Phase equilibrium problems

This is usually thought of as a phase equilibrium problem. Earlier, we indicated that a phase equilibrium is nothing more than a simple chemical equilibrium. This problem is one such example. 

The great utility of the chemical potential in phase equilibrium problems arises in the following way. In open systems in which the number of moles of any component may increase or decrease, any change in the Gibbs free energy of the system as a whole may be expressed as the sum of the following contributions ... 

Perhaps the most significant of the partial molar properties, because of its application to equilibrium thermodynamics, is the chemical potential, i. This fundamental property, and related properties such as fugacity and activity, are essential to mathematical solutions of phase equilibrium problems. The natural logarithm of the liquid-phase activity coefficient, lny, is also defined as a partial molar quantity. For liquid mixtures, the activity coefficient, y describes nonideal liquid-phase behavior. 

The same reference (standard) state, f is chosen for the two phases, so that it cancels on both sides of equation 39. The products stffi and y" are referred to as activities. Because equation 39 holds for each component of a liquid—liquid system, it is possible to predict liquid—liquid phase splitting when the activity coefficients of the individual components in a multicomponent system are known. These values can come from vapor—liquid equilibrium experiments or from prediction methods developed for phase-equilibrium problems (4,5,10). Some binary systems can be modeled satisfactorily in this manner, but only rough estimations appear to be possible for multicomponent systems because activity coefficient models are not yet sufficiendy developed in this area. 

By using the procedures just outlined, the reduced standard-state chemical potential can be estimated for all compounds. This value of Gy is valid for any solid-liquid phase equilibrium problem that contains the compound... 

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. 

Finally, coupled with the phase equilibrium problem of concentrating the cosolvent, perhaps the composition of cosolvent around the probe is not sufficient to facilitate proton transfer. This implies that if the cosolvent is indeed "clustering" around the solute, there is insufficient structural integrity to solvate the proton. 

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. 

We have applied a global optimization technique, based on interval analysis, to the high-pressure phase equilibrium problem (INTFLASH). It does not require any initial guesses and is guaranteed, both mathematically and computationally, to converge to the correct solution. The interval analysis method and its application to phase equilibria using equation-of-state... 

It may be evident (it should certainly not be surprising) that the extended MC framework needed to solve the phase-equilibrium problem entails exploration of a path that links the macrostates of the two competing phases, for the desired physical conditions ((i. The generic choices here are distinguished by the way in which the path is routed in relation to the key landmark in the configuration space, namely, the two-phase region which separates the macrostates of the two phases and which confers on them their (at least meta-) stability. Figure 2 depicts four conceptually different possibilities. 

The application of Eq. (10.3) to specific phase-equilibrium problems requires use of models of solution behavior, which provide expressions for G or for the Hi as functions of temperature, pressure, and composition. The simplest of such expressions are for mixtures of ideal gases and for mixtures that form ideal solutions. These expressions, developed in this chapter, lead directly to Raoult s law, the simplest realistic relation between the compositions of phases coexisting in vapor/liquid equilibrium. Models of more general validity are treated in Chaps. 11 and 12. 

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. 

Cotterman and Prausnitz (1991) have reviewed the approximate methods that have been used in the solution of phase equilibrium problems within the framework of a continuous description (or, often, a semicontinuous one, where a few components are dealt with as discrete ones). These methods are based on relatively trivial extensions of classical methods (which make use of Gibbs free energy equations, equations of state, and the like) to a continuous description. 

There still remains the problem of reducing these fugacity expressions to equations explicit in temperature, pressure, and composition. This is done in different ways for different phase-equilibrium problems. For VLE we present the method in some detail in the next section. 

Using one of these activity coefficient equations it is possible to calculate liquid-liquid equihbrium (LLE) behavior of multicomponent hquid systems. Consider, for example, the ternary system of Figure 1. A system of overall composition A splits into two liquid phases B and C. The calculation of compositions of B and C is analogous to the flash ciculation of vapor-liquid equilibrium problems. By using the UNIQUAC equations to obtain the partition coefficients, Kj, this problem can be solved for any composition A of the overall system. The calculations are lengthy but computer programs for this purpose (2) have been published. In this paper simpler approximate methods for phase equilibrium problems of environmental interest is sought. For the moment it is sufficient to note that the activity coefficients provide the means of complete liquid-liquid equihbrium computations. 

The brief survey presented here must necessarily begin with a discussion of thermodynamics as a language must of Section 1.2 is concerned with the definition of thermodynamic terms such as chemical potential, fiigacity, and activity. At the end of Section 1.2, (he phase-equilibrium problem is clearly staled in several thermodynamic forms each of these forms is particularly anited for a particular situation, as Indicated in Sections 1.5, 1.6, and 1.7. 

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. 

An alternative method of obtaining a solution to the multiple-reaction, single-phase equilibrium problem is to use the method of Lagrange multipliers. Here one first rewrites the constraints as... 

The starting point for the description of this phase equilibrium problem is again the equality of fugacities of each species in all phases in which that species appears,... 

I/i ge/ieral, considering the process to be a combined chemical and phase equilibrium problem will result in more complicated calculations, but will yield somewhat more information—here the concentration of ammonia in the liquid phase. 

Chemical potential was introduced as a partial property of the Gibbs free energy to solve the phase equilibrium problem. Similar partial properties may be considered for other extensive properties, as volume, enthalpy, entropy, etc. It would be useful to generalise the approach. Let s consider again that M represents the mean molar value of a property. For the whole system we have nM = f P,P,n,n2,nj,... The derivation of the nM) as function of T, P, and composition gives ... 

In addition to its generality, the form (7.1.48) is important because it leads to a computational strategy for analyzing phase-equilibrium situations. In that strategy, a phase-equilibrium problem is treated as a multivariable optimization in which the Ihs of (7.1.48) is the quantity to be minimized. An alternative strategy, in which the computational problem is to solve a set of coupled nonlinear algebraic equations, arises from the constraints on open-system processes developed in 7.2. 

Ultimately, we want to develop a computational procedure for determining the observability of a state proposed for a binary fluid. The motivation is that we want to avoid trying to solve phase-equilibrium problems that do not exist. Therefore we first test for observability, and if multiphase situations are observable, then we solve for phase compositions, if they are required. In this section we consider situations in which the proposed state is identified by specifying values for T, P, and Xp Such a state could be in any one of three observable conditions (a) a stable single phase, (b) a stable multiphase equilibrium, or (c) a metastable single phase. Some metastable phases can only relax to a stable single phase, but other metastable phases can split into multiple phases. Multiphase equilibria in binaries are predominantly two-phase situations, so we will restrict our attention to those possibilities here however, three and four-phase binaries are also possible. 

Therefore, of the alternatives a, P, and yin Figure 8.18, the stable one-phase mixture is that which has the lowest value for f2- We would compute f2 from an appropriate equation of state. If two of those states had the same value of f2, then a two-phase equilibrium situation could occur. The condition (8.4.9) together with minimization of f2 give us sufficient tools for determining the stability of states proposed for binary mixtures. Note we can make such judgements without solving the phase-equilibrium problem. We illustrate with an example. 

Step 7. Identify the root having the lowest value of f2 as the stable one-phase mixture at the proposed T, P, and From Table 8.2 we see that the stable one-phase mixture is root p. Therefore root a, which is our proposed mixture, is not a stable one-phase mixture. Further, Figure 8.18 shows that root a satisfies the requirement on the derivative (8.4.8), so the proposed mixture is not unstable. Hence, it must be metastable it might be observed, but more likely it will split into two phases. To find the compositions of those phases, we would solve the phase-equilibrium problem. Other procedures for identifying stable one-phase mixtures include the tangent-plane method which originates with Gibbs [15] and has been fully developed by Michelsen, especially for multi-component mixtures [16]. 

Although methods for identifying phase splits generally involve more that just differential stability criteria, they do not require us to solve the phase-equilibrium problem for the compositions of any new phases. Such methods are particularly useful when we only need to know whether or not a one-phase fluid can separate. Even when we need to compute equilibrium compositions, it is wise to precede the calculations with a determination as to whether a phase separation can actually be observed. In such cases, the phase stability tests presented in this chapter can serve as informative preliminaries to solving phase-equilibrium problems. 

Situations can arise in which we have apparently specified values for enough properties, and yet the state is still not uniquely identified. We follow Prigogine and Defay [2] and call these indijferent states. The existence of these situations can frustrate some trial-and-error procedures for solving phase-equilibrium problems. 

Here is the azeotropic pressure at absolute temperature T, while A and B are parameters whose values are obtained by fits to azeotropic data. This correlation works well for both positive and negative azeotropes, as shown in Figure 9.12. The principal drawback to (9.3.22) is that the azeotropic compositions remain implicit to find those compositions, we must solve the phase-equilibrium problem. 

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