Phase equilibrium vapor-liquid equilibrium

Distillation columns can be used to separate chemical components when there are differences in the concentrations of these components in the liquid and vapor phases. These concentration differences are analyzed and quantified using basic thermodynamic principles covering phase equilibrium. Vapor-liquid equilibrium (VLE) data and analysis are vital components of distillation design and operation. 

Since the vast majority of chemical engineering systems involve liquid and vapor phases, many vapor-liquid equilibrium relationships are used. They range from the very simple to the very complex. Some of the most commonly used relationships are listed below. More detailed treatments are presented in many thermodynamics texts. Some of the basic concepts are introduced by Luyben aM... 

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. 

A system consisting of n-butane and propane exists as two phases in vapor/liquid equilibrium at lObar and 323 K. The mole fraction of propane is about 0.67 in the vapor phase and about 0.40 in the liquid phase. Additional pure propane is added to the system, which is brought again to equilibrium at the same temperature and pressure, with both liquid and vapor phases still present. What is the effect of the addition of propane on the mole fractions of propane in the vapor and liquid phases ... 

In Sec. 11.3 we showed that phases at the same T and P are in equilibrium when the fugacity of each species is the same in all phases. For vapor/liquid equilibrium, this requirement is written... 

The separation in a distillation process is governed by a difference in the composition of a liquid and vapor phase. This difference is usually characterized by a difference in actual vapor pressures, or volatilities, of the liquid-phase components. Vapor-liquid equilibrium data for the mixture components are, therefore, an important element for design and... 

Obtain a Vapor Phase for Vapor-Liquid Equilibrium This situation arises quite frequently when high-boiling-point materials need to be distilled. An exanple is the distillation of crude oil in which the bottom of the atmospheric column is typically operated in the region of 310°C to 340°C (590°F to 645°F). 

Properties of multicomponent systems includes solubility, activity coefficients, diffusion coefficients, phase diagrams, vapor-liquid equilibrium data on mixtures, etc. These are generally functions of temperature, pressure, and composition. 

Using UNIQUAC, Table 2 summarizes vapor-liquid equilibrium predictions for several representative ternary mixtures and one quaternary mixture. Agreement is good between calculated and experimental pressures (or temperatures) and vapor-phase compositions. ... 

Figure 15 shows results for a difficult type I system methanol-n-heptane-benzene. In this example, the two-phase region is extremely small. The dashed line (a) shows predictions using the original UNIQUAC equation with q = q. This form of the UNIQUAC equation does not adequately fit the binary vapor-liquid equilibrium data for the methanol-benzene system and therefore the ternary predictions are grossly in error. The ternary prediction is much improved with the modified UNIQUAC equation (b) since this equation fits the methanol-benzene system much better. Further improvement (c) is obtained when a few ternary data are used to fix the binary parameters. 

Unfortunately, many commonly used methods for parameter estimation give only estimates for the parameters and no measures of their uncertainty. This is usually accomplished by calculation of the dependent variable at each experimental point, summation of the squared differences between the calculated and measured values, and adjustment of parameters to minimize this sum. Such methods routinely ignore errors in the measured independent variables. For example, in vapor-liquid equilibrium data reduction, errors in the liquid-phase mole fraction and temperature measurements are often assumed to be absent. The total pressure is calculated as a function of the estimated parameters, the measured temperature, and the measured liquid-phase mole fraction. 

VPLQFT is a computer program for correlating binary vapor-liquid equilibrium (VLE) data at low to moderate pressures. For such binary mixtures, the truncated virial equation of state is used to correct for vapor-phase nonidealities, except for mixtures containing organic acids where the "chemical" theory is used. The Hayden-0 Connell (1975) correlation gives either the second virial coefficients or the dimerization equilibrium constants, as required. 

Vapor/liquid equilibrium (XT E) relationships (as well as other interphase equihbrium relationships) are needed in the solution of many engineering problems. The required data can be found by experiment, but such measurements are seldom easy, even for binaiy systems, and they become rapidly more difficult as the number of constituent species increases. This is the incentive for application of thermodynamics to the calculation of phase-equilibrium relationships. 

This equation may be applied separately to the liquid phase and to the vapor phase to yield the pure-species values ( ) and ( ) For vapor/ liquid equilibrium (Eq. [4-280]), these two quantities are equal. Given parameters Oj and bj, the pressure P in Eq. (4-230) that makes these two values equal is the equihbrium vapor pressure of pure species i as predicted by the equation of state. 

Data on the gas-liquid or vapor-liquid equilibrium for the system at hand. If absorption, stripping, and distillation operations are considered equilibrium-limited processes, which is the usual approach, these data are critical for determining the maximum possible separation. In some cases, the operations are are considerea rate-based (see Sec. 13) but require knowledge of eqmlibrium at the phase interface. Other data required include physical properties such as viscosity and density and thermodynamic properties such as enthalpy. Section 2 deals with sources of such data. 

Note that this equation holds for any component in a multi-component mixture. The integral on the right-hand side can only be evaluated if the vapor mole fraction y is known as a function of the mole fraction Xr in the still. Assuming phase equilibrium between liquid and vapor in the still, the vapor mole fraction y x ) is defined by the equilibrium curve. Agitation of the liquid in tire still and low boilup rates tend to improve the validity of this assumption. 

K-factors for vapor-liquid equilibrium ratios are usually associated with various hydrocarbons and some common impurities as nitrogen, carbon dioxide, and hydrogen sulfide [48]. The K-factor is the equilibrium ratio of the mole fraction of a component in the vapor phase divided by the mole fraction of the same component in the liquid phase. K is generally considered a function of the mixture composition in which a specific component occurs, plus the temperature and pressure of the system at equilibrium. 

Liquid and vapor are in equilibrium when the pressure of the vapor phase is the vapor pressure. When the (vapor + liquid) equilibrium mixture is exposed to the atmosphere, the mixture will boil at a temperature where the vapor pressure equals the external (atmospheric) pressure. This temperature is known as the boiling temperature. At the normal boiling temperature, the substance has a vapor pressure of exactly one atmosphere (0.101325 MPa) and hence, boils at this external pressure. 

We are interested in comparing the effectiveness of the various equations of state in predicting the (p. V. T) properties. We will limit our comparisons to Tr > 1 since for Tr < 1 condensations to the liquid phase occur. Prediction of (vapor + liquid) equilibrium would be of interest, but these predictions present serious problems, since in some instances the equations of state do not converge for Tr< 1. 

Both liquid and vapor phases are totally miscible. Conventional vapor/liquid equilibrium. Neither phase is pure. Separation factors are moderate and decrease as purity increases. Ultrahigh purity is difficult to achieve. No theoretical limit on recovery. Liquid phases are totally miscible solid phases are not. Eutectic system. Solid phase is pure, except at eutectic point. Partition coefficients are very high (theoretically, they can be infinite). Ultrahigh purity is easy to achieve. Recovery is limited by eutectic composition. 

Traditionally, the binary interaction parameters such as the ka, kb, k, ki in the Trebble-Bishnoi EoS have been estimated from the regression of binary vapor-liquid equilibrium (VLE) data. It is assumed that a set of N experiments have been performed and that at each of these experiments, four state variables were measured. These variables are the temperature (T), pressure (P), liquid (x) and vapor (y) phase mole fractions of one of the components. The measurements of these variables are related to the "true" but unknown values of the state variables by the equations given next... 

Figure 14.1 Vapor-liquid equilibrium data and calcidated values for the n-pentane-acetone system, x andy are the mole fractions in the liquid and vapor phase respectively [reproduced with permission from Canadian Journal of Chemical Engineering].

Experimental values for the activity coefficients for components 1 and 2 are obtained from the vapor-liquid equilibrium data. During an experiment, the following information is obtained Pressure (P), temperature (T), liquid phase mole fraction (x, and x2=l-X ) and vapor phase mole fraction (yi and y2=l—yi). 

Galivel - Solastiuk F., S. Laugier and D. Richon, "Vapor-Liquid Equilibrium Data for the Propane-MethanoI-C02 System", Fluid Phase Equilibria, 28, 73-85 (1986). 

Hong, J.H. and Kobayashi, R., "vapor-Liquid Equilibrium Studies for the carbon Dioxide-Methanol System", Fluid Phase Equilibria. 41,269-276 (1988). 

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). 

This expression provides the basis for vapor-liquid equilibrium calculations on the basis of liquid-phase activity coefficient models. In Equation 4.27, thermodynamic models are required for cf>y (from an equation of state) and y, from a liquid-phase activity coefficient model. Some examples will be given later. At moderate pressures, the vapor phase becomes ideal, as discussed previously, and fj = 1. For... 

Before an equation of state can be applied to calculate vapor-liquid equilibrium, the fugacity coefficient < />, for each phase needs to be determined. The relationship between the fugacity coefficient and the volumetric properties can be written as ... 

In the case of vapor-liquid equilibrium, the vapor and liquid fugacities are equal for all components at the same temperature and pressure, but how can this solution be found In any phase equilibrium calculation, some of the conditions will be fixed. For example, the temperature, pressure and overall composition might be fixed. The task is to find values for the unknown conditions that satisfy the equilibrium relationships. However, this cannot be achieved directly. First, values of the unknown variables must be guessed and checked to see if the equilibrium relationships are satisfied. If not, then the estimates must be modified in the light of the discrepancy in the equilibrium, and iteration continued until the estimates of the unknown variables satisfy the requirements of equilibrium. 

If the K-value requires the composition of both phases to be known, then this introduces additional complications into the calculations. For example, suppose a bubble-point calculation is to be performed on a liquid of known composition using an equation of state for the vapor-liquid equilibrium. To start the calculation, a temperature is assumed. Then, calculation of K-values requires knowledge of the vapor composition to calculate the vapor-phase fugacity coefficient, and that of the liquid composition to calculate the liquid-phase fugacity coefficient. While the liquid composition is known, the vapor composition is unknown and an initial estimate is required for the calculation to proceed. Once the K-value has been estimated from an initial estimate of the vapor composition, the composition of the vapor can be reestimated, and so on. 

Thus, by knowing aAB from vapor-liquid equilibrium and by specifying xA, A can be calculated. Figure 4.3a also shows a typical vapor-liquid equilibrium pair, where the mole fraction of benzene in the liquid phase is 0.4 and that in the vapor phase is 0.62. A diagonal line across the x-y diagram represents equal vapor and liquid compositions. The phase equilibrium behavior shows a curve above the diagonal line. This indicates that benzene has a higher concentration in the vapor phase than toluene, that is,... 

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