Phase equilibrium first criterion

The following criterion of phase equilibrium can be developed from the first and second laws of thermodynamics the equilibrium state for a closed multiphase system of constant, uniform temperature and pressure is the state for which the total Gibbs energy is a minimum, whence... 

Equilibrium in a multiphase system implies thermal, mechanical, and material equilibrium. Thermal equilibrium requires uniformity of temperature throughout the system, and mechanical equilibrium requires uniformity of pressure. To find the criterion for material equilibrium, we treat a two-phase system and consider a transfer of dn moles from phase p to phase a. First, we regard each phase as a separate system. Because material enters or leaves these phases, they are open systems and we must use Eq. (4) to write their change in internal energy ... 

Examples. Figure 2.1 illustrates the first criterion. The system in Fig, 2.1a has a vapor product and a liquid product and therefore obeys this criterion, The systems in Fig, 2,16 and c have no vapor products and therefore are not equilibrium stages. Generating a vapor phase in these systems (Fig, 2,1 d and e) renders them equilibrium stages. Figure 2.1c and e depict a total and a partial condenser, respectively. The total condenser is not a distillation stage, whereas the partial condenser is. 

This fundamental criterion of equilibrium may be applied to practical situations such as determining phase equilibrium conditions. The relationship 1.8 is combined with the first law to arrive at the following expression ... 

Unstable states defined by negative curvature of Ag ixing vs. y2 at constant T and p, between the spinodal points. In this region, (9 Agmixmg/9y2)7 ,p < 0. Single-phase equilibrium states of this nature are completely disallowed even if the first stability criterion is satisfied. 

The combination of successive substitution and Newton s method is a good choice and has the desirable features of both. In this approach, the successive substitution comprises the first few iterations and later, when a switching criterion is met, Newton s method is used. To our knowledge, some commercial reservoir simulation models have adopted the combined successive substitution-New ton approach after the experience with various methods of solving nonlinear flash calculation including Powell s method (1970). The application of a reduction method to phase equilibrium calculations has also been proposed (Michelsen, 1986 Hendriks, and Van Bergen, 1992). In this approach, the dimensionality of phase equilibrium problems for multicomponent mixtures can be drastically reduced. The application of reduction methods and its implementation in reservoir compositional models is under evaluation. 

Use thus of the first criterion for equilibrium reached at constant T and P, leads to a framework of equations for obtaining numerical answers in the case of phase and/or chemical equilibrium under these constraints. 

A plot of 2 vs. -t2 for symmetrical systems (i.e., ii vo) is shown in Fig. 1 for a series of values of the heat lerm, It shows how the partial vapor pressure of a component of a binary solution deviates positively from Raoult s law more and mure as the components become more unlike in their molecular attractive forces. Second, the place of T in die equation shows that tlic deviation is less die higher the temperature. Third, when the heat term becomes sufficiently large, there are three values of U2 for the same value of ay. This is like the three roots of the van der Waals equation, and corresponds to two liquid phases in equilibrium with each other. The criterion is diat at the critical point the first and second partial differentials of a-i and a are all zero. 

We now want to consider the extent to which a solid is soluble in a liquid, a gas. or a supercritical fluid. (This last case is of interest for supercritical extraction, a new separation method.) To analyze these phenomena we again start with the equality of the species fugacities in each phase. However, since the fluid (either liquid, gas, or supercritical fluid) is not present in the solid, two simplifications arise. First, the equilibrium criterion applies only to the solid solute, which we denote by the subscript 1 and second, the solid phase fugacity of the solute is that of the pure solid. Thus we have the single equilibrium relation... 

For a first-order transition the criterion for equilibrium between two phases is the equality of the Gibbs free energy G... 

In the preceding chapters we first considered the primary forces acting on a fluidized particle in a bed in equilibrium, and then the elastic forces between particles that come into play under non-equilibrium conditions. These two effects provide closure for the particle bed model, formulated in terms of the particle-and fluid-phase conservation equations for mass and momentum. Up to now, applications have focused on the stability of the state of homogeneous particle suspension, in particular for gas-fluidized systems for which the condition that particle density is much greater than fluid density enables the particle-phase equations to be decoupled and treated independently. The analysis has involved solely the linearized forms of these equations, and has led to a stability criterion that broadly characterizes fluidized systems according to three manifestations of the fluidized state always stable - the usual case for liquids always unstable - the usual case for gases and transitional behaviour - involving a switch, at a critical fluid flux, from the stable to the unstable condition. This characterization has... 

Consider first point 8. Since this point does not meet the second stability criterion (Ineq. 12.8.2), it cannot represent a stable homogeneous phase. The system must move to points 2 or 10. (Or, it could break down into two phases represented by points 2 and 10. If these two phases, however, are to be in equilibrium, they must meet the criterion that was developed in Section 12.4 the chemical potential of the fluid must be the same in both phases. For this to occur, according to Problem 9.57, the two areas / and II - below and above the constant pressure line - must be equal. This pressure - not equal to Pq of course - represents the vapor pressure of the fluid at the specified temperature according to the EoS used.) This applies to all states described by the line segment from point 5 to point 9 and they are referred to as unstable. 

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