Vapor-liquid-solid model

In the previous chapter the necessary conditions for equilibrium were introduced in terms of the chemical potentials of the constituent species in the various phases. In this chapter we will relate these chemical potentials to a more convenient form, that of the equilibrium constant. Furthermore, we will discuss the application of equilibrium constants to the three types of equilibria which occur in our overall vapor-liquid-solid model ... 

Apparently, the direct transition from vapor to solid is less common than the double transition vapor — liquid — solid, see, e.g., Refs.158-160). From the rate of solidification of metal droplets (average diameter near 0.005 cm) at temperatures 60° to 370° below their normal melting points, the 7sl was concluded158) to be, for instance, 24 for mercury, 54 for tin, and 177 erg/cm2 for copper. For this calculation it was necessary to assume that each crystal nucleus was a perfect sphere embedded in the melt droplet the improbability of this model was emphasized above. 

It is evident from the title of this symposium that as a result of recent requirements to reduce pollutant levels in process wastewater streams, improved techniques for predicting the vapor-liquid-solid equilibria of multicomponent aqueous solutions of strong and/or weak electrolytes are needed. In addition to the thermodynamic models necessary for such predictions, tools have to be developed so that the engineer or scientist can use these thermodynamic models correctly and with relative ease. 

Equations of State. Equations of state having adjustable parameters are often used to model the pressure—volume—temperature (PVT) behavior of pure fluids and mixtures (1,2). Equations that are cubic in specific volume, such as a van der Waals equation having two adjustable parameters, are the mathematically simplest forms capable of representing the two real volume roots associated with phase equilibrium, or the three roots (vapor, liquid, solid) characteristic of the triple point. 

For example, when we consider the design of specialty chemical, polymer, biological, electronic materials, etc. processes, the separation units are usually described by transport-limited models, rather than the thermodynamically limited models encountered in petrochemical processes (flash drums, plate distillations, plate absorbers, extractions, etc.). Thus, from a design perspective, we need to estimate vapor-liquid-solid equilibria, as well as transport coefficients. Similarly, we need to estimate reaction kinetic models for all kinds of reactors, for example, chemical, polymer, biological, and electronic materials reactors, as well as crystallization kinetics, based on the molecular structures of the components present. Furthermore, it will be necessary to estimate constitutive equations for the complex materials we will encounter in new processes. 

Although most of the studies of this model have focused on the fluid phase in connection with the theory of electrolyte solutions, its solid-fluid phase behavior has been the subject of two recent computer simulation studies in addition to theoretical studies. Smit et al. [272] and Vega et al. [142] have made MC simulation studies to determine the solid-fluid and solid-solid equilibria in this model. Two solid phases are encountered. At low temperature the substitutionally ordered CsCl structure is stable due to the influence of the coulombic interactions under these conditions. At high temperatures where packing of equal-sized hard spheres determines the stability a substitutionally disordered fee structure is stable. There is a triple point where the fluid and two solid phases coexist in addition to a vapor-liquid-solid triple point. This behavior can be qualitatively described by using the cell theory for the solid phase and perturbation theory for the fluid phase [142]. Predictions from density functional theory [273] are less accurate for this system. 

Figure 2.4 Three typical growth models for the catalytic synthesis of nanosilica, (a) VLS = vapor-liquid-solid ...

The maximum-likelihood method is not limited to phase equilibrium data. It is applicable to any type of data for which a model can be postulated and for which there are known random measurement errors in the variables. P-V-T data, enthalpy data, solid-liquid adsorption data, etc., can all be reduced by this method. The advantages indicated here for vapor-liquid equilibrium data apply also to other data. 

Transition matrix estimators have received less attention than the multicanonical and Wang-Landau methods, but have been applied to a small collection of informative examples. Smith and Bruce [111, 112] applied the transition probability approach to the determination of solid-solid phase coexistence in a square-well model of colloids. Erring ton and coworkers [113, 114] have also used the method to determine liquid-vapor and solid-liquid [115] equilibria in the Lennard-Jones system. Transition matrices have also been used to generate high-quality data for the evaluation of surface tension [114, 116] and for the estimation of order parameter weights in phase-switch simulations [117]. 

We present results describing the solid-liquid and the vapor-liquid equilibria in the NaCl-HCl-HpO system. In the first part, purely empirical relations are used to describe the activity coefficients and the second part includes use of a semi-empirical model (Z) to describe the compositional dependence of the activity coefficients. 

In applying equation 33, Cpsl (the constant-pressure molar heat capacity of the stoichiometric liquid) is usually extrapolated from high-temperature measurements or assumed to be equal to Cpij of the compound, whereas the activity product, afXTjafXT), is estimated by interjection of a solution model with the parameters estimated from phase-equilibrium data involving the liquid phase (e.g., solid-liquid or vapor-liquid equilibrium systems). To relate equation 33 to an available data base, the activity product is expressed... 

Atomistic MD models can be extended to the coarse-grained level introduced in the previous section, which is determined by the dimension of the backbone chain and branch. For the precise description of water molecular behavior, simple point charge (SPC) model was adopted (Krishnan et al., 2001), which can be used to simulate complex composition systems and quantitatively express vibrational spectra of water molecules in vapor, liquid, and solid states. The six-parameter (Doh, o , fi, Lye, Lyy, and Lee) SPC potential used for the water molecules is shown in Equation (24) ... 

Physical property data for many of the key components used in the simulation for the ethanol-from-lignocellulose process are not available in the standard ASPEN-Plus property databases (11). Indeed, many of the properties necessary to successfully simulate this process are not available in the standard biomass literature. The physical properties required by ASPEN-Plus are calculated from fundamental properties such as liquid, vapor, and solid enthalpies and density. In general, because of the need to distill ethanol and to handle dissolved gases, the standard nonrandom two-liquid (NRTL) or renon route is used. This route, which includes the NRTL liquid activity coefficient model, Henry s law for the dissolved gases, and Redlich-Kwong-Soave equation of state for the vapor phase, is used to calculate properties for components in the liquid and vapor phases. It also uses the ideal gas at 25°C as the standard reference state, thus requiring the heat of formation at these conditions. 

In addition to handling the conventional vapor/liquid process operations, the ASPEN library of process models includes solids handling and separation units, a set of generalized reactors, improved flash and distillation unit models and process models from the FLOWTRAN simulator. The user can also include his or her own model or key elements of a model, such as the reaction kinetics, in FORTRAN code. 

Figure 10. Vapor-liquid equilibria for an argon-krypton mixture (modeled as a Lennard-Jones mixture) for the bulk fluid (R = >) and for a cylindrical pore of radius R = / /Oaa = 2.5. The dotted and dashed lines are from a crude form of density functional theory (the local density approximation, LDA). The points and solid lines are molecular dynamics results for the pore. Reprinted with permission from W. L. Jorgensen and J. Tirado-Rives, J. Am. Chem. Soc.

Contact angle — The contact angle is the angle of contact between a droplet of liquid and a flat rigid solid, measured within the liquid and perpendicular to the contact line where three phases (liquid, solid, vapor) meet. The simplest theoretical model of contact angle assumes thermodynamic equilibrium between three pure phases at constant temperature and pressure [i, ii]. Also, the droplet is assumed to be so small that the force of gravity does not distort its shape. If we denote the - interfacial tension of the solid-vapor interface as ysv. the interfacial tension of the solid-liquid interface as ySL and the interfacial tension of the liquid-vapor interface as yLV, then by a horizontal balance of mechanical forces (9 < 90°)... 

While in the air compartment, the contaminant solubilizes in the vapor-liquid phase or is associated with aerosol particles by adsorption. It is also prone to desorption from the aerosol particles into the vapor phase. Relevant properties of the air used to model transport of partitioning of a contaminant in the air compartment include temperature, turbulence, wind speed, size and composition of aerosol particles, etc.16,19 Relevant properties of the contaminant that measure its tendency to partition among the vapor, liquid, and solid phases in the air include its aqueous solubility (Saq), vapor pressure (VP), Henry s constant... 

The simplest models for vapor/liquid equilibrium, based on Raoult s law and Henry s law, are presented in Chap. 10, largely from an empirical point of view. The calculations by modified Raoult s law, described in Sec. 10.5, are adequate for many purposes, but are hmited to low pressures. The initial sections of this chapter therefore present two general calculational procedures for VLB, the first by an extension of modified Raoult s law and the second by equations of state. The theoreticalfoundationforbothproceduresispresentedinChap. 11. The remainder of tliis clrapter deals more generally witlr plrase equilibria, with consideration given in separate sections to liquid/liquid, vapor/liquid/liquid, solidlliquid, solid/vapor, adsorption, and osmotic equilibria. 

The MD simulations showed liquid-solid phase transitions, at a constant temperature, with the variation in the equilibrium vapor-phase pressure below saturated one, and prove the importance of the tensile effect on freezing in nanopores. The capillary effect on shift in freezing point was successfully described by a model based on the concept of pressure felt by the pore fluid. 

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