Van der Waals and Platteeuw model

A systematic determination of both hydration number (Cady, 1983) and relative cage occupancies (Davidson and Ripmeester, 1984) shows that molecules such as CH3CI and SO2 are the most nonstoichiometric. Although theoretical calculations using the van der Waals and Platteeuw model provides some rationale for the nonstoichiometry, experimental quantification of nonstoichiometry as a function of guest/cavity size ratio has yet to be determined. 

Several methods have arisen to correct the assumptions in the above van der Waals and Platteeuw model, to address the inaccuracies at the high pressures of current applications. The two most prominent modern correction methods are (1) to use ab initio quantum mechanical corrections to relate to first principles as much possible, as briefly discussed in Section 5.1.9, and (2) to fit the existing... 

Figure 5.3 Filling the empty cages without distortion. Van der Waals and Platteeuw model without distortion of hydrate due to guests (7 = 7). Process (1) in Figure 5.3 is given by the summation term in Equation 5.23.

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. 

The van der Waals and Platteeuw model represents the largest major advance in the thermodynamic history of hydrate technology. With this model, single hydrate formation data could not only be interpolated and extrapolated but also they could be extended to accurate mixture formation condition predictions. In Section 4, the details of this model are considered in light of modem hydrate phase measurements and modeling. 

NMR success motivated other spectroscopic studies to measure the hydrate phase directly. This work represented an experimental departure, because previously only the fluid phases (vapor and liquid(s)) were measured, and any experimental error was incorporated in the solid-phase model of van der Waals and Platteeuw, However, with modem solid-phase measurements, the errors in the van der Waals and Platteeuw model could be clarified and corrected. Raman spectroscopy and diffraction (X-ray and neutron, supplemented by Rietveld analysis ) have been successful the first method to measure the relative occupation of single guest cages, and the second to extend the work to hydrate isothermal, adiabatic, and isobaric compressibilities. As shown in Section 4, these measurements combine with spectroscopic hydrate phase measurements to enable improvements of the model. 

The major advance in hydrate thermodynamics was the generation of the van der Waals and Platteeuw model bridging the normal macroscopic and microscopic domains. Only a brief overview is given here to provide a basis for model improvements the reader interested in more details should refer to another source. The essence of the van der Waals and Platteeuw model is the equation for the chemical potential of water in the hydrate phase ... 

Figure 3 Concepts of the van der Waals and Platteeuw model and improvement, (a) original model that does not allow for distortion by guests (b) corrected model...

The initial model was generated by Barrer and Stuart (1957), with a more accurate method by van der Waals and Platteeuw (1959), who are considered the founders of the method. In the present section the latter model is substantially expanded by Ballard (2002), as follows ... 

In particular, the extension of the van der Waals and Platteeuw method addresses the first assumption listed at the beginning of Section 5.1.1—namely that encaged molecules do not distort the cavity. In the development of the statistical thermodynamic hydrate model (Equation 5.23), the free energy of water in the standard hydrate (empty hydrate lattice), gt, is assumed to be known at a given temperature (T) and volume (v). Since the model was developed at constant volume, the assumption requires that the volume of the empty hydrate lattice, 7, be equal to the volume of the equilibrium hydrate, v11, so that the only energy change is due to occupation of the hydrate cavities, as shown in Figure 5.3. 

In Figure 5.5, process (1) is given by Equations 5.30 and 5.31 and process (2) by the van der Waals and Platteeuw statistical model, since it is done at constant volume. Note that, since chemical potential is a state function, Figures 5.4 and 5.5 are equivalent processes. 

The idea of a fixed crystal structure in which single cages contained at most one guest proved irresistible to statistical thermodynamicists. After an initial effort by Powell, Royal Dutch Shell workers van der Waals and Platteeuw generated a method that still stands today as a principal, regular industrial use of statistical thermodynamics. However, the model was not suitable for manual calculations (as were the methods of Katz in item 3 above), but required access to then-scarce computers, which limited its application to large companies or major universities. Widespread adoption of the model awaited the proliferation of personal computers. 

The modem era of hydrate research is marked by the industrial adoption of the van der Waals and Platteeuw statistical thermodynamics model for the hydrate phase. The spectroscopic measurement of the hydrate phase, abetted with molecular simulation, led to accuracy improvements, and industrial applications to energy, seafloor stability, and climate change. With the above historical advances, consider modem thermodynamics of the hydrate phase itself. 

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 addition to salts, alcohols, and polymers, the presence of a porous medium will also have an inhibiting effect on the formation of gas hydrates. Clarke, Pooladi-Darvish, Bishnoi derived the following expression for the activity of water in a pore of radius r, and used it along with the model of van der Waals and Platteeuw to model the incipient conditions for... 

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. 

The van der Waals and Platteeuw (vdW-P) statistical thermodynamic model was based on the following assumptions ... 

In the statistical model developed by van der Waals and Platteeuw (2), the difference in chemical potential between a clathrate and empty host lattice is expressed as... 

The statistical thermodynamic model for hydrate equilibria developed by van der Waals and Platteeuw in 1959 (2) was generalized by Parrish and Prausnitz for prediction of hydrate dissociation pressures (42). The method of... 

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. 

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. 

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]. 

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