Statistical Association Fluid Theory SAFT

Various equations of state have been developed to treat association in supercritical fluids. Two of the most often used are the statistical association fluid theory (SAFT) (60,61) and the lattice fluid hydrogen bonding model (LFHB) (62). These models include parameters that describe the enthalpy and entropy of association. The most detailed description of association in supercritical water has been obtained using molecular dynamics and Monte Carlo computer simulations (63), but this requires much larger amounts of computer time (64—66). 

The non-cubic equations of state are characterized by the use of a repulsive term that is based on the Camahan-Starling or on the HCB expressions already reported in Table 3. The attractive part is generally based on that derived from the perturbed hard chain theory (PHCT) [49], or from the statistical associating fluid theory SAFT [50, 51]. These approaches were the precursors of many theoretical attractive terms and consequently of different equations of... 

The Statistical Associating Fluid Theory (SAFT) (Section 16.6)... 

Very many noncubic equations of state have been proposed for polymers. Most of them are rather complicated and a few of them, such as several of the models discussed previously, are based on the GC concept, like the GC-Flory and the GCLF equation of state. Several of these equations of state are reviewed elsewhere, and we will restrict here our presentation to one equation of state that is very promising for polymer systems and has already found widespread acceptance. This is the Statistical Associating Fluid Theory (SAFT). 

The statistical-associated fluid theory (SAFT) of Chapman et al. [25, 26] is based on the perturbation theory of Wertheim [27]. The model molecule is a chain of hard spheres that is perturbed with a dispersion attractive potential and association potential. The residual Helmholtz energy of the fluid is given by the sum of the Helmholtz energies of the initially free hard spheres bonding the hard spheres to form a chain the dispersion attractive potential and the association potential,... 

For the calculations, different EoS have been used the lattice fluid (LF) model developed by Sanchez and Lacombet , as well as two recently developed equations of state - the statistical-associating-fluid theory (SAFT)f l and the perturbed-hard-spheres-chain (PHSC) theoryt ° . Such models have been considered due to their solid physical background and to their ability to represent the equilibrium properties of pure substances and fluid mixfures. As will be shown, fhey are also able to describe, if not to predict completely, the solubility isotherms of gases and vapors in polymeric phases, by using their original equilibrium version for rubbery mixtures, and their respective extensions to non-equilibrium phases (NELF, NE-SAFT, NE-PHSC) for glassy polymers. 

The statistical associating fluid theory (SAFT) is the first and the most popular approach that uses real hard-chain reference fluids, ineluding chain-bonding contributions. Its basic ideas have been developed by Chapman et Without going into details, the fi-... 

The best-known model of this kind is the Statistical Associated Fluid Theory (SAFT) model [58-61]. Here, a non-spherical molecule (solvent or polymer) is assumed to be a chain of identical spherical segments. Starting from a reference system of m hard spheres (A ), this model considers three perturbation contributions, which are assumed to effect independently attractive interactions of the (non-bonded) segments (A ), hard-sphere chain formation (A ), and association (A ° ) ... 

For the application of supercritical carbon dioxide as a medium for the production of polyolefins, it is important to have rehable thermodynamic data for the systems involved. Knowledge of the phase behavior of the reaction mixture is crucial to properly choose process variables such as temperature and pressure in order to achieve maximum process efficiency. For this reason, the ethylene-poly (ethylene-co-propylene) (PEP)-C02 system has been taken as a representative model system [3]. The effect of molecular weight as well as the influence of CO2 on the phase behavior has been studied experimentally by cloud-point measurements. In addition, the Statistical Associating Fluid Theory (SAFT) has been applied to predict the experimental results. 

The model assumes that the reaction mixture consists of a polymer phase swollen with ethylene and CO2, and an ethylene-C02 phase. The assumption that no polymer dissolves in the ethylene-C02 phase within the experimental conditions has previously been confirmed experimentally for a similar polymer [3]. The swollen polymer phase, i.e. the polymer-ethylene-C02 system, is modeled using the Statistical Associating Fluid Theory (SAFT) eos [6, 7], see Section 8.2.2. The supercritical phase, i.e. the ethylene-C02 system, is either modeled with the Lee-Kessler-Plocker (LKP) eos [30] or with the Peng-Robinson (PR) eos [31], because the use of the SAFT eos for the simulation of both phases results in physically inconsistent behavior [9]. The temperatures used in this work are just above the critical tern-... 

This work is far from complete because only a few of the systems in the database were tested and no contemporary EOS models, such as Statistical Associating Fluids Theory (SAFT) (Muller and Gubbins 2001 Economou 2002), Perturbed... 

The polymer solutions warrant use of a special class of lattice models such as Florry-Huggins. For correlation purposes Sanchez-Lacombe method is sufficient but one may also use Statistical Association Fluid Theory (SAFT) models to obtain a better representation. 

The phase behavior of polymer/SCF mixtures can be described using versions of the lattice fluid (LF) model such as that developed by Sanchez and Lacombe [17]. The LF equation of state is relatively simple, and has been successfully used to describe either polymers dissolved in SCFs, or SCFs dissolved in polymers [18,19], including phenomena such as retrograde vitrification. The statistical associating fluid theory (SAFT) [20] can also describe the phase behavior of polymers dissolved in SCFs. The SAFT model, while somewhat more cumbersome to implement than the LF model, is especially well-suited for polymers with varying backbone architecture, such as branched polymers or copolymers. Both the Sanchez-Lacombe and SAFT models have been incorporated into commercially available modeling software [21]. 

A number of equation of state theories have been used to model phase behavior of polymers in supercritical fluids. For example the lattice-fluid theory of Sanchez and Lacombe[4U 42] includes holes on the lattice in order to model compressibility. The lattice-fluid theory has been applied to model phase behavior of both homopolymers and copolymers in supercritical fluids[32, 38, 43, 44]. The statistical associating fluid theory (SAFT)[43,45-48] and corresponding state models[49] have also been employed to model compressible polymer-solvent mixtures. Figure 1 gives the pressure-concentration phase diagram for poly(dimethyI siloxane) in CO2 modeled with the lattice-fluid equation of state[50]. 

Modelling Separators. In many polymerization processes pressure has been applied for several decades to control the thermodynamic state in the reactor as well as to effect downstream separations of products and reactants. As is shown above, recent developments in thermodynamic modelling, such as the Statistical Associating Fluid Theory (SAFT) and the Mean Field Lattice Gas (MFLG) model, make it possible to draw up adequate correlations of the influence of p,T, overall composition and copolymer composition on the phase behaviour, even in systems involving many components. 

The above-mentioned deficiencies of the Flory-Huggins theory can be alleviated, in part, by using the local-composition concept based on Guggenheim s quasichemical theory for the random mixing assumption and replacing lattice theory with an equation-of-state model (Prausnitz et al., 1986). More sophisticated models are available, such as the perturbed hard sphere chain (PHSC) and the statistical associating fluid theory (SAFT) (Caneba and Shi, 2002), but they are too mathematically sophisticated that they are impractical for subsequent computational efforts. 

The main characteristics of type V phase behaviour compared to type I are the region of liquid-liquid immiscibility, the so-called cloud curve, close to the critical point of the more volatile component (the short-chain n-alkane), and the appearance of a three-phase line (L-L-V) close to the vapour pressure curve of the same component (see figure 1). Since demixing of the two components is the main difference between these two types of phase behaviour, a criterion to identify liquid-liquid immiscibility is developed. Such a criterion is introduced in the next section. A simplified version of the statistical associating fluid theory (SAFT-HS) is used to model the n-alkane molecules. This approach offers a good representation of the entire n-alkane series (Galindo et al., 1996) incorporating an intermolecular parameter, which describes the n-alkane size. In this... 

The last term in the r.h.s. of eq 4.21 is based on the results of Statistical Associating Fluid Theory (SAFT) where X is the fraction of molecules not hydrogen bonded at site A, given by the expression ... 

The last type of equation of state that we will consider is the Statistical Associating-Fluid Theory (SAFT) first proposed by Chapman et and Huang and Radosz, and for which there are now many variants as discussed in Chapter 8. 

However, equations of state, too, will be an essential component of chemical engineering theory and practice for the foreseeable future, and as ever, the balance will need to be struck between rigorous theory and engineering applicability. One equation of state, which seems to have done an admirable job of bridging the gap between molecular theory and engineering application, is statistical associating fluid theory (SAFT) and it is with this equation of state and its spin-offs that the remainder of this discussion is concerned. 

Deviations of real molecules from the reference system may occur, e.g., due to attractive interactions (dispersion), formation of hydrogen bonds (association), or the nonspherical shape of the molecules (which can be understood as the formation of chains from spherical segments). These contributions are usually assumed to be independent of each other and are accounted for by different perturbation terms. Depending on the kind of considered perturbation and on the expression used for its description, different models have been developed. One of the first models derived from that idea was the Statistical-Associating-Fluid Theory (SAFT) (Chapman et al. [12, 13] Huang and Radosz [14, 15]). 

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