Vapor-liquid equilibrium phase rule

Schwartzentruber J., F. Galivel-Solastiuk and H. Renon, "Representation of the Vapor-Liquid Equilibrium of the Ternary System Carbon Dioxide-Propane-Methanol and its Binaries with a Cubic Equation of State. A new Mixing Rule", Fluid Phase Equilibria, 38,217-226 (1987). 

When applying an equation of state to both vapor and liquid phases, the vapor-liquid equilibrium predictions depend on the accuracy of the equation of state used and, for multicomponent systems, on the mixing rules. Attention will be given to binary mixtures of hydrocarbons and the technically important nonhydrocarbons such as hydrogen sulfide and carbon dioxide -Figures 6-7. 

Thus, one specifies either T or P and either the liquid-phase or the vapor-phase composition, fixing 1 + (N - 1) or N phase-rule variables, exactly the number required by the phase rule for vapor/liquid equilibrium. All of these calculations require iterative schemes because of the complex functionality implicit in Eqs. (12.1) and (12.2). In particular, we have the following functional relationships for low-pressure VLE ... 

The most commonly encountered coexisting phases in industrial practice are vapor and liquid, although liquid/liquid, vaporlsolid, and liquid/solid systems are also found. In this chapter we first discuss the nature of equilibrium, and then consider two rules that give the lumiber of independent variables required to detemiine equilibrium states. There follows in Sec. 10.3 a qualitative discussion of vapor/liquid phase behavior. In Sec. 10.4 we introduce tlie two simplest fomiulations that allow calculation of temperatures, pressures, and phase compositions for systems in vaporlliquid equilibrium. The first, known as Raoult s law, is valid only for systems at low to moderate pressures and in general only for systems comprised of chemically similar species. The second, known as Henry s law, is valid for any species present at low concentration, but as presented here is also limited to systems at low to moderate pressures. A modification of Raoult s law that removes the restriction to chemically similar species is treated in Sec. 10.5. Finally in Sec. 10.6 calculations based on equilibrium ratios or K-values are considered. The treatment of vapor/liquid equilibrium is developed further in Chaps. 12 and 14. 

Example 3 Detv and Bubble Point Calculations As indicated by Example 2a, a binary system in vapor/liquid equilibrium has 2 degrees of freedom. Thus of the four phase rule variables T, P, x, and t/i, two must be fixed to allow calculation of the other two, regardless of the formulation of the equilibrium equations. Modified Raoults law [Eq. (4-307)] may therefore be applied to the calculation of any pair of phase rule variables, given the other two. 

Boukouvalas, C., Spiliotis, N., Coutsikos, P., and Tzouvaras, N., 1994. Prediction of vapor-liquid equilibrium with the LCVM model. A linear combination of the Huron-Vidal and Michelsen mixing rules coupled with the original UNIFAC and the t-mPR equation of state. Fluid Phase Eq., 92 75-106. 

For a system consisting of C components, the phase rule indicates that, in the two-phase region, there are F=C-2 + 2 = C degrees of freedom. That is, it takes C independent variables to define the thermodynamic state of the system. The independent variables may be selected from a total of 2C intensive variables (i.e., variables that do not relate to the size of the system) that characterize the system the temperature, pressure, C - 1 vapor-component mole fractions, and C - 1 liquid-component mole fractions. The number of degrees of freedom is the number of intensive variables minus the number of equations that relate them to each other. These are the C vapor-liquid equilibrium relations, Yj = K,X, i=l,. .., C. The equilibrium distribution coefficients, AT, are themselves functions of the temperature, pressure, and vapor and liquid compositions. The number of degrees of freedom is, thus, 2C - C = C, which is the same as that determined by the phase rule. 

Keshtkar, a., Jalali, F. Moshfeghian, M. 1997. Evaluation of vapor-liquid equilibrium of CO2 binary systems using UNIQUAC-based Huron-Vidal mixing rules. Fluid Phase Equilibria, 140(1/2), 107-128. 

Calculations of vapor-liquid equilibrium involving a gas dissolved in a liquid are performed using the Lewis-Randall rule for the fugacity of the liquid phase and Henry s law for the vapor phase. Using the subscript s for the solvent, and i for the gas, the equilibrium criterion for the two components is. 

The vapor-liquid equilibrium data usually consist of temperature, pressure and composition of both phases. According to the phase rule, a binary system has two degrees of freedom. This means that having two of the above mentioned values the other two can be calculated. A data set containing temperatures, pressures and compositions of both phases is thermodynamically over determined. This extra information can be used to verify whether the measured values are eonsistent. The test is derived from fundamental thermodynamic properties. 

In a binary mixture, if two of the loop curves intersect, i.e., if the vapor curve of one crosses the liquid curve of the other, then the two compositions determine a vapor-liquid equilibrium point. This is due to the fact that, for a binary system of two phases, the phase rule allows two degrees of freedom. However, the value may not be unique, t.e., in the higher pressure region, particularly very near the critical, it is possible for a given vapor to have two possible equilibrium liquids of different compositions. These two conditions can be at the... 

The various rules developed for the vapor phase for miscible liquids apply equally well to this case. In the case of the liquid phase, if the liquids are completely immiscible, at equilibrium each component would exert its own vapor pressure independent of the others present. For this case, the vapor-liquid equilibrium expression will be... 

Could all three phases of water coexist over some finite range of temperatures Could the vapor-liquid equilibrium exist over a range of pressures at one temperature, instead of at just one pressure for any given temperature We have all been told, in previous courses, that the answer is no. But how would you prove that The answer is that we would use the phase rule, often called Gibbs phase rule after Josiah Willard Gibbs (1790-1861). 

The vapor-liquid equilibrium data usually consist of temperature, pressure and composition of both phases. According to the phase rule, a binary system has two degrees of freedom. This means that having two of the... 

Finally, because of its rigorous mixing rules, and the success of the empiric correlations for the estimation of Bjf and By, the virial equation finds extensive use in the estimation of vapor fugacities for low pressure vapor-liquid equilibrium calculations, especially in systems containing polar components. The liquid phase fugacities, in such cases, are calculated using the standard state fugacity approach that will be discussed in Section 11.10. 

Consider a vapor-liquid equilibrium mixture containing three components. According to the phase rule P = 3 + 2- 2 = 3, but according to Duhem s theorem, specification of two of these variables suffices to completely describe this system. Is there a paradox here ... 

The two degrees of freedom for this system may be satisfied by setting T and P, or T and t/j, or P and a-j, or Xi and i/i, and so on, at fixed values. Thus, for equilibrium at a particular T and P, this state (if possible at all) exists only at one liquid and one vapor composition. Once the two degrees of freedom are used up, no further specification is possible that would restrict the phase-rule variables. For example, one cannot m addition require that the system form an azeotrope (assuming this possible), for this requires Xi = i/i, an equation not taken into account in the derivation of the phase rule. Thus, the requirement that the system form an azeotrope imposes a special constraint and reduces the number of degrees of freedom to one. 

The term ff denotes the number of independent phase variables that should be specified in order to establish all of the intensive properties of each phase present. The phase variables refer to the intensive properties of the system such as temperature (T), pressure (P), composition of the mixture (e.g., mole fractions, x ), etc. As an example, consider the triple point of water at which all three phases—ice, liquid water, and water vapor—coexist in equilibrium. According to the phase rule,... 

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