5045
278.23
0.5167
278.23
0.5262
278.43
0.5408
278.43
0.5501
278.62
0.5654
278.63
0.5774
a
Uncertainty for hydrates temperature measurements using the calibrated PRT
was estimated to be ± 0.1 K.
b
Uncertainty for the hydrates pressure measurements was estimated to be
±0.0069 MPa.
58
Table 3.5. Experimental dissociation conditions for 99.999 mol % C3H8 hydrates along the Lw–
H–C3H8(l) phase boundary.
T / Ka
p / MPac
278.64
278.65
18.2622
15.9072
278.65
13.3852
278.64
13.3478
278.68
12.4553
278.67
11.9547
278.68
11.6402
278.69
10.5883
278.69
9.4884
278.68
7.2316
278.72
5.5831
278.73
4.2907
278.74
2.2420
278.75
2.0535
278.75
1.0952
278.75
0.8096
278.75
0.7855
278.75
0.5717
a
Uncertainty for hydrates temperature measurements using
the calibrated PRT was estimated to be ± 0.1 K.
c
Uncertainty for the hydrates pressure measurements was
estimated to ± 0.001MPa.
The experimental data for the two phase boundaries also were used to fit a semi-empirical
correlation based on the Clausius-Clapeyron relation for the rapid calculation of the hydrate
formation conditions. The Lw-H-C3H8(g) locus can be calculated using:
ln p  0.5778T 
27150.7
 259.0014 .
T
(3.42)
59
Similarly, the Lw–H–C3H8(l) phase boundary also can be calculated from the relationship:
p = 36704 – 131.668T.
(3.43)
where p and T are pressure and temperature in MPa and K respectively.
100.00
p / MPa
10.00
1.00
0.10
275.0
275.5
276.0
276.5
277.0
277.5
278.0
278.5
279.0
T/K
Figure 3.3. Pressure versus temperature for the Lw–H–C3H8(g) and Lw–H–C3H8(l) phase
boundaries (experimental and model). ____, model; ----, vapour pressure of pure C3H8 calculated
with the reduced Helmholtz energy equation using REFPROP 9.1,115 , this study (99.5 % C3H8);
, this study (99.999 % C3H8); □, Reamer et al.(1952);70; +, Tumba et al.(2014);71 *, Verma
(1974);72 ♦, Engelos and Ngan (1993);73 ●, Robinson and Mehta (1976);74 +, Patil (1987),75 ■;
Kubota et al.(2003), 76; ♦, Deaton and Frost (1946);77 ■, Thakore and Holder (1987);78 ◊, Den
Heuvel et al. (2001);79▲, Nixdorff (1997);80 ○,Wilcox et al.(1941);81 ▬, Miller and Strong
(1946);82 ∆, Makogon(2003);83 ●, Maekawa (2008).84
60
3.4.1 Model comparison to experimental and literature data along the Lw–H–C3H8(g) region.
The experimental data for 99.999 mol % C3H8 were used to calibrate and validate the
thermodynamic model along the Lw–H–C3H8(g) phase boundary. The deviations between the
temperatures estimated by the model ( Tcalc ) and experimental temperature ( Texp ) for pressure
p = 0.2524 - 0.5654 MPa are reported in Table 3.6.
Table 3.6. Model comparison to the experimental data along the Lw–H–C3H8(g) locus.
p exp
Texp
Tcalc
Texp-Tcalc
0.2524
275.02
275.07
-0.05
0.2637
275.22
275.26
-0.04
0.2758
275.43
275.47
-0.04
0.2882
275.63
275.66
-0.03
0.3005
275.83
275.85
-0.02
0.3135
276.03
276.04
-0.01
0.3280
276.22
276.24
-0.02
0.3434
276.42
276.44
-0.02
0.3594
276.62
276.64
-0.02
0.3754
276.82
276.83
0.01
0.3927
277.03
277.02
0.01
0.4109
277.22
277.22
0.00
0.4302
277.42
277.41
0.01
0.4501
277.63
277.60
-0.03
0.4709
277.83
277.79
0.04
0.4939
278.03
277.99
0.04
0.5167
278.23
278.18
0.05
0.5408
278.43
278.37
-0.06
61
The Texp  Tcalc for each experimental datum compared was found to be within the ± 0.1 K
estimated uncertainty for experimental temperature measurements for pressures on the Lw–H–
C3H8(g) phase boundary. The number of literature data, purities, average deviations (AD),
pressure and temperature ranges compared to the model in this study are presented in Table 3.7.
The pressure versus temperature plot of these experimental results, model, empirical correlations
and literature data for the Lw–H–C3H8(g) locus is shown in Figure 3.4. The visual representation
of the deviation between the model and the other data (literatures and correlations) for pressures
p = 0.2524 - 0.5408 MPa is shown in Figure 3.5.70-84
62
Table 3.7. Summary of literature data along the Lw-H-C3H8(g) phase boundary compared.
T/K
p / MPa
Purity
(mol %)
AD* (K)
Source
0.305 - 0.414
No. of
data
compared
3
275.7 - 278.6
>99
-0.14
Reamer et al.70
274.6 – 278.0
0.2069 - 0.5100
4
99.5
-0.20
Patil75
276.4 - 278.1
0.3415 - 0.5006
2
99.5
0.01
Tumba et al.71
276.37 - 278.87
0.3309 - 0.5516
3
99.5
0.13
Robinson and Mehta74
274.95 - 278.30
0.2656 - 0.5353
5
99.5
-0.20
Engelos and Ngan73
274.63 - 278.63
0.2489 - 0.5501
21
99.5
-0.18
This study
275.1 - 278.4
0.2500 - 0.5620
7
>99.5
-0.01
Verma72
276.15 - 278.45
0.3230 - 0.5520
7
>99.5
-0.02
Kubota et al.76
275.37 - 277.04
0.1827 - 0.3861
3
99.8
0.05
Deaton and Frost77
275.15 - 278.15
0.2169 - 0.5099
3
99.9
0.08
Thakore and Holder78
276.77 - 278.55
0.3840 - 0.5658
10
99.95
-0.08
Den Heuvel et al.79
275.49 - 278.43
0.2774 - 0.5490
7
>99.995
0.02
Nixdorff80
275.3 - 278.1
0.2670 - 0.5090
10
99.999
-0.03
Maekawa84
276.8 – 278.0
0.3650 - 0.4720
5
-
0.13
Miller and Strong82
275.0 - 278.6
0.2513 - 0.5616
19
-
-0.14
John Carrol8
275.0 - 278.6
0.2513 - 0.5616
19
-
-0.29
Kamath correlation69
275.0 - 278.6
0.2513 - 0.5616
19
99.999
-0.02
Maekawa84
275.0 - 278.6
0.2513 - 0.5616
19
99.999
-0.01
This study correlation
* AD 
1
 (Texp  Tcalc ) , n is the number of data point compared.
n n
63
0.65
0.60
0.55
0.50
p / MPa
0.45
0.40
0.35
0.30
0.25
0.20
275.0
275.5
276.0
276.5
277.0
277.5
278.0
278.5
T/K
Figure 3.4. Pressure versus temperature plot of experimental results, model, empirical
correlations and literature data along the Lw–H–C3H8(g) locus. , this study (99.5 % C3H8); ,
this study (99.999 % C3H8); □, Reamer et al.(1952);70 +, Tumba et al.(2014);71 *, Verma,
(1974);72 ♦, Engelos and Ngan (1993);73 ●, Robinson and Mehta (1976);74 +, Patil (1987);75 ■,
Kubota et al.(2003);76 ♦, Deaton and Frost (1946);77 ■, Thakore and Holder (1987);78 ◊, Den
Heuvel et al.(2001);79▲, Nixdorff (1997);80 ○, Wilcox et al.(1941);81 ▬, Miller and Strong
(1946);82 , Maekawa (2008);84……., Kamath correlation (2008);69 ––––, this study model;
-----, Carrol correlation (2003);8 -------, this study Clausius-Clapeyron equation; ------, Maekawa
correlation (2008).84
64
0.30
0.20
Texp− Tcalc / K
0.10
0.00
-0.10
-0.20
-0.30
-0.40
-0.50
0.26
0.31
0.36
0.41
0.46
0.51
0.56
0.61
p / MPa
Figure 3.5. Temperature difference between the model and experimental data, literature data and
correlations along the Lw-H-C3H8(g) locus. , this study (99.5 % C3H8); , this study( 99.999 %
C3H8); □, Reamer et al.(1952);70 + , Tumba et al. (2014);71 *, Verma (1974);72 ♦, Engelos and
Ngan (1993);73 ●, Robinson and Mehta (1976);74 +, Patil (1987);75 ■, Kubota et al.(2003);76
, Deaton and Frost (1946);77 ■, Thakore and Holder (1987);78 ◊, Den Heuvel et al.(2001);79 ▲,
Nixdorff (1997);80 ○,Wilcox et al. (1941);81 ▬, Miller and Strong (1946);82 , Maekawa
(2008);84 ……., Kamath correlation (2008);69 ––––, this study model;-------, Carrol correlation
(2003);8 -----, this study Clausius-Clapeyron equation;
, Maekawa correlation (2008).84
65
The relationship between the purities and variance of the literature data to the model is shown in
Figure 3.6.70-84 Generally, the higher the purity of C3H8 reported in the literature for
measurements along the Lw–H–C3H8(g) region, the lower the deviation in pressure and
temperature from this study’s model, the only exception to this trend was observed for the data
reported by Tumba et al. (2014) for the 2 experimental points.71 The temperature deviation
between their two values and the model presented in this study was unusually small and lower
than the deviations observed for other literature data with similar purities, i.e., 99.5 mol % C3H8
and some higher purities. Engelos and Ngan (1993), Robinson and Mehta (1976), Patil (1987)
and this study reported dissociation data using 99.5 % mol C3H8; however, those authors did not
report any analysis of impurities.73-75 The data all show a similar AD of between -0.13 and -0.27
K for pressures, p = 0.2069 - 0.5516 MPa when compared to the model in this study. The highest
purity of C3H8 in Lw–H–C3H8(g) phase boundary reported in literature was 99.999 mol % by
Maekawa (2008), which is the same as that used in this study. As expected, the data were
comparable to this model presented in this study to within AD = 0.02 K.84 Also, similar
deviations were observed for data obtained by Nixdorff and Oellrich (1997), for > 99.995 mol %
C3H8.80 The temperatures predicted by Carroll’s and Kamath’s empirical correlations deviate
from this model as the pressure increases from 0.2513 to 0.5616 MPa; although, the Kamath
correlation tends to predict a higher dissociation average temperature T = ~ 0.25 K than the
Carroll correlation for the same pressure.8,69 The Maekawa correlation compared favourably with
this model to within an average temperature of within ± 0.02 K.84
66
1.0
Texp− Tcalc / K
0.5
0.0
-0.5
-1.0
98.5
99.0
99.5
100.0
Propane purity / mol %
Figure 3.6. Relationship between C3H8 purities and variance of the literature data along the LwH-C3H8(g) locus to the model presented in this study ──, this study model; , this study (99.5
mol % C3H8); , this study (99.999 mol % C3H8); □, Reamer et al. (1952);70 ∆; Tumba et al.
(2014);71 ■, Verma (1974);72 ×, Engelos and Ngan, (1993);73 ●, Robinson and Mehta (1976);74 +,
Patil (1987);75 ♦, Kubota et al.(2003); 76 ♦, Deaton and Frost (1946);77 ■, Thakore and Holder
(1987);78 ◊, Den Heuvel et al. (2001);79▲, Nixdorff (1997);80 ○, Maekawa (2008).84
3.4.2 Model comparison to experimental and literature data along the Lw–H–C3H8(l) region.
The measured and calculated conditions with their corresponding deviations along the Lw–H–
C3H8(l) phase boundary are presented in Table 3.8. The model predicts the dissociation
temperature to an AD = 0.01 K and to within the uncertainty of the experimental temperature
measurements (± 0.1 K). The pressure versus temperature plot of this study’s dissociation
conditions, including the model and literature data, on the Lw–H–C3H8(l) locus are presented in
Figure 3.7.
67
Table 3.8. Model comparison to the experimental data along the Lw–H–C3H8(l) phase boundary.
pexp
Texp
Tcalc
Texp-Tcalc
18.2622
278.64
278.63
0.01
15.9072
278.65
278.64
0.01
13.3852
278.65
278.66
-0.01
13.3478
278.64
278.66
-0.02
12.4553
278.68
278.66
-0.02
11.9547
278.67
278.66
0.01
11.6402
278.68
278.67
0.01
10.5883
278.69
278.67
0.02
9.4884
278.69
278.67
0.02
7.2316
278.68
278.68
0.00
4.2907
278.71
278.69
0.02
2.2420
278.74
278.70
0.04
2.0535
278.75
278.70
0.05
1.0952
278.75
278.69
0.06
0.8096
278.75
278.69
0.06
0.7855
278.75
278.69
0.06
0.5717
278.75
278.69
0.06
68
40
35
30
p / MPa
25
20
15
10
5
0
277.0
277.5
278.0
278.5
279.0
T/K
Figure 3.7. The pressure versus temperature plot of this study’s dissociation conditions, model
and literature data along the Lw–H–C3H8(l) locus. , this study model; , this study (99.999 mol
% C3H8); *, Verma (1974);72 ◊, Den Heuvel et al.(2001);79 ○, Wilcox et al.(1941);81 ∆, Makogon
(2003);83 -----, this study Clausius-Clapeyron equation.
This locus leans towards lower temperature as pressure increases, as opposed to higher
temperatures as is reported in some literature.79,89 This similar pattern also was observed by
Dyadin et al. (2001) and Makogon (2003), although, at lower temperatures than the temperatures
reported for this study.84,131 This behaviour can be attributed to lower density of formed C3H8
hydrates in the liquid water and liquid C3H8 phase which causes the hydrates to remain afloat in
these coexisting phases. This behaviour is similar to the behaviour of hexagonal ice in liquid
69
water phase, whereby the ice melting line tends towards lower temperatures due to its lower
density (increase in volume upon solidification).
The number of data points, purities, average deviation, pressure and temperature ranges
compared to the model presented in this study along the Lw–H–C3H8(l) locus are presented in
Table 3.9, while the deviations in temperature of the model presented in this study to the
literature data are presented in Figure 3.8.70,72,79,81,83
Table 3.9. Summary of literature data and corresponding purities along the Lw–H–C3H8(l) phase
boundary.
T/K
p / MPa
Purity (%)
AD (K)
Source
0.555 - 35.00
No. of
data points
9
278.05 - 278.28
99.95
-0.46
Makogon83
278.55 - 278.88
0.643 - 9.893
17
99.95
0.08
Den Huevel et al.79
278.2 - 278.6
0.562 - 11.30
4
> 99.50
-0.20
Verma72
278.6 - 278.8
0.684 - 2.046
3
> 99.00
0.01
Reamer et al.70
278.6 - 279.2
0.807 - 6.115
7
-
0.14
Wilcox et al.81
70
0.80
0.60
0.40
Texp −Tcalc / K
0.20
0.00
-0.20
-0.40
-0.60
-0.80
0
5
10
15
20
25
p / MPa
Figure 3.8. Hydrate dissociation temperature difference between the model in this study to the
literature data along the Lw-H-C3H8(l) locus. ____, this study model; , this study (99.999 mol %
C3H8); *, Verma (1974);72 ◊, Den Heuvel et al. (2001);79 +, Wilcox et al.(1941);81 ∆, Makogon
(2003),83; ----, this study Clausius-Clapeyron equation.
The Clausius-Clapeyron equation compares favourably to the model presented in this study to
within an AD = 0.01 K for pressure ranges p = 0.5717 – 18.2622 MPa. The literature data in the
Lw–H–C3H8(l) region were all within an AD = 0.2 K compared to this model except for the data
reported by Makogon (2003).70,72,79,81,83 As opposed to the Lw–H–C3H8(g) region, temperature
deviation from this model does not correlate with the C3H8 purity. For example, Makogon (2003)
and den Huevel et al’s (2001) reported using 99.995 mol % C3H8 purity. While den Huevel et
al., data shows a significantly lower deviation, comparable to this model. The Makogon (2003)
data show a very large deviation to the model presented in the study.79,83
71
The Reamer et al. (1952) data show the lowest deviation to this model for > 99 mol % C3H8. The
purity of C3H8 used by Wilcox et al. (1941) was not reported but the data were still comparable
to the model of this study to with AD = 0.2 K.70,81
3.4.3 Comparison of Upper Quadruple points of this study and literature.
The upper quadruple point, Q2, for this study was calculated from the point of intersection of
Lw-H-C3H8(g) and Lw-H-C3H8(l) loci.
1.00
Lw – H – C3H8(l) locus
0.90
p / MPa
0.80
0.70
0.60
C3H8 vapour pressure
0.50
Upper quadruple point (Q2)
0.40
278
278.2
278.4
278.6
T/K
278.8
279
Figure 3.9. A graphical representation of upper quadruple point determination from the point
of intersection of the Lw-H-C3H8(g) and Lw-H-C3H8 (l ) loci.
72
The calculated pressure, p Q 2 = 0.5591 MPa, and other quadruple pressures are shown in Table
3.10 below.8,72,74,79,83 All the p Q 2 in the literature fall within the uncertainty of this study
measurements ( p   6.9 × 10-3 MPa) except the pressure reported by den Heuvel et al.
(2001).79 Similarly, most of the quadruple point temperatures, TQ 2 , reported in the literature
falls within 95 % confidence interval, TQ 2 =278.68 ± 0.1 K except that of Robinson and Mehta
(1974).
Table 3.10. Quadruple points conditions from this study and literature.
Source
% purity
p / MPa
T/K
Makogon83
99.95
0.555
278.3
Robinson and Mehta74
99.5
0.5516
278.87
Den Huevel et al.79
99.95
0.6
278.62
Carroll8
-
0.556
278.75
Verma72
-
0.562
278.4
This study
99.999
0.5591 ± 0.0069
278.68 ± 0.10
73
CHAPTER FOUR: Conclusion, Recommendation and Future work
4.1
Conclusion
The measured formation conditions for C3H8 hydrate in equilibrium with gaseous and liquid
C3H8 were reviewed, where it was found that the liquid C3H8 region showed a large variance in
the literature measurements. Due to the industrial importance of C3H8 hydrates and because C3H8
hydrate is a reference material for other sII hydrates, the C3H8 hydrate dissociation conditions
were independently measured using the phase boundary dissociation method for T = 273.63 278.75 K and p = 0.1887 - 18.2622 MPa. Along the Lw-H-C3H8(g) phase boundary, two
different purities of 99.5 and 99.999 mol % C3H8 were used. The higher pressure dissociation
data reported for 99.5 mol % C3H8 were attributed to stabilization of the fluid phase with respect
to the hydrate due to the presence of fluid phase impurities. C3H8 with a listed purity of 99.999
mole % was used for study of the Lw-H-C3H8(l) locus. A thermodynamic-based model was
optimized using the high-purity data. The model agrees with the experimental data to within the
estimated uncertainty of T  ± 0.1 K.
The optimized thermodynamic model also was compared to the available literature data along
the two phase boundaries.7,69-84 Even small amount of impurities were found to be important
when studying C3H8 hydrate dissociation conditions, where larger deviations from the model
reported in this study were observed for the studies that used lower purity C3H8 for
measurements on the Lw-H-C3H8(g) locus. The curve of the Lw-H-C3H8(l) locus also shows an
inclination towards lower temperatures with increasing pressure as opposed to higher
temperatures which has been reported by other studies.84,132 This inclination is expected to be
towards lower temperatures when the densities of C3H8 hydrates are lower than the densities of
74
the other two coexisting phases (liquid water and liquid C3H8). Similar to the literature data
along the Lw-H-C3H8(g) locus, most of the literature data on the Lw-H-C3H8(l) phase boundary
compare favourably to within ± 0.2 K of this model except for Makogon’s (2003) data.
70,72,79,81,83
This can be attributed to the techniques used for measuring the dissociation point
which relies on visual determination of phase transition from one phase to another and the
floating hydrates which can easily get into the pressure transducer.
4.2
Recommendation
The model presented in this study does not converge at hydrate dissociation conditions at low
pressure (< 0.25 MPa). This can be attributed to the Helmholtz energy equation used for
calculating the fugacity in the fluid phase. Here the equations with the GERG mixing rules do
not converge easily at low pressures. Further code development will improve the calculation of
most thermodynamic parameters involving mixtures, including fugacity, at these conditions. As
discussed in section 3.2.2.1, it is assumed that the volume of ice is not changing under 20 MPa.
Higher pressure studies can be carried out to: (i) check the accuracy this present model and
possibly recalibrate some of the parameters used or (ii) to account for the change in volume of
the hydrate as suggested by Ballard.89
4.3
Future work
Normally, laboratory hydrate solid formation conditions are measured in the presence of three
phases (hydrate phase, non-aqueous fluid phase and liquid water phase). In many ways, sub-
75
saturated hydrate formation (no dense phase water) is more applicable to the transportation of
compressed fluids, because they have been previously partially dehydrated. Experimental
measurements are needed for C3H8 in the two phase regions (C3H8(l)–H, C3H8(g) –H)) to
recalibrate the current model and extend its uses for calculations at these conditions, i.e., water
content measurements above hydrate would make these models much more applicable to
industrial issues associated with flow assurance.
Of all the known hydrate formers, H2S can form hydrates at very low pressures and can remain
stable up to a temperature of 303.15 K. H2S also increases the hydrate formation temperature of
hydrocarbons.62 To the best of my knowledge, there are no equilibrium data in the literature
containing more than 50 mol % H2S with any hydrocarbon in the presence of a liquid water
(saturated) phase or water content data above mixed hydrates. High H2S concentrations with
some C3H8, C4H10 or C2H6 impurities in variable amounts over a range of temperature and
pressure would be of interest models applied to in the oil and gas industry. These data also could
be useful in designing a hydrate based gas separation processes to separate H2S in sour gas
streams, where hydrate separation is energy demanding and has the potential to partially replace
some amine processes currently used.
76
Appendix A
Calibrations and Results
A.1.1 Pressure calibration
The Paroscientific Inc. Digiquartz 410KR-HT-101 Pressure Transducer was used for the study of
the dissociation conditions in the Lw–H–C3H8(g) phase boundary, but for measurements in the
Lw–H–C3H8(l) region a Keller druckmesstechnik PA-33X Pressure Transducer was used. There
are two methods which have been used for calibrating these transducers: (i) primary calibration
against a deadweight tester and (ii) secondary calibration against another well calibrated
transducer.
A.1.1.1 Primary transducer calibration through the use of Deadweight Testers.
Deadweight Testers are the primary standard used for calibrating any pressure measuring
transducers and gauges above ambient conditions. There are three primary components of a
Deadweight Tester device: a weight and piston used to apply the pressure, a clamp to attach the
gauge or transducer and a calibrating fluid (isopropanol) for pressure transmission.106 Weights
are used to apply a known force on an accurately determined area on the piston thereby exerting
a pressure on the fluid; this pressure is transferred to the gauge to be calibrated. The pressure at
the piston face, therefore, is equal to the pressure throughout the calibrating fluid in the tester and
is given as107-108
mi g
F 
p  i
A
A
(A.1.1)
77
where F, A,
m
i
and g represents the force of the weight on the piston, cross sectional area of
i
the occupied weight, sum total of the masses of the applied weight and acceleration due to
gravity respectively. To ensure an accurate calibration, the applied force needs to be corrected
for factors such as the local gravity, the buoyancy of the weight on the fluid, the local
temperature and the thermal expansion of the tester, expansion of the effective area due to the
applied pressure, and any additional static head pressure caused by a height difference between
the transducer and piston.106-109 Applying these corrections, equation A.1.1 becomes:
pcorr 
m g 
i
i
Ap
a 
1 
  gl  f

m


(A.1.2)
where A p is the buoyancy correction factor, g l represents the local acceleration due to gravity
which was recorded as 9.8082 m.s-2 while  f ,
 m and  a represent the densities for the fluid,
weight and air which were 785 kg.m-3, 7300 kg.m-3 and 1.22 kg.m-3 respectively.109 The
corrected pressure, pcorr , measured for the different weights were then plotted against the pressure
obtained from the gauge to obtain a calibration equation.
A.1.1.2 Secondary transducer calibration
This type of calibration was achieved by comparing the measurements from a primarily
calibrated transducer to the uncalibrated transducer in hydraulic communication with each other.
It is easier and faster to calibrate a pressure measuring device using this method versus of going
through the deadweight test procedure for an uncalibrated transducer. The uncalibrated
transducers (Paroscientific Inc. Digiquartz 410KR-HT-101 and Keller Druckmesstechhnik PA33X) were compared to a primary calibrated transducer Paroscientific Inc. Digiquartz 410KR78
HT-101 which was calibrated using the Pressurements Limited T 3800/4 Deadweight Tester by
Connor Deering.109 The Paroscientific Inc. Digiquartz 410KR-HT-101 Pressure Transducer was
initially calibrated by Zachary Ward through a secondary calibration, another secondary check
for this work was also done to confirm the calibration was still valid.105 Different pressures of
nitrogen ranging from 3.39 to 14.00 MPa and under vacuum were used as reference points for
calibrations, where pressurized nitrogen reduces any hydraulic head difference. The calibrated
and uncalibrated transducers were placed in hydraulic communication, the pressure
measurements from the calibrated gauge were plotted against measurements by the uncalibrated
devices to obtain a linear calibration equation.
A.1.1.3 Results and discussion
The Paroscientific and Keller Pressure Transducers can measure pressure up to 20.84 and 100.00
MPa respectively. The measured pressures by the uncalibrated Paroscientific Pressure
Transducer ( p meas ) were compared to the pressures ( p cal ) from the calibrated primary
transducer in the range p = 3.3881 - 13.9339 MPa and under vacuum with Table A.1 showing the
differences between the transducers. The mean average of the differences in pressure
measurements ( p meas  pcal ) between the transducers before calibration was 9.99 10 3 MPa with a
95 % standard error of 0.016 observed at 5.8 MPa. A calibration equation was obtained from the
linear regression of p cal versus pmeas :
pcal  p meas (1.00002 ± 0.00032 )
(A.1.3)
79
Table A.1 Comparison of the pressures measured by the calibrated primary Paroscientific
Transducer, p cal , and the uncalibrated Paroscientific Pressure Transducer, p meas .
p cal / MPa
p meas / MPa
p meas  pcal / MPa
13.9339
13.9460
0.0121
10.3479
10.3577
0.0098
6.8819
6.8909
0.0089
3.3881
3.3964
0.0084
0.0000017
0.0107
0.0107
Average
0.0099 ± 0.0015
Similarly, pressure measurements from the uncalibrated Keller transducer were also compared to
the primary calibrated transducer from pressure p = 5.1173 - 19.0525 MPa and under vacuum.
The mean average for the differences before calibration was -0.0402 MPa (Table A.2). The
linear regression of p cal against p meas shows
p cal  p meas (1.0058 ± 0.001 )
(A.1.4)
Table A.2. Comparison of the pressures measured by the calibrated primary Paroscientific
Transducer, p cal , and the uncalibrated Keller Pressure Transducer, p meas .
p cal / MPa
p meas / MPa
p meas  pcal / MPa
19.0362
19.0525
0.0163
7.8011
7.7124
-0.0887
5.1173
5.0287
-0.0886
0.0017
0.0019
0.0002
Average
-0.0402 ± 0.0564
80
A.1.2 Temperature calibration
A.1.2.1 The International Temperature Scale
The International Temperature Scale (ITS) was adopted by the seventh Conference Generales
des Poids et Mesures in 1927 to overcome the difficulties and variation in measurement of
thermodynamic temperature by use of gas thermometry.110 The temperature scale was amended
and updated at various points through the years, in 1948, 1960, 1968, 1975, 1976 and finally in
1990. The International Temperature Scale of 1990 (ITS-90) supersedes any previously amended
scales and it is the currently accepted standard. The ITS-90 defines temperature in terms of
Kelvin (T90) and Celsius (t90) with the relationship:
110
t 90 / º C  T90 / K  273 .15 . The ITS-90
also provides a temperature scale between the 0.65 K to the highest temperature practically
measurable in terms of the Planck radiation law for monochromatic radiation, between defined
fixed points and specified references for different temperature ranges.110-111 These fixed points
and specified references are the primary and secondary standards respectively used for the
accurate calibration of a thermometer by comparing the temperature measured to the
standards.111 The fixed points are usually triple point temperatures for pure substances while the
reference points consist of melting and boiling temperatures of various pure substances. Between
the triple point of hydrogen (13.803 K) to the freezing point of silver (1234.93 K), T90 is
measured by means of a platinum resistance thermometer (PRT) calibrated at specified sets of
defining fixed point and specified references provided by ITS-90.110 The ITS-90 provides a
number of secondary reference points whose temperatures also have been accurately determined
from the primary standards.111 These secondary references point can be used in calibrating a
thermometer in place of the primary standards because, in most cases, they are easily
reproduced.
81
A.1.2.2 Calibration procedure
The autoclave is rated for a large temperature range, although the temperature range for C3H8
hydrate dissociation study was only between 271.15 to 280.15 K. ITS-90 recommends that
within that region, thermometers can be calibrated using triple or the melting point temperatures
of H2O at T = 273.15 K and 273.16 K respectively.110 Here only a single point calibration
assumes a constant offset. The 100 ohm, four-wire PRT used for temperature measurement
inside the autoclave was calibrated by the melting point of ice water for a single point
calibration. The temperature was regulated with the PolyScience circulating bath to an precision
of δT = ± 0.004 K using 50:50 ethylene glycol:water as the circulating fluid. The water used was
purified using a EMD Millipore Milli-Q water treating system to a resistance of 18 MΩ·cm
followed by degassing for several hours under vacuum. Prior to loading with degassed water, the
autoclave was evacuated overnight to a vacuum of 2.5  10 7 MPa. About 15.00 cm3 of the
degassed water was injected into the autoclave by suction. The system was first cooled to
T = 278.15 K rapidly for 2 minutes and then sub-cooled to T = 263.15 K to form ice water for 7
hours. This is to allow the ice to anneal before increasing the temperature back to 278.15 K for
60 minutes to obtain the melting point temperature of ice at 273.15 K. This procedure was
repeated three times and the average of the horizontally leveled region (Figure A.1) was obtained
and recorded as the melting point temperature of ice water.
The other PRT (used inside the circulating water bath) was compared to the primary calibrated
PRT. The PRTs (i.e., already calibrated and the one to be calibrated) were used to measure the
temperature of a thermal equilibrated circulating water bath for a period of 2 minutes at different
temperature from T = 273.15 to 333.15 K.
82
273.30
273.28
273.26
Temperature / K
273.24
273.22
273.20
273.18
273.16
273.14
273.12
273.10
Time
Figure A.1. A representative temperature-time plot showing the water freezing points for the
PRT probe calibration.
A.1.2.3 Result and discussion
Although the melting point of H2O is a secondary reference, it is easier to obtain than the triple
point primary reference. In order to achieve satisfactory isothermal phase transitions, two
requirements were needed: (i) a sharp, identifiable initial inflection in the measured temperature,
and (ii) a stable, constant temperature while the transition is occurring.109 To achieve these
conditions, the heating rate was set to 0.167 K min-1. The average of all the temperatures
recorded during the inflection (see Figure A.1) was taken and the procedure was repeated three
times. The results of the trials are presented in Table A.3 below.
83
Table A.3. The experimentally measured melting points of H2O with the corresponding
deviations.
Trial
1.
T/K
δT / K
273.124 ± 0.003
2.
273.118 ± 0.005
3.
273.127 ± 0.001
Average T / K
273.123 ± 0.002
The results of the other PRT, T2, used inside the water bath are presented in Table A.4 The
average of the differences observed between the measurements was 0.159 ± 0.0483 K.
Table A.4. Comparison of the measured temperatures from the calibrated, Tcal, and uncalibrated
PRT, Tmeas (used inside the water bath).
Step T / K
Tmeas / K
Tcal / K
Tmeas  Tcal / K
273.15
273.231
273.118
0.112
283.15
283.362
238.098
0.264
293.15
293.251
293.108
0.143
303.15
303.278
303.133
0.145
313.15
313.298
313.146
0.152
323.15
323.298
323.148
0.149
333.15
333.438
333.292
0.146
Average
0.159 ± 0.048
84
A.1.3 Volume calibration
The volume (V) of the autoclave was calibrated by volume difference. The autoclave cell was
filled with degassed water using a high-pressure syringe pump (Teledyne ISCO Model 260D) at
a constant pressure of p = 25.00 MPa. Before loading with degassed water, the setup was
evacuated for 12 hours. The volume of degassed water (V1) inside the syringe pump was first
recorded before filling the autoclave cell and feed lines that connects the autoclave to the syringe
pump. The water was pumped at pressure p = 25.00 MPa up to the outlet valve on the autoclave
(see Figure 2.1) while closed. When the volume remained constant as indicated by the pump, the
new volume inside the pump was recorded as V2. The difference in the volume (V1 – V2) before
loading and after loading up to valve, VA2, gives the volume of the feedline. Valve, VA2, was
then opened to allow water from the syringe pump, still at pressure p = 25.00 MPa, into the cell.
The system was then left for a period of 48 hours to ensure that all volume was filled with water
and that the pressure remained constant, after which the new volume (V3) of the pump was then
recorded. The volume of the autoclave cell is given as
V = V3 – V2 = 165.72 – 119.51 = 46.21 cm3
(A.1.5)
This method, though accurate enough for this study, had two disadvantages: (i) setup took a very
long time to completely dry after the procedure and (ii) the precision is limited to the volume
measurement of the pump which is ± 0.01 cm3.
85
Appendix B
Pressure versus temperature plots of the experimental run for the dissociation points along
Lw-H-C3H8(g) and Lw-H-C3H8(l) phase boundaries reported in this study.
0.6
0.5
p / MPa
0.4
0.3
0.2
0.1
0.0
270
275
280
285
290
295
300
T/ K
Figure B.1. Pressure versus temperature profile for 99.999 mol % C3H8 + H2O showing the
cooling, hydrate formation and heating stages along the Lw-H-C3H8(g) locus.
0.7
0.6
p / MPa
0.5
0.4
0.3
0.2
0.1
0.0
270
275
280
285
290
295
300
T/ K
Figure B.2. Pressure versus temperature profile for 99.5 mol % C3H8 + H2O showing the
cooling, hydrate formation and heating stages along the Lw-H-C3H8(g) locus.
86
p / MPa
p / MPa
9
8
7
6
5
4
3
2
1
0
275
T/ K
280
285
272
274
276
278
280
282
278
T/ K
280
282
T/ K
24
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
22
p / MPa
p / MPa
270
21
20
19
18
17
16
15
14
13
12
20
18
16
14
12
272
274
276
278
T/ K
280
282
272
274
276
Figure B.3. Pressure versus temperature profiles for C3H8 + H2O showing the cooling, hydrate
formation and heating stages along the Lw-H-C3H8(l) locus.
87
2.5
2.0
p / MPa
p / MPa
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
1.5
1.0
0.5
272
274
276 278
T/ K
280
282
0.0
272
274
276
278
280
282
T/ K
20
p / MPa
p / MPa
18
16
14
12
272
274
276 278
T/ K
280
282
10
9
8
7
6
5
4
274
276
278
280
T/ K
282
Figure B.3. Pressure versus temperature profiles for C3H8 + H2O showing the cooling, hydrate
formation and heating stages along the Lw-H-C3H8(l) locus cont’d.
88
16
9
15
8
14
p / MPa
p / MPa
10
7
6
13
12
5
11
4
10
274
276
278
280
282
274
276
T/ K
0.5
p / MPa
p / MPa
0.6
0.4
0.3
0.2
274
276
278
T/ K
280
282
T/ K
0.7
272
278
280
282
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
270
275
280
T/ K
285
290
Figure B.3. Pressure versus temperature profiles for C3H8 + H2O showing the cooling, hydrate
formation and heating stages along the Lw-H-C3H8(l) locus cont’d.
89
p / MPa
p / MPa
9
8
7
6
5
4
3
2
1
0
275
T/ K
280
285
272
274
276
278
280
282
T/ K
24
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
22
p / MPa
p / MPa
270
21
20
19
18
17
16
15
14
13
12
20
18
16
14
12
272
274
276
278
T/ K
280
282
272
274
276
278
T/ K
280
282
Figure B.3. Pressure versus temperature profiles for C3H8 + H2O showing the cooling, hydrate
formation and heating stages along the Lw-H-C3H8(l) locus cont’d.
90
Appendix C
Parameters and coefficients used in the reduced energy Helmholtz EOS for calculation of
thermodynamic properties of C3H8 in equation 3.13.112
k
k
k
k
1
0.963
2.33
0.684
1.283
2.5
1
1.977
3.47
0.829
0.693
0.22228777
2.75
1
1.917
3.15
1.419
0.788
15.
-0.23219062
3.05
2
2.307
3.19
0.817
0.473
16.
-0.09220694
2.55
2
2.546
0.92
1.5
0.857
17.
-0.47575718
8.4
4
3.28
18.8
1.426
0.271
18.
-0.01748682
6.75
1
14.6
547.8
1.093
0.948
k
Nk
tk
dk
1.
0.042910051
1
4
2.
1.7313671
0.33
1
3.
-2.4516524
0.8
1
4.
0.34157466
0.43
2
5.
-0.46047898
0.9
2
6.
-0.66847295
2.46
1
1
7.
0.20889705
2.09
3
1
8.
0.19421381
0.88
6
1
9.
-0.22917851
1.09
6
1
10.
-0.60405866
3.25
2
2
11.
0.06668065
4.62
3
2
12.
0.01753462
0.76
13.
0.33874242
14.
lk
91
Appendix D
First derivative of

and the reducing function  r and Tr with respect to n i .113

  (n r ) 
1

n 
= r 1 
 ni T ,v ,n j
  r
 
.n r
 ni
N
 
1  T 
   r .n r    xri   xk xrk ,
T  ni  n, j
k 1
 n, j 
where
 T 
n r  =
 ni  n, j
 Tr

 xi
 N  Tr
   xk 
 k 1  xk
 

 and n r
xj
 ni

   N   
 =  r    xk  r  .
 n , j  xi  k 1  xk  x j
92
Appendix E
Copyright permissions
93
94
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