Van der Waals and Platteeuw

Double hydrates was initially reserved for structure II hydrates in which one component is hydrogen sulfide or hydrogen selenide. It has come to mean hydrates in which each size cage is primarily occupied by a different type of molecule. Von Stackelberg proposed that double hydrates were stoichiometric due to their almost invariant composition. Van der Waals and Platteeuw (1959) suggested this invariance was caused instead by azeotropic composition (i.e., hydrate and gas phase compositions are the same). 

The advantage of the method in addition to accuracy is that, in principle, it enables the user to predict properties of mixtures from parameters of single hydrate formers. Since there are only eight natural gas components (yet an infinite number of natural gas mixtures) that form hydrates, the method represents a tremendous saving in experimental effort for the natural gas industry. The modified van der Waals and Platteeuw method is detailed in Chapter 5. 

McKoy and Sinanoglu (1963) and Child (1964) refined the van der Waals and Platteeuw method using different intermolecular potentials such as the Kihara potential. Workers at Rice University, such as Marshall et al. (1964) and Nagata and Kobayashi (1966a,b), first fit simple hydrate parameters to experimental data for methane, nitrogen, and argon. Parrish and Prausnitz (1972) showed in detail how this method could be extended to all natural gases and mixed hydrates. 

Efforts to improve the original assumptions by van der Waals and Platteeuw were detailed in a review by Holder et al. (1988). Erbar and coworkers (Wagner et al., 1985) and Anderson and Prausnitz (1986) presented improvements to inhibitor prediction. Robinson and coworkers introduced guest interaction parameters into their prediction scheme, as summarized by Nolte et al. (1985). At Heriot-Watt University, the group of Tohidi and Danesh generated another prediction extension, with emphasis on systems containing oil or condensate (Avlonitis et al., 1989 Avlonitis, 1994 Tohidi et al., 1994a). 

The van der Waals and Platteeuw method has been extended to flash programs by a number of researchers (Bishnoi et al., 1989 Cole and Goodwin, 1990 Edmonds et al., 1994, 1995 Tohidi et al., 1995a Ballard and Sloan, 2002). These flash calculations predict the equilibrium amount of the hydrate phase relative to associated fluid phases. 

Fundamentals of phase equilibria (i.e., phase diagrams, early predictive methods, etc.) are listed in Chapter 4, while Chapter 5 states the more accurate, extended van der Waals and Platteeuw predictive method. Chapter 6 is an effort to gather most of the thermodynamic data for comparison with the predictive techniques of Chapters 4 and 5. Chapter 7 shows phase equilibria applications to in situ hydrate deposits. Chapter 8 illustrates common applications of these fundamental data and predictions to gas- and oil-dominated pipelines. 

Chapter 5 details the modified statistical thermodynamic prediction method of van der Waals and Platteeuw (1959). The application of molecular simulation methods to hydrates is outlined in Section 5.3. 

Hydrate Nonstoichiometry. The cause of the nonstoichiometric properties of hydrates has been considered. Evidence for the view that only a fraction of the cavities need to be occupied is obtained from both the experimental observations of variation in composition, and the theoretical success of the statistical thermodynamic approach of van der Waals and Platteeuw (1959) in Chapter 5. Typical occupancies of large cavities are greater than 95%, while occupancy of small cavities vary widely depending on the guest composition, temperature, and pressure. 

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. 

While a first approach to phase diagrams is given here, Section 5.2 extends the phase diagrams in this portion of Chapter 4 to single, binary, and ternary mixtures of methane, ethane, and propane. The reader may wish to consult Section 5.2 for a more enlightening discussion that applies the van der Waals and Platteeuw method to the most common components of natural gases. 

Brown, and the statistical thermodynamic method of van der Waals and Platteeuw (1959a) was substituted for the three-phase hydrate line prediction by the gas gravity chart of Katz. 

In hydrate equilibrium, it may seem slightly unusual to apply it to binary systems (water and one guest component) of three-phase (Lw-H-V or I-H-V) equilibrium to obtain the heats of dissociation. As van der Waals and Platteeuw (1959b) point out, however, the application of the Clapeyron equation is thermodynamically correct, as long as the system is univariant, as is the case for simple hydrates. 

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 ... 

The first two of the above sections are a simplification and slight expansion of the derivation from the review article by van der Waals and Platteeuw (1959). They were written assuming that the reader has a minimal background in statistical thermodynamics on the level of an introductory text, such as that of Hill (1960), McQuarrie (1976), or Rowley (1994). The reader who does not have an interest in statistical thermodynamics may wish to review the basic assumptions in Sections 5.1.1 and 5.1.4 before skipping to the final equations and the calculation prescription in Section 5.2. 

Multiplying all three factors of Equations 5.1, 5.2b, and 5.3 together over type i cavities, the canonical partition function was obtained by van der Waals and Platteeuw ... 

However, it should be remembered that the fractional filling is a function of the product Cjjfj, rather than either factor in the product. Finally, in the original van der Waals and Platteeuw approach the Langmuir constants for both adsorption and enclathration were only functions of temperature for each molecule type retained at the individual site or cavity. In the modified approach below, the Langmuir constants are also a function of cage size, or the unit cell volume, which is a function of the hydrate guests, temperature, and pressure. 

Following van der Waals and Platteeuw (1959, pp. 26ff) the individual particle partition function is related to the product of three factors (1) the cube of the de Broglie wavelength, (2) the internal partition function, and (3) the configurational triple integral, as... 

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.

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 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. 

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 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. 

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