Phase equilibria problems, formulation

Determination of T y. In the formulation of the phase equilibrium problem presented earlier, component chemical potentials were separated into three terms (1) 0, which expresses the primary temperature dependence, (2) solution mole fractions, which represent the primary composition dependence (ideal entropic contribution), and (3) 1, which accounts for relative mixture nonidealities. Because little data about the experimental properties of solutions exist, Tg is usually evaluated by imposing a model to describe the behavior of the liquid and solid mixtures and estimating model parameters by semiempirical methods or fitting limited segments of the phase diagram. Various solution models used to describe the liquid and solid mixtures are discussed in the following sections, and the behavior of T % is presented. 

The completely reliable computational technique that we have developed is based on interval analysis. The interval Newton/generalized bisection technique can guarantee the identification of a global optimum of a nonlinear objective function, or can identify all solutions to a set of nonlinear equations. Since the phase equilibrium problem (i.e., particularly the phase stability problem) can be formulated in either fashion, we can guarantee the correct solution to the high-pressure flash calculation. A detailed description of the interval Newton/generalized bisection technique and its application to thermodynamic systems described by cubic equations of state can be found... 

The phase equilibrium problem consists of two parts the phase stability calculation and the phase split calculation. For a particular total mixture composition, the phase stability calculation determines if that feed will split into two or more phases. If it is determined that multiple phases are present, then one performs the phase split calculation, assuming some specified number of phases. One must then calculate the stability of the solutions to the phase split to ascertain that the assumed number of phases was correct. The key to this procedure is performing the phase stability calculation reliably. Unfortunately, this problem—which can be formulated as an optimization problem (or the equivalent set of nonlinear equations)— frequently has multiple minima and maxima. As a result, conventional phase equilibrium algorithms may fail to converge or may converge to the wrong solution. 

This formulation, while of absolutely general validity, is so complicated that approximate methods of solution of phase equilibrium problems have to be developed. A few essential aspects of these approximate methods are discussed in the next subsection. 

Equation (1,2-15) is a basis for the formulation of phase-equilibrium problems. However, since the chemical potential has some practical and conceptual shortcomings, it is useful to replace it, with a related quantity.i, the fugacity. Equation (1,2-15) is then replaced by the equivalent criterion for phase equilibrium. 

The difficulties encountered in the Chao-Seader correlation can, at least in part, be overcome by the somewhat different formulation recently developed by Chueh (C2, C3). In Chueh s equations, the partial molar volumes in the liquid phase are functions of composition and temperature, as indicated in Section IV further, the unsymmetric convention is used for the normalization of activity coefficients, thereby avoiding all arbitrary extrapolations to find the properties of hypothetical states finally, a flexible two-parameter model is used for describing the effect of composition and temperature on liquid-phase activity coefficients. The flexibility of the model necessarily requires some binary data over a range of composition and temperature to obtain the desired accuracy, especially in the critical region, more binary data are required for Chueh s method than for that of Chao and Seader (Cl). Fortunately, reliable data for high-pressure equilibria are now available for a variety of binary mixtures of nonpolar fluids, mostly hydrocarbons. Chueh s method, therefore, is primarily applicable to equilibrium problems encountered in the petroleum, natural-gas, and related industries. 

Problem Formulation. The conditions of equilibrium require the equivalence in each phase of temperature, pressure, and chemical potential for each component that is transferable between the phases and are subject to constraints of stoichiometry. A statement of the equivalence of chemical potential is identical to equations 8 and 9. An example is the AJB C D quaternary system. This system contains four binary compounds, AC, BC, AD, and BD, and the conditions of equilibrium allow three equations (of the type given by equation 8) to be written. The fourth possible equation is redundant as a result of the stoichiometric constraint (i.e., equal number of atoms on each sublattice). 

A general formulation of the problem of solid-liquid phase equilibrium in quaternary systems was presented and required the evaluation of two thermodynamic quantities, By and Ty. Four methods for calculating Gy from experimental data were suggested. With these methods, reliable values of Gy for most compound semiconductors could be determined. The term Ty involves the deviation of the liquid solution from ideal behavior relative to that in the solid. This term is less important than the individual activity coefficients because of a partial cancellation of the composition and temperature dependence of the individual activity coefficients. The thermodynamic data base available for liquid mixtures is far more extensive than that for solid solutions. Future work aimed at measurement of solid-mixture properties would be helpful in identifying miscibility limits and their relation to LPE as a problem of constrained equilibrium. 

If chemical reactions occur, then we must introduce a new variable, the i coordinate e for each independent reaction, in order to formulate the mate balance equations. Furthermore, we are able to write a new equilibrium rela [Eq. (15.8)] for each independent reaction. Therefore, when chemical-rea equilibrium is superimposed on phase equilibrium, r new variables appear r new equations can be written. The difference between the number of va and number of equations therefore is unchanged, and Duhem s theorem originally stated holds for reacting systems as well as for nonreacting syste Most chemical-reaction equilibrium problems are so posed that it is 1 theorem that makes them determinate. The usual problem is to find the corn-tion of a system that reaches equilibrium from an initial state of fixed an of reacting species when the two variables T and P are specified. 

Since Pf is a function of temperature only, Raoult s law is a set of N equations in the variables T, P, y,, and, . There are, in fact, N - 1 independent vapor-phase mole fractions (the y,- s), N - 1 independent liquid-phase mole fractions (the x, s), and T and P. This makes a total of 21V independent variables related by N equations. The specification of N of these variables in the formulation of a vapor/liquid equilibrium problem allows the remaining N variables to be determined by the simultaneous solution of the N equilibrium relations given here by Raoult s law. In practice, one usually specifies either T or P and either the liquid-phase or the vapor-phase composition, fixing 1 + N - 1) = N variables. 

In principle, knowing the molar entropy of the perfect gas (Section 1.17), and by measuring the change of equilibrium gas pressure as a function of temperature, one can determine the molar entropy of the adsorbed phase. The problem here is that the experiment has to be carried out at constant 0, a problematic task. Methods for circumventing this difficulty are shown below. Meanwhile, for completeness, we observe that at equilibrium the chemical potentials of the gas and adsorbate must match then Hg — Hs — T(Sg — Ss), so that we obtain the alternative formulation... 

The A"-factor formulation introduced in this calculation is frequently useful in solving vapor-liquid equilibrium problems. The procedure is easily generalized to nonideal liquid and vapor phases as follows ... 

The unsteady state balance equations must be completed with constitutive equations, which are relations between some state variables, usually expressing natural laws or the kinetics of transport phenomena. Examples are PVT relations, as equations of state, or kinetics expressions, phase-equilibrium factors, etc. Specifying the initial and boundary conditions completes the problem formulation. 

The impact of these liquid phase reactions on the phase equilibrium properties is thus an increased solubility of NH3, CO2, H2S and HCN compared with the one calculated using the ideal Henry s constants. The reason for the change in solubility is that only the compounds present as molecules have a vapour pressure, whereas the ionic species have not. The change thus depends on the pH of the mixture. The mathematical solution of the physical model is conveniently formulated as an equilibrium problem using coupled chemical reactions. For all practical applications the system is diluted and the liquid electrolyte solution is weak, so activity coefficients can be neglected. 

Equilibrium, stability, and criticality arc important concepts that are closely related. In this chapter, after the formulation of simple methods for phase-equilibria calculations, the concepts of stability and criticality are introduced, and the application of the Gibbs free energy surface analysis to phase-equilibrium calculations is demonstrated. Then the stability and criticality concepts are presented in detail. These concepts are useful for a broad range of problems in engineering and physics. 

The example discussed suggests another more realistic formulation of the equilibrium problem. Under the conditions of the previous problem, instead of P, and Pg, we specify the pressure in one phase, say, Pg. The system of Eqs. (8) and (9) is completed by the Laplace equation in the form of Eqs. (23). In the second equation, (23), the dependence /(/ ) is supposed to be known from the geometry of the porous space. The surface tension and the wetting angle are defined as known functions of the thermodynamic conditions (e.g., the surface is assumed to be wet by the condensate and the surface tension is calculated by the parachor method). The volumetric hquid... 

The G-L problem arises, for example, when the capillary transition zone in a thick gas-condensate reservoir is evaluated [42]. The physical interpretation of this problem is as follows We measure pressure and composition of the gas phase after the porous sample has been introduced into a gas vessel and would hke to determine the parameters related to the capillary condensate in the porous medium. The difference between normal and eapUlary equilibrium, related to the additional degree of freedom, is especially pronounced in this case. For ordinary equilibrium, the formulated problem would be overdefined, because it would include an additional condition of the equality of phase pressures. 

We will take thermal and mechanical equilibrium for granted in the following discussion hence, in formulating the phase equilibria problem, we need only measure the temperature and pressure of one phase, and these values must apply to the entire system. This concept is illustrated schematically in Figure 6.1, where temperature and pressure measurements that are made only to phase a apply to the entire system. A piston-cylinder assembly is used to remind us that the system must be able to change in volume to accommodate thermal and mechanical equilibrium. 

If our interest is restricted to those sharp characteristics—in particular where a phase transformation occurs in an ideal experiment —we may formulate (and in principle solve) the problem entirely within the framework of the equilibrium statistical mechanics of the competing phases (the framework which implies that sharpness ). 

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