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Prediction of gas hydrate equilibrium Amir-Sardary, Babak 2012

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Prediction of Gas Hydrate Equilibrium  by  BABAK AMIR-SARDARY B.A.Sc., Sharif University of Technology, 2003   A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF   MASTER OF APPLIED SCIENCE  in  The Faculty of Graduate Studies (Chemical and Biological Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)  March 2012 © Babak Amir Sardary, 2012 ii  Abstract  This thesis studies the application of Statistical Association Fluid Theory (SAFT) in the prediction of hydrate formation conditions. The main objective is to develop a robust, reliable and purely predictive model for calculating the formation of single hydrates former gases. The current study is based on the use of the algorithm proposed by Englezos et al. (1991).  Simplified SAFT (Fu & Sandler 1995) was employed to model the vapor and liquid phases as well as the van der Waals-Platteew model to represent the hydrate phase. The predictive ability of the model was investigated on single hydrate formers in the presence of inhibitors. With this end in mind, the inhibiting effects of methanol and ethylene glycol on methane, ethane, propane and carbon dioxide incipient hydrate forming were studied. The calculated results were compared to the experimental data obtained from the literature. A deviation of less than     in pressure or    in temperature was desired. Additionally, the phase equilibria of water-methanol, methanol-methaen, methanol-ethane and methanol-propane were also studied. Excellent results were obtained from incipient hydrate calculations and the SAFT equation of state was found to be highly capable of tackling non-ideal mixtures such as water-alcohol and water-alcohol-hydrocarbon systems. Estimation of the SAFT pure component parameters and the temperature range over which the SAFT parameters are estimated was found to be crucial. To overcome this issue, several parameters were estimated over various different temperature ranges, and the one which provided the smallest average absolute deviation was selected. iii  Table of Contents Abstract ..................................................................................................................................... ii Table of Contents ..................................................................................................................... iii List of Tables ............................................................................................................................ v List of Figures .......................................................................................................................... vi Nomenclature ......................................................................................................................... viii Acknowledgments ................................................................................................................... xi 1 Introduction ....................................................................................................................... 1 1.1 Motivation .................................................................................................................. 1 1.2 Knowledge Gap ......................................................................................................... 5 1.3 Scope of This Work ................................................................................................... 7 1.4. System of Interest ...................................................................................................... 8 2 Clathrate Hydrates ........................................................................................................... 10 2.1 Hydrate Structure ..................................................................................................... 10 2.2 Thermodynamics of Gas Hydrates ........................................................................... 13 2.2.1 Evaluation of Cell Partition Function ............................................................... 19 3 Statistical Associating Fluid Theory ............................................................................... 22 3.1 Introduction .............................................................................................................. 22 3.2. Simplified-SAFT ...................................................................................................... 24 3.2.1. Pure Components .............................................................................................. 26 3.2.2. Mixtures ............................................................................................................ 31 4 Methodology ................................................................................................................... 35 4.1 Estimating the SAFT Parameters ............................................................................. 36 4.2 Implementation of the van der Waals-Platteeuw Model into MATLAB Code ....... 37 4.3 Performing the Incipient Hydrate Formation Calculation ....................................... 40 iv  4.4 Investigating the Accuracy of the Results................................................................ 43 5 Results and Discussion .................................................................................................... 44 5.1 Estimating the Simplified-SAFT Parameters ........................................................... 44 5.2. Prediction of Vapor Liquid Equilibrium for Binary Systems .................................. 46 5.3. Evaluating the Proposed Model in Hydrate Formation Calculations ...................... 49 5.3.1. Inhibiting Effect of Ethylene Glycol ................................................................ 52 5.3.2. Inhibiting Effect of Methanol ........................................................................... 57 6 Conclusions and Recommendations ................................................................................ 61 6.1. Conclusions .............................................................................................................. 61 6.2. Recommendations .................................................................................................... 62 References............................................................................................................................... 63 Appendices: ............................................................................................................................ 70 Appendix A: Helmholtz Free Energy ................................................................................. 70 Appendix B: Driving Compressibility Factor from Helmholtz Free Energy ..................... 72           v  List of Tables Table ‎2.1: Geometry of cages (adapted from Bagherzadeh Hosseini (2010), by permission from the author) ...................................................................................................................... 12 Table ‎4.1: Thermodynamic reference properties for gas hydrates (Englezos et al. 1991),  by permission from the author ..................................................................................................... 38 Table ‎5.1: Simplified-SAFT parameters obtained in this work .............................................. 45 Table ‎5.2: Prediction of hydrate formation pressure .............................................................. 50 Table ‎5.3: Required parameters for the van der Waals-Platteew model selected for this work  ................................................................................................................................................ 54            vi  List of Figures Figure ‎1.1:  A gas hydrate block from 1200 metres under water (source: http://commons.wikimedia.org/wiki/File:Gashydrat_mit_Struktur.jpg) .................................. 3 Figure ‎2.1:  Gas hydrate structure (source: http://commons.wikimedia.org/wiki/File:Clathrate_hydrate_cages.jpg) ............................... 11 Figure ‎2.2: Comparison between different potential models.................................................. 20 Figure ‎3.1: Procedure for forming a molecule in SAFT (adapted from Al-Saifi (2012), by permission from the author) ................................................................................................... 25 Figure ‎3.2: Molecular shape in SAFT (adapted from Al-Saifi (2012), by permission from the author) ..................................................................................................................................... 25 Figure ‎4.1: Computational flow diagram (P. Englezos et al. 1991), by permission from the author ...................................................................................................................................... 42 Figure ‎5.1:  Predicted results of vapor-liquid equilibrium (   =0) by SSAFT for the methanol (1)/ water (2) system at    ,    ,   ,     and     ..................................................... 46 Figure ‎5.2: Predicted results of vapor-liquid equilibrium (   =0) by SSAFT for the methane (1)/ methanol (2) system at                               and                                   .................................................................................... 48 Figure ‎5.3: Predicted results of vapor-liquid equilibrium (   =0) by SSAFT for the ethane (1)/ methanol (2) system at                                  and                                .................................................................................... 48 Figure ‎5.4: Hydrate formation prediction (   =0) by SSAFT for methane hydrate in presence of ethylene glycol aqueous solution,      ,      ,       and      ...................... 52 vii  Figure ‎5.5:  Hydrate formation prediction (   =0) by SSAFT for ethane hydrate in presence of ethylene glycol aqueous solution,      ,      ,       and      ...................... 55 Figure ‎5.6: Hydrate formation prediction (   =0) by SSAFT for propane hydrate in presence of ethylene glycol aqueous solution,      ,      ,       and      ...................... 55 Figure ‎5.7: Hydrate formation prediction (   =0) by SSAFT for carbon dioxide hydrate in presence of ethylene glycol aqueous solution,      ,      ,       and       ....... 56 Figure ‎5.8: Hydrate formation prediction (   =0) by SSAFT for methane hydrate in presence of methanol aqueous solution,      ,      ,      ,      ,          and  ................................................................................................................................................ 58 Figure ‎5.9: Hydrate formation prediction (   =0) by SSAFT for ethane hydrate in presence of methanol aqueous solution,      ,      ,      ,      ,          and       .. 58 Figure ‎5.10: Hydrate formation prediction (   =0) by SSAFT for propane hydrate in presence of methanol aqueous solution,      ,     ,          and       ............................ 59 Figure ‎5.11: Hydrate formation prediction (   =0) by SSAFT for carbon dioxide hydrate in presence of methanol aqueous solution,      ,      ,       and       ................ 60        viii  Nomenclature   Molar Helmholtz free energy per mole of molecules    Molar Helmholtz free energy per mole of segments   Temperature-dependent segment diameter,   Number of segments   Number of association sites on the molecules     Avogadro’s number   ⁄  Temperature-dependent dispersion energy of interaction between segments,   ⁄ Temperature-independent dispersion energy of interaction between segments,   Total volume    Temperature-dependent segment volume,     ⁄     Temperature-independent segment volume,     ⁄   Mole fraction    Monomer mole fraction   Compressibility factor     Langmuir constant,  1/MPa    Fugacity,  MPa   Enthalpy,  J/mole   Boltzman’s‎constant,  J/K   Number of components    Number of hydrate forming components   Pressure,  MPa   Radial distance from centre of hydrate cavity,  m   Universal gas constant,  J/mole K    Type m spherical cavity radius,  m   Temperature,  K   Molar volume,  /mole ix       Cell potential function,  J  Greek Letters     Volume of interaction between site   and     Strength of interaction between site   and  ,     ⁄  Association energy of interaction between site   and  ,   Pure component reduced density   Molar density,     ⁄   Polynomial defined by equation 7   Chemical potential,  J/mole    Number of cavities type m per water molecule in hydrate   Lenard-Jones segment diameter,  Subscripts   Component i   Component j   Type of cavity   Water  Superscripts     Residual     Segment       Associating    Hard-sphere       Ideal gas           Association sites   Hydrate   Liquid    Pure liquid water x    Empty lattice   Reference conditions, 273.15 K and zero absolute pressure   Vapour                  xi  Acknowledgments I would like to thank my supervisor Professor Peter Englezos for providing me with valuable supervision and continuous guidance and encouragement. Thank you also for the wonderful time I had working in your research group. I owe particular thanks to Dr. Nayef Al-Saifi for his unwavering support, valuable input, and immense assistance in the process of my research. Thank you for your guidance during the thesis course - it could have not been done without your backing.  I also want to thank my colleagues Alireza Bagherzadeh, Negar Mirvakili, Nagu Daraboina, Iwan Townson and Sima Motiee for their friendship and encouragement throughout this project. I would like to express heartfelt thanks to my wife Sheida Sharifi for her understanding, consideration, and companionship, as well as her consistent help and encouragement during the course of this Master’s Program. Thank you for being there when I needed you most. Finally, I offer sincere gratitude to my family, particularly my mother Shahin Goodarzi, my sisters Anahita and Mandana and my parents-in-law Hassan and Nadereh for their unconditional support and love.   xii          Dedicated to my parents and to my wife         1  1 Introduction 1.1 Motivation Clathrate hydrates are non-stoichiometric crystalline compounds that consist of a hydrogen-bonded network of water and encaged molecules. Davy (1811) first observed clathrate hydrate while he was working on mixing chlorine with water. However, clathrate hydrates were not extensively studied until Hammerschmidt (1934) found that the natural gas pipelines could be blocked by the formation of gas hydrates. This observation raised a great deal of attention in the oil and gas industry, prompting increasing research on the gas hydrates of natural gas. Ever since, the hydrate formation condition and its prevention have been of special interest to those creating chemical technology, especially in the natural gas industry, with the overall aim of avoiding it from  plugging gas pipelines. In the early 1970s, Russian researchers reported natural gas trapped in hydrate form in northern Russia (Makogon 1981). Following these findings, large amounts of hydrates, mostly methane, were also discovered below the seafloor and in regions of permafrost such as in northern Alaska, Siberia and Canada (Haq 1999; Max 2003). This amount of methane has been estimated to be in an order of magnitude greater than the methane in all known reservoirs all around the world (Collett 2002). It has been proposed that this gas be used as an energy resource (Yamazaki 1997).  2  Recent discoveries show that hydrates are connected with environmental concerns. Gas hydrates may play a crucial role with regard to global warming. Gas hydrates may increase global warming. The increase in temperature of the planet’s‎ surface layer, which results from increasing amounts of greenhouse gases such as methane, might cause hydrate decomposition, consequently releasing methane into the atmosphere (Englezos, 1993). Studies of gas hydrate thermodynamics have concentrated on measuring the pressure- temperature conditions at which all existing phases - water, ice, solid hydrate, vapour and/or liquid hydrocarbon - are in equilibrium. Attempts to accurately predict the equilibrium conditions for this multi-phase system depend on the thermodynamic model which describes each phase. Through the use of  x-ray analysis in the late 1940s and early 1950s (Stackelberg 1949; Claussen 1951; Pauling & Marsh 1952), van der Waals and Platteew (1959) gained an understanding of the structure of gas hydrate, and thus were able to derive statistical thermodynamic equations for gas hydrates. In their work, an expression for the chemical potential of water in any hydrate structure was developed using an approach analogous to Langmuir adsorption. This model was later used to represent hydrate phase behavior. Some authors have made slight changes to the van der Waals and Platteew model by, for instance, applying modifications in the Langmuir constant approximation (Klauda & Sandler 2000; Klauda & Sandler 2002; John & Holder 1982; John & Holder 1985; Sparks et al. 1999) or through modifying the original assumptions 1  (Ballard & Sloan Jr 2002a; Ballard & Sloan Jr 2002b; Jager et al. 2003; Ballard & Sloan, et al. 2004; Ballard & Sloan, et al. 2004). Others, meanwhile have adapted the original model (Parrish & Prausnitz 1972;  Englezos et al. 1991).  1  These assumptions will be covered in Chapter 2. 3  Subsequent to the work of van der Waals and Platteuw, many studies were conducted to develop a predictive model adopting their model to represent the hydrate phase and an empirical or semi-empirical model for the fluid phases. Parrish and Prausnitz (1972) developed an iterative scheme, using the Redlish-Kwong (RK) (Redlich & Kwong 1949) equation for the vapour phase. They used the Krichevsky and Kasarnovsky equation and Morrison’s‎ method‎ to‎ estimate‎ gas‎ solubilities‎ during‎ the‎ liquid‎ phase.‎ Anderson‎ and‎ Prausnitz (1986) used the Redlish-Kwong equation (Redlich & Kwong 1949) for the vapor phase and the UNIQUAC model for the liquid one. Du & Guo (1990) modeled the inhibiting effect of methanol on the formation of gas hydrates. In their study, both the vapour and liquid phases were modeled based on the Peng-Robinson equation of state (PR) (Peng & Robinson 1976) and‎on‎Mollerup’s‎random‎and‎non-random (PNR) theory (Mollerup 1983).   Figure ‎1.1:  A gas hydrate block from 1200 metres under water (source: http://commons.wikimedia.org/wiki/File:Gashydrat_mit_Struktur.jpg)  4  Englezos et al. (1991) proposed an algorithm for calculating incipient hydrate formation conditions in the presence of inhibitors like methanol. They used the Trebble- Bishnoi equation of state (Trebble & Bishnoi 1988; 1987) for the both liquid and the vapour phases. Englezos et al. (1991) showed that with the aid of an accurate model for the vapor and liquid phases, one could obtain a good agreement with regard to the hydrate formation calculation‎using‎van‎der‎Waals‎and‎Platteew’s‎statistical‎thermodynamic‎model.         5  1.2 Knowledge Gap Due to the regular structure of gas-hydrate, the van der Waals-Platteeuw statistical thermodynamic model has been used to quantitatively represent gas hydrate thermodynamic properties. As disscused in last section, the next essential action in developing a predictive model for the incipient hydrate formation conditions involves utilizing a reliable model to represent the fluid phases. Many empirical and semi emprical equation of states and activity coefficent model such as Soave, Peng-Robinson, Redlish-Kowang, Trebble-Bishhoi, UNIFAQ and UNIQUAQ have been used in the literature. Despite their widespread use in industry and in the scientific community, empirical and semi-empirical models have certain limitations which must be taken into account. For instance, the accuracy of these equations requires a large experimental database over the entire P-T range for which the model is intended to be used. Generally, it cannot be safely extrapolated. Furthermore, because of the applied simplifications, it is often impossible to reproduce experimental data with the required accuracy for these models. Moreover, these models do not take association interactions into account, and hence fail to predict the fluid properties of polar and hydrogen bounded fluids (Churakov & Gottschalk 2003). Therefore, empirical and semi-empirical equations may fail to accurately calculate the dissociation pressure of hydrates, particularly in the presence of inhibitors such as methanol and ethylene glycol. Molecular-based equations however, are more reliable for tackling difficult systems of associative fluids. One of the most successful molecular-based models is the Statistical 6  Association Fluid Theory (SAFT) which has the ability to deal successfully with associative fluids. However, it is necessary to adjust experimental data to the SAFT in order to obtain good results for associating mixtures. In order to render the SAFT predictive for such mixtures, Al-Saifi et al. (2008) exploited the fact that the association phenomena are result of both hydrogen bonding as well as dipole-dipole interactions which were not considered in the original SAFT. Thus, along with hydrogen bonding interactions, they incorporated the dipole-dipole interactions into the SAFT. They demonstrated that their model was capable of predicting several water-alcohol-hydrocarbon systems. Using this approach, the required parameters were estimated solely according to pure component data.        7  1.3 Scope of This Work The excellent results of the work of Al-Saifi et al. (2011) motivated us to employ a new approach for calculating hydrate formation conditions in order to overcome the non- ideal nature of the mixture and to avoid using binary interaction parameters. Based on the algorithm presented by Englezos et al., and adopting the work of Al-Saifi et al., the new approach is able to accurately predict the incipient hydrate formation conditions, and binary data are not needed for this prediction. In other words, SAFT is the model used for the vapour and liquid phases, and the solid hydrate phase is represented by the van der Waals- Platteew model. Specifically, this work focuses on evaluating the SAFT model, as improved by Al-Saifi et al., with regard to the conditions of incipient hydrate formation. Single component hydrate former gases in the presence of inhibitors such as methanol and ethylene glycol, are examined in the present work, and multi-component mixtures will be left for future studies.         8  1.4. System of Interest The study of gas hydrates has attracted the attention of the natural gas transportation industry for nearly a century. Hydrates are considered an inconvenience due to pipelines blocking, foul heat exchangers and plug columns and expanders valves (Sloan Jr. 1991). The most common approach used in the natural gas industry to prevent the unwanted effects of gas hydrates is to use a variety of thermodynamic inhibition techniques (Englezos et al. 1991). These techniques provide thermodynamically unstable conditions for hydrate formation through the introduction of a less-structured water molecule organization which results from inhibitor-water and inhibitor-hydrocarbon interactions (Englezos et al. 1991). Natural gas consists of a flammable mixture of light hydrocarbon gases. The composition of a natural gas may vary according to its components. However, it is made up primarily of methane (70-90%), ethane and propane although it might also contain iso- butane, normal butane (typical natural gas may contain 0–20% ethane, propane, normal butane and iso-butane), iso-pentane, normal pentane and carbon dioxide (0-8%).  This study focuses primarily on systems that contain single hydrate former gases in the presence of an inhibitor. These types of systems were selected for particular attention because of their industrial importance as well as because of the complexity of their phase behavior. Water-alcohol-hydrocarbon mixtures present behaviours that are far from ideal. Additionally, some interactions, such as that of hydrogen bonding and polar interactions are difficult to describe. The literature has not, to this date, revealed any thermodynamic model 9  that is able to provide accurate phase equilibrium of these mixtures unless the models are correlated to experimental data.         10  2 Clathrate Hydrates 2.1 Hydrate Structure Depending on the size of the gas molecules, natural gas hydrates are categorized according to three basic structure classes: structure   (  )   tac elberg‎  ‎ M ller‎      , structure    (   ) (Claussen 1951), and structure   (  ) (Ripmeester et al. 1987). The common building block of each of these structures is a 12-sided pentagonal-faced polyhedral, pentagonal dodecahedron (   ).  The role of guest molecules is to stabilize the cages which are held by the hydrogen-oxygen bonds and to prevent them from collapsing. This cage accommodates small molecules. Depending on the guest gas molecules and ultimately the hydrate structure, more complex cages might be present. For the sI hydrates, tetrakaidecahedron cages that have 12 pentagonal and 2 hexagonal faces (      ) accommodate the guest molecules. For sII hydrates, hexakaidecahedron cages are formed; 12 pentagonal and 4 hexagonal faces (     ). For sH hydrates, two new cages are formed, using the previous nomenclature for a cage,        and      . Table 2.1, summarizes the number and types of cages, as well as the number of water molecules for each structure. Under normal conditions, only one molecule can occupy each cavity (Sloan & Koh 2007). It is obvious that size of the guest molecule should be at least equal to, or less than, the‎cavity’s‎diameter.‎For instance,    ,     ,     and     form   , and    and    form    . As opposed to the two other structures,    needs larger molecules, like methyl cyclohexane, to fill its larger cavity and smaller molecules, such as     or    , 11  to play a helping role and to occupy the medium and/or small cavities so as to stabilize the structure and prevent it from collapsing.   Figure ‎2.1:  Gas hydrate structure (source: http://commons.wikimedia.org/wiki/File:Clathrate_hydrate_cages.jpg)   Gas hydrates are formed when a gas mixture is brought into contact with water, generally at low temperatures and at high pressure. Studies of gas hydrate thermodynamics have concentrated upon measuring the pressure-temperature conditions at which all existing phases - water, ice, solid hydrate, vapour and/or liquid hydrocarbon - are in equilibrium. Attempts at accurately predicting the equilibrium conditions for this multi-phase system are dependent on the thermodynamic model which describes each phase.  12  Table ‎2.1: Geometry of cages (adapted from Bagherzadeh Hosseini (2010), by permission from the author) Hydrate crystal structure I  II  H Cavity  Small Large  Small Large  Small Medium Large Description Shape  Number of cavities/unit cell  2 6  16 8  3 2 1 Average cavity radius (Å)  3.95 4.33  3.91 4.73  3.94 4.04 5.79 Number of water molecules/cavity  20 24  20 28  20 20 36 Repeating Unit               13  2.2 Thermodynamics of Gas Hydrates The statistical thermodynamic equations for gas hydrates derived by van der Waals and Platteew are based on six assumptions: I. The contribution of the host molecules (water) to the free energy is independent of the mode of occupation of the cavities. II. The encaged molecules (solute) are localized in the cavities, and a cavity never holds more than one guest III. The interaction of the solute molecules is neglected IV. Classical statistics are valid V. The solute molecules can rotate freely in their cages VI. Based on x-ray analysis, the potential energy of a solute molecule is given by the spherically symmetrical potential      proposed by Lennard-Jones and Devonshire As did van der Waals and Platteew, the system in this work is defined as a clathrate crystal containing    molecules of Q in equilibrium with the solutes            , at temperature T and occupying volume V. In this system,           ‎2-1 In the equation 2.1,    stands for the absolute activity of solute J. The independent variables in our system are as follow: 14                 ‎2-2 Before we describe our system with a generalized partition function, we need to define an ordinary partition function. If the cavities of type   contain     molecules of each species  , the ordinary partition function PF would be (Waals 1956):         (   )∏[   ∏   ]   ‎2-3 in which,   is the number of cavities,        ‎2-4 and   is the number of guest molecules occupying cavities,   ∑      (∑  )    ∏   ‎2-5  ‎2-6 15  In ‎2-3,    is the free energy of the empty   lattice for the system described by       . The combination2 part computes the number of distinct ways that solute (guest) molecules may occupy the host cavities, and     is the partition function of encaged molecule    when trapped in cavities of type  . By replacing 2.5 and 2.6 into 2.3 and multiply it by the product of absolute activities, we may obtain,         ∏∏    ‎2-7 And by summing the result over all possible values of     , we obtain the grand partition function  ,      (   )∑∏[ (    ) (     ∑     ) ∏ ∏    ]   ‎2-8 Utilizing the multinomial3 theorem, we may express the above equation as,  2 This combination is a way of selecting several things out of a larger group and can be described mathematically as follows: (   )       The above equation computes k-combinations of a set that has n elements. 3  For any positive integer  and any nonnegative integer  ,    ∑ (   ) ∏    16       (   )∏(  ∑  )    ‎2-9 Now that we have the grand partition function of our system, we can obtain other properties from that partition function 4 . The grand partition function is connected to other thermodynamic properties,          ‎2-10 From classical thermodynamics5, we have,             ∑   ‎2-11 combining 2.10 and 2.11,                  ∑   ‎2-12 now by replacing 2.1 into 1.12 one may get,  4  For more information about Partition Function, the readers are referred to “Equilibrium Statistical Mechanics” (Jackson 2000). 5  See Appendix A. 17              ⁄         ∑     ‎2-13 From equation 2.13 we can simply obtain the composition of the clathrate hydrate,    ∑     (   )   ∑    ∑   ‎2-14 therefore,     ∑         ∑  ‎2-15 ‎2-16  In ‎2-16,     is the fraction of type   cavities occupied by guest molecules  . The chemical potential of the host molecules   , is immediately obtained from 2.9 and 2.13,     (   )       ∑    (  ∑  )   ‎2-17  18    , chemical potential of solvent molecules at empty   lattice, comes directly from the definition of chemical potential which is:    (   ) ‎2-18 On the other hand, we have,     ‎2-19 In the above equation,    is the fugacity and       is the molecular partition function of solute vapour,  . Now, by replacing ‎2-19 into ‎2-16 and ‎2-17 and defining a new parameter,    , as:        ‎2-20 the following important equations are obtained: 19      ∑       ∑    (  ∑  )   ‎2-21  ‎2-22 By constructing these parameters, the thermodynamic behaviour of clathrate might be predicted relative to empty   lattice.  2.2.1 Evaluation of Cell Partition Function Based on assumptions V and VI, van der Waals and Platteew showed the following expression for the Langmuir constant    ,    ∫     (   )    ‎2-23 In their work, the Lennard-Jones 12-6 potential was used to study the force field in the cavity. Even though this model is suitable for monoatomic gases, it fails in the case of non- spherical molecules like      and    . To overcome this problem, McKoy and  inano lu 20  (1963) utilized the Kihara potential with a spherical core 6  in which the molecules are assumed to have impenetrable (hard) cores surrounded by penetrable (soft) electron clouds (Prausnitz et al. 1998). They summed all the guest-host interactions in the cell and obtained the spherically symmetric cell potential      as follows,  Figure ‎2.2: Comparison between different potential models    6       {    [(   )   (   )  ]   -7 -5 -3 -1 1 3 5 7 0 1 2 3 4 5 SW LJ 12-6 Kihara HS LJ 28-7 21          [   (      )   (     )] ‎2-24  Where    [(      )   (      )  ] ‎2-25  Comparing three different potential models - the Lennard-Jones 12-6 Potential, the Kihara Potential and the Lennard-Jones 28-7 Potential - in order to predict hydrate dissociation pressure, McKoy and  inano lu concluded that while Lennard-Jones 12-6 Potential might be satisfactory for hydrates of monoatomic molecules such as   ,    and spherical molecules like    , the Kihara Potential Model is more suitable for nonspherical molecules like     ,      and    .       22  3 Statistical Associating Fluid Theory 3.1 Introduction The Association Fluid Theory was first developed by Wertheim (1984a; 1984b; 1986a; 1986b). According to Prausnitz et al. (1998), Wertheim’s‎ idea‎ was‎ “brilliant but almost incomprehensible.”‎ Wertheim demonstrated that Helmholtz free energy can be expressed as series of integrals obtained from cluster expansion. Based on physical arguments, Wertheim showed that many of the integrals were zero. He then used Perturbation Theory to solve these integrals, which therefore can be simplified and truncated. The essential result of this theory involves an expression for Helmholtz free energy which accounts for the effect of intermolecular association and/or solvation forces, for instance, hydrogen bonding, and the effect of molecular shapes in addition to the effects of the repulsion and dispersion forces. Based‎ on‎ the‎Wertheim’s‎ theory,‎ Chapman‎ et‎ al. (1989; 1990) developed an equation of state model for associating fluids and named it the Statistical Associating Fluid Theory (SAFT). The concept of the SAFT equation of states, and its ability to deal with highly associated fluids like water and alcohols, attracted the attention of many researchers. It was not long before a number of SAFT equation of states were developed, e.g., CK-SAFT (Huang & Radosz 1990; 1991), PC-SAFT (Gross & Sadowski 2001; 2002) and simplified SAFT (Fu & Sandler 1995). In this chapter, a brief description of the simplified SAFT (SSAFT), which is utilized throughout this thesis, will be given. Readers are referred to the work of Al-Saifi (2012)  for more information about Statistical 23  Association Fluid Theory and to view a complete comparison between the different versions of SAFT.           24  3.2. Simplified-SAFT  According to SAFT, the molecules are assembled from a chain of   hard sphere segments. The number of segments, , is not necessarily an integer due to the fact that it is determined by fitting the SAFT equation of state into the vapor pressure and liquid density data (Al-Saifi 2012). It is assumed that these segments are attached by covalent-like bonds. The procedure for forming a molecule of a pure fluid in SAFT is illustrated in Figure 3.1. Initially, the fluid is composed of hard sphere segments. Then, attractive forces are added to these segments. A proper potential model like Lennard-Jones or Square-Well may be utilized at this stage. In the next step, the chain sites are added to each segment and the chain molecules appear by joining the segments through their defined chain sites. Finally, the association sites are added to the associating compounds, for instance water and alcohols, and molecules form association complexes. Each of these four steps contributes to the residual Helmholtz free energy,                                        ‎3-1    25          To replace the complicated dispersion term in earlier versions of SAFT, Fu and Sandler (1995) proposed a new simpler dispersion term based on an attraction term developed originally by Lee et al.(1985). The new SAFT version, therefore, was named simplified SAFT (SSAFT).     1 2 3 m    Figure 3.1: Procedure for forming a molecule in SAFT (adapted from Al-Saifi (2012), by permission from the author)  Figure 3.2: Molecular shape in SAFT (adapted from Al-Saifi (2012), by permission from the author) 26  3.2.1. Pure Components Based on Carnahan‎and‎ tarling’s‎(1969) expression, the Helmholtz free energy for one mole of hard-sphere fluid,  , is defined (Fu & Sandler 1995) as:       ‎3-2 Therefore, the hard-sphere Helmholtz free energy for one mole of pure fluid consisting of molecules with  segments is,    ‎3-3 in above equation,   is the reduced fluid density and can be expressed as,     ‎3-4 where   is the molar density and   is the effective temperature-dependent segment diameter .   could also be expressed based on segment molar volume in a close-packed7 arrangement,   ,                                  ‎3-5  7  Volume occupied by   closely packed segments (Huang & Radosz 1990) 27  combining ‎3-4 and ‎3-5, one may obtain,    (   ) ‎3-6 Because of    it is clear that    is temperature-dependent. As a result, it might be useful to introduce the temperature-independent segment molar volume     analogous to (Huang & Radosz 1990),      (   ) ‎3-7 where   is the segment temperature-independent diameter. The temperature-independent parameters in the above equations are related to the following temperature-dependent parameters  (Huang & Radosz 1990),    [      (   )]       [      (   )]   ‎3-8  ‎3-9 in equation‎3-8 and ‎3-9,   is a parameter which is set at       and   is the temperature- independent square-well depth.  28   For one mole of pure fluid, the dispersion Helmholtz free energy is (Fu & Sandler 1995),    ‎3-10 with             (   ) ‎3-11    is the maximum coordination number which is 36, and    is the molar volume of the segment,     ‎3-12 In equation ‎3-11,      (   ) ‎3-13 where   is the parameter that describes the segment-segment interactions, and is the temperature-dependent depth of the square-well potential, 29     [  (   )] ‎3-14 In above equation,   ⁄   is equal to      for all the molecules (Fu & Sandler 1995).  The chain formation contribution and the association contribution to the residual Helmholtz free energy,                   respectively, are (Fu & Sandler 1995),     ‎3-15          ∑ [(   )   ]   ‎3-16   is the number of association sites, and    is the mole fraction of unbounded molecules which is,     {     ∑     [   (   )   ]      }   ‎3-17     and      are the association volume and association energy specified to the interaction between two energy sites,   and  . 30  Obtaining all the compartments of the residual Helmholtz free energy, the SAFT equation of states are good to go by driving the compressibility factor from the volume derivative of the Helmholtz free energy 8 ,     ‎3-18 where,       [   ] ‎3-19           (   ) ‎3-20     ‎3-21         ∑[      ]     ‎3-22    8  See Appendix B 31  3.2.2. Mixtures  The procedure that we considered for the pure components is also followed for the mixtures. We shall start with equation 3-1, as before. Based on Mansoori’s‎    7   results, and due to the fact that in developing the hard-sphere compartment of equation ‎3-1, it has been assumed that the hard-spheres are not bonded, the expression of the Helmholtz free energy for the mixture of hard-sphere might be as follows (Fu & Sandler 1995),    [      [    ]         ] ‎3-23 with    ∑    ‎3-24 In the above equation,   is the molecule density,    is the mole fraction,   is the number of segments per molecule and    is the temperature-dependent diameter of the segment.  The accounted contribution for the dispersion interaction may be obtained by extending equation ‎3-11 for the mixtures (Fu & Sandler 1995),              (     〈   〉 ) ‎3-25 32     is the molar volume defined in equation ‎3-12 and  is expressed by the following mixing rules,   ∑   ‎3-26 and 〈   〉    ∑ ∑         (   √ ) [   (    )   ] ∑ ∑  ‎3-27 where      (     )√ ‎3-28     ‎3-29  The contribution of chain Helmholtz free energy is shown in equation ‎3-30.         , the pair correlation function, is derived from the work of Mansoori et al. (1971) for molecules with same size segments. 33           ∑          (        )   ‎3-30 where            [   ]     ‎3-31  Finally, the association term derived by Chapman et al. (1990) can be utilized to describe the association contribution to the residual Helmholtz free energy in a mixture (Fu & Sandler 1995),          ∑   [∑[    ]     ]   ‎3-32 with the mole fraction of unbounded molecules,    ,given by,     [     ∑∑           [   (   )   ]   ]   ‎3-33  34  Again, the compressibility factor may be obtained from the volume derivative of the Helmholtz free energy,    [          ] ‎3-34            ( 〈   〉    〈   〉 ) ‎3-35         ∑        (      )   ‎3-36        ∑  [∑(      )    ]   ‎3-37       35  4 Methodology To accomplish the objectives of this work the following approach was followed in current study: 1. Parameter estimation was performed to estimate the required SAFT parameters from pure component vapour pressure and density data 2. A code in MATLAB was prepared to employ the van der Waals-Platteew statistical thermodynamic model to calculate the fugacity of solid hydrate phases 3. The fugacity of each component in the aqueous and the gaseous phases was calculated by using the SAFT in-house program developed by Al-Saifi in Gas Hydrates Group at University of British Columbia 4. The incipient hydrate formation pressure at a given temperature was then calculated by following the algorithm proposed by Englezos et al. (1991) 5. The calculated equilibrium hydrate formation pressures (predictions) were compared with the experimental data. It was examined whether the predictions are within      (pressure) or    (temperature) when compared to experimental data.     36  4.1 Estimating the SAFT Parameters The SAFT equation of state requires the following six adjustable parameters: the segment number ( ), the segment diameter ( ), the segment dispersion energy (  ⁄ ), the energy of association (    ⁄ ), the volume of association (   ) and the fraction of the dipolar segment in a molecule (  ). Optimum parameter values are obtained by fitting the dipolar SAFT to the pure component vapour pressure and liquid density data. The parameters are optimized based solely on pure component data and the following objective function :       ∑    ∑    ‎4-1       37  4.2 Implementation of the van der Waals-Platteeuw Model into MATLAB Code Based on  the gas hydrates statistical thermodynamic model derived by van der Waals and Platteeuw, the fugacity of water in the hydrate phase can be described as follows (Englezos et al. 1991):            (    ) ‎4-2 In the above equation,              (    ) ‎4-3  where,         ∫            ‎4-4        is defined as the difference between , a property of water, in state a and in state b. In equation ‎4-2,  , the difference between the chemical potential of water in an empty lattice ( ) and in a hydrate lattice ( ), is formulated as follows:  38      ∑    (  ∑   )   ‎4-5     Table ‎4.1: Thermodynamic reference properties for gas hydrates (Englezos et al. 1991),  by permission from the author Property Structure I Structure II References        ⁄ 1235  10        (Holder et al. 1980) 1297 937      (Dharmawardhana et al. 1980) 1264 883      (Parrish & Prausnitz 1972) 1299.5  10        (Holder, Malekar & Sloan 1984a) 1287 1068      (Handa & Tse 1986)        ⁄ -4327        (Ng & Robinson 1985) -4622 -4986      (Dharmawardhana et al. 1980) -4860 -5203.5      (Parrish & Prausnitz 1972) -4150        (Holder, Malekar & Sloan 1984b) -5080 -5247      (Handa & Tse 1986)          ⁄ -38.13 -38.13      (Parrish & Prausnitz 1972) -34.583 -36.8607      (Holder & John 1983; John et al. 1985)            ⁄ 0.141 0.141      (Parrish & Prausnitz 1972) 0.189 0.1809      (Holder & John 1983; John et al. 1985)    39  where, the Langmuir constant,    , account for gas-hydrate interactions and are obtained from the following equation:     ∫     (   )    ‎4-6  where,          [   (      )   (     )] ‎4-7 where,     [(      )   (      )  ] ‎4-8 The required hydrate structure parameters were taken from the literature and are summarized in Table 4.1.    40  4.3 Performing the Incipient Hydrate Formation Calculation In a mixture at phase equilibrium and at a constant temperature and pressure, the fugacity of each component is equal in all coexisting phases.  Regarding the question of interest in this work, the phase equilibrium of a system containing solid hydrate (H), vapour (V) and liquid (L) may be represented using the following criteria:     ‎4-9                                ‎4-10  where,   is the number of components and    is the number of hydrate forming components, including water. The SAFT model improved by Al-Saifi et al. (2008) was used to compute the fugacities of substances in both the liquid and vapour phases.  In order to calculate the incipient hydrate formation pressure (or temperature) for a given mixture containing hydrate, vapour and liquid phases at a constant temperature (or pressure), the flash calculations are performed at the assumed pressure (or temperature). First, under these conditions, the fugacity of water in the hydrate phase is calculated using equation 4-2. This fugacity is then compared to the fugacity of water in the liquid phase which is computed by flash calculations. If these two fugacities are equal, the assumed pressure (or temperature) is the hydrate formation pressure (or temperature) (Englezos et al. 41  1991). The computation scheme is designed as shown in Figure 4.1. It should be noted that the fugacities of hydrate formation gases in the vapour phase,  , are incorporated in calculating the fugacity of water in the hydrate phase by using equation ‎4-5.      42  Start  Enter T (or P), feed composition and initial guess for P (or T) Perform TP flash Incipient hydrate formation P (or T) Stop Update P or (1/T)Hwf TOL f f L w H w  2 ln        Figure ‎4.1: Computational flow diagram (P. Englezos et al. 1991), by permission from the author      43  4.4 Investigating the Accuracy of the Results After the completion of the programing and the procurement of the calculated data, a comparison of the experimental data is required. Since single component hydrate former gases in the presence of inhibitors such as methanol and ethylene glycol are studied in the present work, our choice of system are based on hydrocarbon-water-alcohol systems. The acceptable deviation is less than 10% in pressure or 1  in temperature.                 44  5 Results and Discussion 5.1 Estimating the Simplified-SAFT Parameters Three adjustable parameters are required for all fluids in the simplified-SAFT equation of state-. These parameters are the segment number ( ), the segment diameter ( ) and the segment dispersion energy (  ⁄ ). Associating fluids such as water and alcohols require two additional parameters; the energy of association (    ⁄ ), and the volume of association (    .‎According‎to‎Jog‎and‎Chapman’s‎dipolar‎term, it is necessary to introduce an extra adjustable parameter, the fraction of dipolar segment in a molecule (  ), when dipolar interactions are included in SSAFT. The required parameters for water and methanol were obtained for this study by fitting the simplified-SAFT to pure component vapour pressure and liquid density data. At the same time, the parameters for some of the components were collected from the literature (Fu &  Sandler 1995). It should be noted that the parameters were optimized based solely on pure component data. Table 5-2 shows a list of fitted parameters obtained in this work. For more information about the model parameter estimation of water and alcohols, readers are referred to the work of Al-Saifi (Nayef Masned Al-Saifi 2012).    45  Table ‎5.1: Simplified-SAFT parameters obtained in this work                     ⁄      ⁄     Propane 44.10 2.710 16.744 92.935     0.08 0.1 200-340K (Glos et al. 2004) Carbon dioxide 44.01 1.839 14.492 80.563     0.36 0.68 216.55-303K (Vargraftik 1975) Tetrahydrofuran 72.11 2.200 24.385 143.51   0.59788 1.7 3.58 0.66 250-330 (Yaws 2003) Water 18.015 (Wagner & Pruss 2002)      SSAFT  1.500 9.1362 189.74 925.78 0.10178   0.04 0.03 273.16-330K      Dipolar-SSAFT  2.407 7.853 64.511 915.85 0.04411 0.37988 1.85 Methanol 32.04           (Smith & Srivastava 1986)      SSAFT  1.250 21.981 224.54 1116.6 0.00967   0.61 0.81 212-300K Ethylene glycol 62.07 (Yaws 2003)      SSAFT  1.900 22.089 235.06 1471.2 0.00891      Dipolar-SSAFT  2.100 19.569 229.77 1433.2 0.00412 0.27378 1.7  46  5.2. Prediction of Vapor Liquid Equilibrium for Binary Systems   Figure ‎5.1:  Predicted results of vapor-liquid equilibrium (   =0) by SSAFT for the methanol (1)/ water (2) system at    ,    ,   ,     and  The capability of SAFT in dealing with highly non-ideal solutions such as polar solvents and hydrogen bonded compounds renders this theoretically-based equation of state a successful model for tackling difficult systems like those of water-alcohol or water-alcohol- hydrocarbon. These types of systems are very interesting and of great interest both from the points-of-view of industry and academia (Nayef Masned Al-Saifi 2012). As discussed in 0 20 40 60 80 100 120 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 P re ss u re  ( k P a ) Mole fraction Kurihara et al. (1995) McGlashan and Williamson (1976) simplified-SAFT 65 60 55 50 35 47  previous sections, it is very common in the oil and gas industry to use alcohols, for example methanol and ethylene glycol, as inhibiting agents for preventing hydrate formation. The phase diagram of the water-methanol system over a wide temperature range is shown in Figure 5.1. As seen, simplified-SAFT is able to predict the phase behavior of this system very well. In this work, it was assumed that water molecules with four association sites interact with three association site methanol molecules through hydrogen bonding. The SAFT parameters used in the vapor-liquid equilibrium calculation are those obtained and used for the prediction of the incipient hydrate formation conditions. The ability of the Statistical Association Fluid Theory (SAFT) to predict vapor-liquid equilibrium was investigated by Al-Saifi (Nayef Masned Al-Saifi 2012). In his work, Al- Saifi employed PC-SAFT and studied the phase behavior of several alcohol-hydrocarbon systems including those of methanol-butane, methanol-pentane and methanol-hexane. In spite of his excellent results, Al- aifi’s‎ predictions‎ proved‎ unsatisfactory‎ for‎ methanol- hydrocarbon systems at low temperatures (     . In this thesis project, we studied the phase equilibria for methanol-methane and methanol-ethane systems. Figure 5.2 and Figure 5.3 show the phase diagrams of these systems respectively. Although the simplified-SAFT was able to predict the overall phase behavior, the quality of prediction was not as good as the one obtained for the water-methanol system. This result could be due to the fact that the SAFT parameters estimated in this work were suitable for temperature ranges that gas hydrates form and that are obtained by employing the liquid density and vapor pressure of pure compounds at low temperatures. 48   Figure ‎5.2: Predicted results of vapor-liquid equilibrium (   =0) by SSAFT for the methane (1)/ methanol (2) system at                                    and   Figure ‎5.3: Predicted results of vapor-liquid equilibrium (   =0) by SSAFT for the ethane (1)/ methanol (2) system at                                      and   0 50 100 150 200 250 300 350 400 450 500 0 0.2 0.4 0.6 0.8 1 1.2 P re ss u re  ( at m ) Mole fraction SSAFT 10 20 30 40 50 60 70 80 0 0.2 0.4 0.6 0.8 1 1.2 P re ss u re  ( at m ) Mole fraction of ethane SSAFT 49  5.3. Evaluating the Proposed Model in Hydrate Formation Calculations As we have seen in the previous chapter, other studies have given strong consideration to the inhibiting effects of methanol and ethylene glycol on incipient hydrate formation calculations. The systems studied in this work are methane-water-methanol, ethane-water-methanol, propane-water-methanol, CO2-water-methanol, methane-water- ethylene glycerol, ethane-water-ethylene glycerol, propane-water-ethylene glycerol and CO2- water-glycerol. Based on the computational scheme presented in Figure 4.1, the equilibrium hydrate formation pressure was calculated at a given temperature and for a given aqueous inhibitor (methanol and ethylene glycol) concentration. The inhibitor concentration is usually reported as the mass fraction of the inhibitor during the water phase. The absolute average deviation of the predicted pressure is defined as follows:    ∑ [|   |]     ‎5-1  Table 5.3 summarizes the results.      50  Table ‎5.2: Prediction of hydrate formation pressure    No. of data points   Methane /Water/Methanol 0   wt% methanol 275.2-291.2 0 3.98 7 (Verma 1974) 10 wt% methanol 266.2-286.4 0 3.49 11 (H.-J. Ng & Robinson1985) (Mohammadi & Richon 2009) 20 wt% methanol 263.3-280.2 0 4.82 11 (H.-J. Ng & Robinson 1985) (Mohammadi & Richon 2009) 35 wt% methanol 250.9-270.1 0 10.0 11 (Robinson et al. 1986)(Mohammadi & Richon 2009) 50 wt% methanol 232.8-259.5 0 2.86 7 (Ng et al. 1987)(Mohammadi & Richon 2009) 65 wt% methanol 214.1-240.3 0 8.89 10 (Ng et al. 1987)(Mohammadi & Richon 2010) Ethane /Water/Methanol 0   wt% methanol 277.8-287.2 0 3.72 10 (Avlonitis 1988) 10 wt% methanol 268.3-280.4 0 5.82 7 (Ng & Robinson 1985) 15 wt% methanol 268.2-278.9 0 7.15 5 (Mohammadi et al. 2008) 20 wt% methanol 263.5-274.1 0 4.07 6 (Ng & Robinson 1985) 35 wt% methanol 252.6-262.2 0 4.77 4 (Ng et al. 1985) 50 wt% methanol 237.5-249.8 0 19.6 4 (Ng et al. 1985)  Propane /Water/Methanol 0   wt% methanol 274.2-278.4 0 1.66 9 (Kubota et al. 1984) 5 wt% methanol 272.1-274.8 0 4.23 5 (H.-J. Ng & Robinson 1985) 10.39 wt% methanol 268.3-271.8 0 3.72 6 (H.-J. Ng & Robinson 1985) 15 wt% methanol 266.3-269.9 0 8.43 4 (Mohammadi et al. 2008)  Carbon dioxide /Water/Methanol 0   wt% methanol 273.7-282.9 0 2.27 18 (Deaton & Frost Jr 1946) 20 wt% methanol 264-268.9 0 11.4 7 (Ng & Robinson 1985) 35 wt% methanol 242-255.1 0 12.8 5 (Robinson & Ng 1986) 50 wt% methanol 232.6-241.3 0 12.7 3 (Robinson & Ng 1986)  Methane /Water/Ethylene glycol 0   wt% methanol 274.3-285.3 0 3.98 16 (Nakamura et al. 2003) 10 wt% methanol 270.2-287.1 0 1.09 4 (Robinson & Ng 1986) 30 wt% methanol 267.6-279.9 0 1.57 5 (Robinson & Ng 1986) 50 wt% methanol 263.4-266.5 0 5.52 3 (Robinson & Ng 1986) 51     No. of data points   Ethane /Water/Ethylene glycol 0   wt% methanol 277.8-285.9 0 3.72 9 (Avlonitis 1988) 10 wt% methanol 271.1-278.5 0 9.33 4 (Mohammadi & Richon 2010) 20 wt% methanol 267.1-275.3 0 8.52 4 (Mohammadi & Richon 2010) 35 wt% methanol 262.1-269.4 0 3.74 4 (Mohammadi & Richon 2010)    Propane /Water/Ethylene glycol 0   wt% methanol 274.2-278.4 0 1.66  (Kubota et al. 1984) 10 wt% methanol 271.5-274.9 0 2.73  (Maekawa 2008) 15 wt% methanol 269.8-273.7 0 3.58  (Mohammadi et al. 2008) 20 wt% methanol 267.5-270.8 0 1.13  (Maekawa 2008)  Carbon dioxide /Water/Ethylene glycol 0   wt% methanol 273.7-282.9 0 2.27 18 (Deaton & Frost Jr 1946) 10 wt% methanol 271.4-277.8 0 11.8 4 (Mohammadi & Richon 2010) 20 wt% methanol 267.5-274.5 0 7.69 4 (Mohammadi & Richon 2010) 35 wt% methanol 261.6-267.4 0 4.33 4 (Mohammadi & Richon 2010)         52  5.3.1. Inhibiting Effect of Ethylene Glycol   Figure ‎5.4: Hydrate formation prediction (   =0) by SSAFT for methane hydrate in presence of ethylene glycol aqueous solution,      ,      ,       and   The     information for methane, ethane, propane and carbon dioxide hydrate in presence of ethylene glycol are provided in Table 5.3. Figure 5.4 to Figure 5.7 also show the experimental data with predictions that result from the use of simplified-SAFT for these systems respectively. It should be noted that these predictions are based on parameters obtained from pure component data, and no binary interaction parameters (   ) have been used.  Furthermore, Figure 5.4 presents the polar SSAFT prediction results for methane gas 0 2 4 6 8 10 12 14 16 18 255 260 265 270 275 280 285 290 P re ss u re  ( M P a)  Temperature (K) Robinson and Ng (1986) Nakamura et al. (2003) SSAFT dipolar-SSAFT 50 wt% 30 wt% 10 wt% 0 wt% 53  hydrates. As seen, excellent agreements were obtained through the use of simplified-SAFT predictions with regard to hydrocarbon/water/ethylene glycol systems. However, the predictions for the carbon dioxide/water/ethylene glycol system were not as satisfactory as those obtained for methane, ethane and propane. In all these cases except for carbon dioxide in presence of ethylene glycol         aqueous solution (    equals to        ), the absolute average deviation (   ) was found to be less than the desired maximum target which is        deviation in pressure. In spite of satisfactory predicted results in most cases, the absolute deviation was found to be insufficient according to the specified criteria. For instance, the AAD for methane in the presence of ethylene glycol         aqueous solution is         which satisfies the pressure criteria; however, at a temperature of       , the predicted pressure has          deviation from the experimental data, and therefore does not satisfy the pressure criteria. The prediction results of polar SSAFT were found to be satisfactory for hydrocarbons in the presence of low concentrations of ethylene glycol aqueous solutions. Figure 5.4 shows the phase diagram produced by polar SSAFT along with experimental data and the calculated results obtained by SSAFT for methane hydrates. As seen, there is satisfactory agreement between calculated and experimental data for concentrations of up to        . Interestingly, at higher concentrations, the polar-SSAFT shows a significant deviation from the experimental data. This is most likely due to the contribution of dipolar interaction in SSAFT which results in an overestimation in the force field and, consequently, an underestimation in the incipient hydrate formation prediction, most specifically at higher 54  concentrations of ethylene glycol. The same deviations were observed for ethane, propane and carbon dioxide. On the other hand, the parameters required by the van der Waals-Platteew model play a role in the quality of the predictions. Various sets of parameters were extracted from the literature and compared with each other. Table 5.4 summarizes the van der Waals-Platteew parameters. Along with different sets of parameters, the following were selected for the current work:  Table ‎5.3: Required parameters for the van der Waals-Platteew model selected for this work  Property Structure I Structure II        ⁄ 1289.5        ⁄ -4327.9          ⁄ -38.13 -38.13            ⁄ 0.141 0.141   The above set of parameters was chosen because, when calculations were made on data available from the literature, these were the parameters that showed the smallest absolute average deviation (   ) .  55   Figure ‎5.5:  Hydrate formation prediction (   =0) by SSAFT for ethane hydrate in presence of ethylene glycol aqueous solution,      ,      ,       and    Figure ‎5.6: Hydrate formation prediction (   =0) by SSAFT for propane hydrate in presence of ethylene glycol aqueous solution,      ,      ,       and   0 0.5 1 1.5 2 2.5 3 3.5 260 265 270 275 280 285 290 P re ss u re  ( M P a)  Temperature (K) Series1 Avlonitis (1988) SSAFT 0 0.1 0.2 0.3 0.4 0.5 0.6 265 267 269 271 273 275 277 279 P re ss u re  ( M P a)  Temperature (K) SSAFT Kubota et al. (1984) Maekawa (2008) Mohammadi et al. (2008) 35 wt% 20 wt% 10 wt% 0 wt% 20 wt% 15 wt% 10 wt% 0 wt% 56   Figure ‎5.7: Hydrate formation prediction (   =0) by SSAFT for carbon dioxide hydrate in presence of ethylene glycol aqueous solution,      ,      ,       and               0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 249 254 259 264 269 274 279 284 289 P re ss u re  ( M P a)  Temperature (K) Mohammadi and Richon (2010) Deaton and Frost (1946) SSAFT 35 wt% 20 wt% 10 wt% 0 wt% 57  5.3.2. Inhibiting Effect of Methanol  The predicted hydrate formation conditions for methane, ethane, propane and carbon dioxide in the presence of methanol aqueous solution are presented in Figure 5.8 to Figure 5.11 respectively. The     information for these systems is also presented in Table 5.3. It must be emphasized that these are pure predictions and that no binary data was used in either the estimation of SSAFT parameters or the calculation of equilibrium conditions. Outstanding agreements were obtained through the use of simplified-SAFT predictions in the hydrocarbon/water/methanol systems. For all systems except ethane in presence of methanol         aqueous solution (    equals to        ), the absolute average deviation (   ) was found to be less than the desired maximum target of deviation in pressure.  In some cases, for instance methane in the presence of methanol         aqueous solution, the deviation of predicted results from one set of experimental data was found to be higher than for another, leading to a higher average deviation. In spite of obtaining excellent results for the predictions in most cases, the absolute deviation was found to be higher than acceptable for some. For instance, the AAD for methane in the presence of ethylene glycol         aqueous solution is         which satisfies the pressure criteria; however, at a temperature of       , the predicted pressure has a          deviation from the experimental data.   58    Figure ‎5.8: Hydrate formation prediction (   =0) by SSAFT for methane hydrate in presence of methanol aqueous solution,      ,      ,      ,      ,          and    Figure ‎5.9: Hydrate formation prediction (   =0) by SSAFT for ethane hydrate in presence of methanol aqueous solution,      ,      ,      ,      ,          and  0 5 10 15 20 200 220 240 260 280 300 P re ss u re  ( M P a)  Temperature (K) SSAFT Verma (1974) Ng and Robinson (1985) Robinson and Ng (1986) Ng et al. (1987) Mohammadi and Richon (2010a) Mohammadi and Richon (2010b) 0 0.5 1 1.5 2 2.5 3 3.5 235.0 245.0 255.0 265.0 275.0 285.0 P re ss u re  ( M P a)  Temperature (K) Ng and Robinson (1985) Mohammadi et al. (2008) Ng et al. (1985) SSAFT 0 wt% 10 wt% 20 wt% 35 wt% 50 wt% 65 wt% 0 wt% 10 wt% 15 wt% 20 wt% 35 wt% 50 wt% 59    Figure ‎5.10: Hydrate formation prediction (   =0) by SSAFT for propane hydrate in presence of methanol aqueous solution,      ,     ,          and   The prediction for the carbon dioxide-water-methanol system, however, was not as good as those calculated for methane, ethane and propane. In the hydrate formation calculation of carbon dioxide in the presence of methanol, the average absolute deviation exceeded the maximum desired values (         for      ,         for       and         for      ). Despite the unsatisfactory predictions calculated for the inhibiting effect of methanol on carbon dioxide hydrate formation, significant improvement was obtained when compared to the results reported by Englezos et al. (1991) for the    -water- methanol system using the Trebble-Bishnoi equation of states and the van der Waals- Platteew model.  0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 265.0 267.0 269.0 271.0 273.0 275.0 277.0 279.0 P re ss u re  ( M P a)  Temperature (K) Kubota et al. (1984) Ng and Robinson (1985) Mohammadi et al. (2008) SSAFT 15 wt% 10.39 wt% 5 wt% 0  wt% 60   Figure ‎5.11: Hydrate formation prediction (   =0) by SSAFT for carbon dioxide hydrate in presence of methanol aqueous solution,      ,      ,       and            0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 200 210 220 230 240 250 260 270 280 290 P re ss u re  ( M P a)  Temperature (K) Deaton and Frost (1946) Ng and Robinson (1985) Robinson and Ng (1986) SSAFT 0 wt% 20 wt% 35 wt% 50 wt% 61  6 Conclusions and Recommendations 6.1. Conclusions This thesis used the simplified SAFT (SSAFT) to in conjunction with the van der Waals Platteeuw model to predict the incipient hydrate formation conditions for natural gas – type systems. The predicted conditions were then compared to experimental data from the literature. It was found that the SSAFT is highly capable of predicting the conditions of incipient hydrate formation for the single hydrate former gases of methane, ethane, propane and carbon dioxide in the presence of the inhibitors methanol and ethylene glycol, even without the introduction of binary interaction parameters (    .‎Jog‎and‎Chapman’s‎dipolar‎ term was then introduced into the simplified SAFT and the results were examined. Despite obtaining satisfactory predictions at lower alcohol concentrations (of less than        ), we were not able to obtain better agreement with polar SSAFT. Pure compound parameters were correlated using vapor pressure and liquid density for water, methanol and ethylene glycol. It was observed that the quality of a prediction is strongly influenced by the temperature range on which these parameters are correlated. It should be noted that in spite of the excellent results obtained in this thesis, the adapted computational scheme was found to be cumbersome compared to its use with conventional equation of states because of the heavy load of computations that are required with the SAFT equation of state.   62  6.2. Recommendations Based on the results and outcomes of the model described above, there are few suggestions that would be beneficial if further investigated: 1. The prediction accuracy of the SAFT modular equations which was developed for the prediction of hydrate formation conditions for single gases (methane, ethane, propane and carbon dioxide), should be explored for gas mixture as well. 2.   The mixture of the inhibitors, methanol and ethylene glycol, which SAFT model was used for examining the degree of   inhibiting effect, should be investigated along with the introduction of other inhibitors such as glycerol and triethylene glycol . 3. Although the use of original form of van der Waals-Platteeuw model employed in the current study resulted in satisfactory to excellent agreements, the improvement on the degree of accuracy by modifying the assumptions of van der Waals-Platteeuw should be explored further. 4. The behavior of introducing quadrupole moments to the SAFT model prediction of carbon dioxide should be examined.    63  References Al-Saifi, Nayef M., Hamad, E.Z. & Englezos, Peter, 2008. Prediction of vapor-liquid equilibrium in water-alcohol-hydrocarbon systems with the dipolar perturbed-chain SAFT equation of state. Fluid Phase Equilibria, 271(1-2), pp.82-93. Al-Saifi, Nayef Masned, 2012. Prediction and computation of phase equilibria in polar and polarizable mixtures using theory-based equations of state. Available at: https://circle.ubc.ca/handle/2429/38970 [Accessed November 17, 2011]. Anderson, F.E. & Prausnitz, J. M., 1986. Inhibition of gas hydrates by methanol. AIChE Journal, 32(8), pp.1321-1333. 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Yaws’ Handbook of Thermodynamic and Physical Properties of Chemical Compounds, Knovel. 70  Appendices: Appendix A: Helmholtz Free Energy For a closed system (Smith et al. 2004):   B-1 we also have,   B-2   B-3 Substituting B-2 and B-3 into B-1, one may obtain,   B-4 Recalling the definition of Helmholtz free energy,   B-5   B-6 71  combining B-4 and B-6,   B-7 and finally, according to the Gibbs/Duhem equation:             ∑    B-8                72  Appendix B: Driving Compressibility Factor from Helmholtz Free Energy   (   )       C-1  Substituting  B-5 and B-7 into C-, we may obtain,  (   )       C-2  from the above equation,     [      ⁄  ]   C-3 and      [      ⁄  ]   C-4  Because       , the compressibility factor and the Helmholtz free energy are related as,     [      ⁄  ]   C-5  As an example, we show how to obtain the hard-sphere compressibility factor,    , from the hard-sphere Helmholtz free energy,    . Recalling equation ‎3-3‎3-4,       and 73      At a constant temperature, for one mole of fluid,        C-6  Therefore,         (   )  (   )  C-7  Performing the volume derivative of the above expression at a constant temperature, [        ⁄  ]   [         (      (   )  (   )  )  ]                (     )  (   )    C-8  Combining C- and C-,       [        ⁄  ]     [    (     )  (   )  ]    (   ) (   )   C-9  Finally, we get,      

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