Thermodynamic Phase Equilibrium

In this chapter the thermodynamic behavior of single- and multiphase systems of pure substances and their mixtures are described in a general way. In the References section some general textbooks and data compilations are recommended. 

Multiphase systems are often found in machinery and apparatuses of the processing industry because most thermal separation processes are based on the transfer of one or more components from one phase to another. 

A phase is the entirety of regions, where material properties either do not change or only change continually, but never change abmptly. However, it makes no difference whether the regions are spatially coherent or not (continuous or dispersed phase). A phase can consist of one or more chemically uniform substances, which are called components. A system can contain one phase (gas, liquid, solid), two phases (e.g., liquid/gas, fluid/sohd, fluid/fluid), or even more (in an evaporative crystallizer, e.g., there are a solid, a liquid and a gaseous phase) This chapter describes the thermo namic equilibrium between phases. 

Gibbs phase-rule states how many degrees of freedom/fully describe a multicomponent and multiphase system  

Here k is the number of components and p is the number of phases. For hquid water,/equals 2, because the water s state is ftrlly described by stating its pressure and temperature. For saturated steam, however, which is a liquid/gas system, statement of either pressure or temperature is sufficient. At the triple point there are three coexistent phases one solid, one hquid, and one gaseous. The system is exactly defined at this point. When deahng with a two-component system or a binary ntixture the degree of freedom increases by one. Then more information, e.g., the concentration, is necessary to fully describe the system. 

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For two-phase flow, additional assumptions are made that thermodynamic phase equilibrium exists before and after the restriction (or expansion), and that no phase change occurs over the restriction. Romie (Lottes, 1961) wrote the equation for the momentum change across an abrupt expansion as... 

These considerable solvent-dependent shifts demonstrate that for molecules with such pronounced conformational ambiguity, the equilibration of conformations has to be taken into account in the calculation of thermodynamic phase-equilibrium data. In order to enable a consistent treatment, we have implemented an automated conformation equilibration scheme in COSMO therm. A compound X can be represented by a set of COSMO-files for the conformers, and a multiplicity ojx(i) can be assigned to each con-former based on geometrical degeneration aspects. Then the population of a conformer, i, in a solvent S is calculated as... 

Strictly speaking, the result obtained from P.q. 14-18 for the mole fraction of dissolved gas is valid for the liquid layer just beneath the interface and not necessarily the entire liquid. The latter will be the case only when thermodynamic phase equilibrium is established throughout the entire liquid body. 

Here, assume that in the range of compositions involved, the thermodynamically phase equilibrium relations between rich and lean streams are linear, and concern with the operating temperature T and pressure P, then we can obtain phase equilibrium equations as Eq. (1). 

In order to calculate the distribution coefficient by Equation 1.29, the activity coefficient Y must be evaluated. The activity coefficients are generally determined from the experimental data and correlated on the basis of thermodynamic phase equilibrium principles. The relationship most often used for this purpose is the Gibbs-Duhem equation (Equation 1.7). At constant temperature and pressure, this equation becomes... 

Alternatively, thermodynamic phase equilibrium in a model system can be evaluated by beginning the simulation with two (or more) phases in the same simulation volume, in direct physical contact (i.e., with a solid-fluid interface). This approach has succeeded [79], but its application can be problematic. Some of the issues have been reviewed by Frenkel and McTague [80]. Certainly the system must be large (recent studies [79,81,82] have employed from 1000 up to 65,000 particles) to permit the bulk nature of both phases to be represented. This is not as difficult for solid-liquid equilibrium as it is for vapor-liquid, because the solid and liquid densities are much more alike (it is a weaker first-order transition) and the interfacial free energy is smaller. However, the weakness of the transition also implies that a system out of equilibrium experiences a smaller driving force to the equilibrium condition. Consequently, equilibration of the system, particularly at the interface, may be slow. 

In the case of true thermodynamic phase equilibrium, in which the absolute minimum is attained for the system Gibbs free energy at given T and p, the solubility calculation is performed following the classical thermodynamic result which imposes the equality between the equilibrium chemical potential of the penetrant in the polymeric mixture and in the external phase Th equilibrium solute content, and... 

As discussed in Section 2.4, the residue curve equation is merely a mass balance, and can be used for any relationship between x and y. In this scenario, however, it is logical and common practice to assume that the streams emerging from a tray, or packing segment, are in equilibrium with each other. This then allows y to be defined by the appropriate thermodynamic phase equilibrium model. 

Scharfer et al. set up a multi-component transport model to describe the diffusion driven mass transport of water and methanol in PEM [170]. For a PEM in contact with liquid methanol and water on one side and conditioned air on the other, the corresponding differential equations and boundary conditions were derived taking into account the polymers three-dimensional swelling. Phase equilibrium parameters and unknown diffusion coefficients for Nafion 117 were obtained by comparing the simulation results to water and methanol concentration profiles measured with confocal Raman spectroscopy. The influence of methanol concentration, temperature and air flow rate was predicted by the model. Although there are indications for an influence of convective fluxes, the measured profiles are ascribed to a Fickean diffusion. Furthermore, the assumption to describe the thermodynamic phase equilibrium as liquid-type equilibrium also at the lower surface of the membrane, which is in contact with a gas phase, can be confirmed by their results. 

Obviously no thermodynamic phase equilibrium method can handle all these mixtures. 

The PDLC system performance depends strongly on the final morphology of the liquid crystal domains dispersed inside the polymer matrix. The size, shape and distribution of liquid crystal domains are generally dictated not only by thermodynamic phase equilibrium, but also by the type of material used and by interfacial interactions [58-62]. 

Following the thermodynamics (phase equilibrium) of CO2 hydrate, which is time-independent, it is also important to know the time-dependent phenomenon of hydrate, namely, how hydrates form and dissociate. Note that the study of such phenomena is much more challenging than that of the thermodynamic properties. 

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