Vapor-liquid equilibrium intensive variables

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. 

For each stream in vapor-liquid equilibrium, there are C + 2 degrees of freedom, where C is the number of chemical species. These degrees of freedom can be satisfied by specifying C speeies flow rates (or C - 1 species mole fractions and the total flow rate) and two intensive variables such as the temperature, pressure, vapor fraction, or enthalpy. For example, when specifying the species flow rates for a stream and its pressure and tempera-... 

In the case of binary vapor-liquid equilibrium for example, Eq. 12.7.1 indicates that knowledge of two intensive variables, say T andxi, suffices to determine P and yj, but not the extensive properties, such as the amounts of liquid and vapor. The condition for their determination is provided by Duhem s Theorem. 

The fundamental idea of this procedure is as follows For a system of two fluid phases containing N components, we are concerned with N — 1 independent mole fractions in each phase, as well as with two other intensive variables, temperature T and total pressure P. Let us suppose that the two phases (vapor and liquid) are at equilibrium, and that we are given the total pressure P and the mole fractions of the liquid phase x, x2,. .., xN. We wish to find the equilibrium temperature T and the mole fractions of the vapor phase yu y2,. .., yN-i- The total number of unknowns is N + 2 there are N — 1 unknown mole fractions, one unknown temperature, and two unknown densities corresponding to the two limits of integration in Eq. (6), one for the liquid phase and the other for the vapor phase. To solve for these N +2 unknowns, we require N + 2 equations of equilibrium. For each component i we have an equation of the form... 

Enthalpy-concentration charts are particularly useful for two-component systems in which vapor and liquid phases are in equilibrium. The Gibbs phase rule (Equation 6.2-1) specifies that such a system has (2 -I- 2 - 2) = 2 degrees of freedom. If as before we fix the system pressure, then specifying only one more intensive variable—the system temperature, or the mass or mole fraction of either component in either phase—fixes the values of all other intensive variables in both phases. An H-x diagram for the ammonia-water system at 1 atm is shown in Figure 8.5-2. 

The intensive property must be uniform throughout the phase. If a system at equilibrium consists of two nonreacting components such as NH3 and H2O, and two phases, liquid and vapor, intensive variables would be temperature (the same in both phases), pressure (the same in both phases), concentration in a phase (different in each phase), specific volume (different in each phase), and so on. The overall concentration (including both phases) for the system would not be an intensive property because it is not a value equal to the actual value of the concentration in either phase. 

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,... 

As mentioned earlier, the state of a pure homogeneous fluid is fixed whenever two intensive thermodynamic properties are set at definite values. However, for more complex systems this number is not necessarily two. For example, a mixture of steam and liquid water in equilibrium at 101.33 kPa can exist only at 100°C. It is impossible to change the temperature without also changing the pressure if vapor and liquid are to continue to exist in equilibrium one cannot exercise independent control over these two variables for this system. The number of... 

The general VLE problem involves a multicomponent system of N constituent species for which the independent variables are T,P,N -I liquid-phase mole fractions, and N - I vapor-phase mole fractions. (Note that = 1 and yi = 1, where Xi and yi represent liquid and vapor mole fractions respectively.) Thus there are 2N independent variables, and application of the phase rule shows that exactly N of these variables must be fixed to establish the intensive state of the system. This means that once N variables have been specified, the remaining N variables can be determined by simultaneous solution of the N equilibrium relations ... 

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