Platteeuw theory

The above advantages remove three of the major assumptions in the van der Waals and Platteeuw model—namely Assumptions 3 and 4 in Section 5.1.1, as well as Assumption 6 in Section 5.1.4. The three theses show that, in principle, the ab initio methods have the potential to compose the largest improvements to the van der Waals and Platteeuw theory in the last half-century. For cases with a few components, it can be shown that ab initio methods represent an improvement over common methods (Anderson et al., 2005), such as the program CSMHYD, which accompanied the second, 1998 edition of this book. 

Kobayashi and co-workers began to use the van der Waals and Platteeuw theory in the 1960s to predict hydrate formation in ternary systems. Parrish and Prausnitz extended the method to prediction of hydrate incipient formation in natural gas systems causing the widespread industrial adoption of the van der Waals and the Platteeuw statistical method. Many academic (e.g.. Holder and co-workers ) and commercial programs (e.g., D.B. Robinson, and Associates ) enabled the gas and oil industry to predict thermodynamic conditions at the incipient formation point, and thereby to prevent hydrate formation in industrial processes. All of the errors in the van der Waals and Platteeuw theory were placed in the solid phase. It may be argued that the theory s unusual success in prediction inhibited motivations for advances in hydrate phase measurements. 

The above two successes have served the energy industry well for the last 40 years. Before the modifications indicated in the next section, the hydrate formation temperature prediction error to within 0.7 K (average absolute error for uninhibited hydrates) was acceptable, approximately twice the experimental error. With modem modifications of the van der Waals and Platteeuw theory, the energy industry feels confident in making multimillion dollar hydrate formation decisions based on the predictions, without obtaining data for important applications, such as ... 

In the current study we are mainly interested in describing the gas solubility in pure water, under two-phase equilibrium (H-L ) conditions. The gas of primary interest to the study is methane. To this purpose we use different published thermodynamic models that are based on Equations of State (EoS) forfugacity calculations that are coupled with the van der Waals-Platteeuw theory from Statistical Thermodynamics, and models of gas solubility in the aqueous phase. 

Tsimpanogiarmis et al., [11] reported a comprehensive comparison of all the experimental data for the solubility of methane in the aqueous phase under hydrate equilibrium conditions against a nttmber of models that were based on the van der Waals-Platteeuw theory. The component-specific EoS for methane reported by Sun and Duan [13] has been considered. In addition, a nttmber of models that describe the gas solubihty in aqueous systems have been considered [14-17]. 

Ravipati, S., Punnathanam, S.N., Analysis of parameter values in the van der Waals and Platteeuw theory for methane hydrates using Monte Carlo molecular simulations, bid. Eng. Chem. Res., 51 (2012) 9419-9426. 

Equilibrium measurements of the solid hydrate phase have been previously avoided due to experimental difficulties such as water occlusion, solid phase inhomogeneity, and measurements of solid phase concentrations. Instead, researchers have traditionally measured fluid phase properties (i.e., pressure, temperature, gas phase composition, and aqueous inhibitor concentrations) and predicted hydrate formation conditions of the solid phase using a modified van der Waals and Platteeuw (1959) theory, specified in Chapter 5. 

The lattice of the host in the form it takes in the clathrate is usually thermodynamically unstable by itself—that is, with the holes empty. It is stabilized by inclusion of the guest molecules, and it is of obvious interest in connection with the nonstoichiometry of clathrates to consider the extent to which the cavities in the host lattice must be filled before the system achieves thermodynamic stability. The cavities in the host lattice may all be identical in size and environment, as in the hydroquinone clathrates, or they may be of more than one kind. The gas hydrates, for example, have two possible structures, in each of which there are two sorts of cavity, van der Waals and Platteeuw (15) have developed a general statistical theory of clathrates containing more than one type of cavity. 

The van der Waals and Platteeuw principal equation (1) has five assumptions, which enable refinement of the theory ... 

Van der Waals and Platteeuw (VDWP) were the first to present a theory that describes the thermodynamic equilibrium of hydrates based on principles from Statistical Mechanics. Several modifications of the original VDWP model have been utilized in order to expand the limits of its applicability and improve its accuracy. A brief presentation of the equations that describe the modification of the VDWP used in this work is presented below. 

In the current study we are mainly interested in describing the gas solubility in pure water, under two-phase equihbrium (H-Lw) conditions. Gases of interest to this study include methane and carbon dioxide, and we report results mainly for the case of methane. To this purpose we couple different published thermodynamic models that are based on (i) the van der Waals-Platteeuw (vdWP) theory [9, 10] from Statistical Thermodynamics to describe three-phase (H-Lw-V) equihbria, (ii) Equations of State (EoS) for fugacity calculations, and (iii) models of gas solubihty in the aqueous phase. The considered approach is described in detail by Tsimpanogiannis et al., [11]. The authors conducted an extensive review of experimental and theoretical studies related to the solubility of gases in the aqueous phase under hydrate equilibrium conditions. Here, we report additional results that were not included in the original publication. 

In order to describe the three-phase (Hydrate - Liquid Water - Vapor) equihbria (H-Lw-V) the theory developed by van der Waals-Platteeuw [9, 10] is traditionally used. The theory is based on Statistical Thermodynamics and according to Sloan and Koh [1] it is probably one of the best examples of using Statistical Thermodynamics to solve successfully a real engineering problem. An excellent description of the theory for the three-phase equilibrium calculations is provided in a number of publications [1,9, 10] and will not be repeated here. In addition, extensive details on the methodology for the calculation of two-phase equilibrium (H-L ) conditions can be foimd in the review papers by Holder et al. [12], and Tsimpanogiannis et al., [11]. 

The hydrate-forming conditions are modelled by the solid solution theory of van der Waals and Platteeuw [2]. The statistical thermodynamic model of van der Waals and Platteeuw [2] provides a bridge between the microscopic properties of the clathrate hydrate structure and macroscopic thermodynamic properties, i.e., the phase behaviour. The hydrate phase is modelled by using the solid solution theory of van der Waals and Platteeuw [2], as implemented by Parrish and Prausnitz [10]. The fligacity of water in the hydrate phase is given by the following equation [11] ... 

In the thermodynamic model presented here, the Cubic-Plus-Association equation of state combined is used to model the fluid phases. The hydrate phase is modelled by the solid solution theory of van der Waals and Platteeuw. Good agreement between the model predictions and experimental data is observed, demonstrating the reliability and robustness of the developed model. The CPA EoS is shown to be a very successful model for multi-phase multi-component mixtures containing hydrocarbons, glycols and water. 

In 1958, van der Waals and Platteenw developed a statistical mechanical theory for predicting the stability region of the clathrate hydrate phase with different guest molecules under different temperature and pressure conditions. The van der Waals-Platteeuw (vdWP) theory is based on the thermodynamic condition of equilibrium between a hydrate phase water/ice ()8) phase and i guest species encapsulated in the hydrate at the phase boundary ... 

Thermodynamic properties of clathrate hydrates have been calculated by the van der Waals and Platteeuw (vdWP) theory [16]. It has been applied to predicting... 

A common thread in all these applications is the need to understand what makes clathrate hydrates stable. This chapter will present some of the evidence that recent computer simulations have contributed to this issue. To provide a context for the simulation results, we begin with a brief description of clathrate hydrates and their experimental properties. This will be followed in Section 3 by a discussion of the current theory of hydrate stability (the cell theory). It is intended that this Section should bring out the main ideas behind the cell theory, as it is these basic principles that have motivated recent simulations for a rigorous derivation of the cell theory the reader is referred to the original work of van der Waals and Platteeuw [2]. The role of computer simulations in elucidating the behaviour of clathrate hydrates will be considered in detail in Section 4. 

Using these assumptions in a very elegant statistical mechanical derivation, van der Waals and Platteeuw were able to calculate the conditions required for the clathrate hydrate to be thermodynamically stable. The central result of this theory shows that for a mixture of gases, the chemical potential of the water molecules is related to y Q - the fraction of cavities of type i (i.e. 12-, 14- or 16-hedra) occupied by a guest of species K - according to the equation... 

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