Vapor/liquid equilibrium relationship

Define equilibrium relationship between liquid and vapor mole fractions at operating pressure... 

Lapidus and coworkers (1961) have studied the dynamics and control of a six-plate absorber controlled by the inlet feed streams. In their work, they assume a linear equilibrium relationship between liquid xm) and vapor um) at each plate ... 

Ideal Vapor/Liquid Equilibrium Systems, Nonideal Vapor/Liquid Equilibrium Systems, Vapor/Liquid Equilibrium Relationships,... 

Figure 15-1 shows the equilibrium relationships between liquid and vapor compositions for butane-pentane mixtures at 100 psia. Upon heating a liquid that contains 70 per cent butane and 30 per cent pentane, no vapor will be formed until the liquid equilibrium line is reached at 163 F. At this point an infinitesimal quantity of vapor will be formed having the composition corresponding to 163 F on the vapor curve, i.e., 88 per cent of butane. If heating is continued, the composition of the liquid... 

The vapor-liquid equilibrium relationship can be deflned in terms of K values by... 

The vapor pressure of a crude oil at the wellhead can reach 20 bar. If it were necessary to store and transport it under these conditions, heavy walled equipment would be required. For that, the pressure is reduced (< 1 bar) by separating the high vapor pressure components using a series of pressure reductions (from one to four flash stages) in equipment called separators , which are in fact simple vessels that allow the separation of the two liquid and vapor phases formed downstream of the pressure reduction point. The different components distribute themselves in the two phases in accordance with equilibrium relationships. 

Enthalpy of Vaporization (or Sublimation) When the pressure of the vapor in equilibrium with a liquid reaches 1 atm, the liquid boils and is completely converted to vapor on absorption of the enthalpy of vaporization ISHv at the normal boiling point T. A rough empirical relationship between the normal boiling point and the enthalpy of vaporization (Trouton s rule) is ... 

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. 

The phase-distribution restrictions reflect the requirement that ff =ff at equilibrium where/is the fugacity. This may be expressed by Eq. (13-1). In vapor-hquid systems, it should always be recognized that all components appear in both phases to some extent and there will be such a restriction for each component in the system. In vapor-liquid-hquid systems, each component will have three such restrictions, but only two are independent. In general, when all components exist in all phases, the uumDer of restricting relationships due to the distribution phenomenon will be C(Np — 1), where Np is the number of phases present. 

Since the boiling point properties of the components in the mixture being separated are so critical to the distillation process, the vapor-liquid equilibrium (VLE) relationship is of importance. Specifically, it is the VLE data for a mixture which establishes the required height of a column for a desired degree of separation. Constant pressure VLE data is derived from boiling point diagrams, from which a VLE curve can be constructed like the one illustrated in Figure 9 for a binary mixture. The VLE plot shown expresses the bubble-point and the dew-point of a binary mixture at constant pressure. The curve is called the equilibrium line, and it describes the compositions of the liquid and vapor in equilibrium at a constant pressure condition. 

Multicomponent distillations are more complicated than binary systems due primarily to the actual or potential involvement or interaction of one or more components of the multicomponent system on other components of the mixture. These interactions may be in the form of vapor-liquid equilibriums such as azeotrope formation, or chemical reaction, etc., any of which may affect the activity relations, and hence deviations from ideal relationships. For example, some systems are known to have two azeotrope combinations in the distillation column. Sometimes these, one or all, can be broken or changed in the vapor pressure relationships by addition of a third chemical or hydrocarbon. 

Equation 4.26 defines the relationship between the vapor and liquid mole fractions and provides the basis for vapor-liquid equilibrium calculations on the basis of equations of state. Thermodynamic models are required for (/) and [ from an equation of state. Alternatively, Equations 4.21, 4.22 and 4.25 can be combined to give... 

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. 

The procedure developed by Joris and Kalitventzeff (1987) aims to classify the variables and measurements involved in any type of plant model. The system of equations that represents plant operation involves state variables (temperature, pressure, partial molar flowrates of components, extents of reactions), measurements, and link variables (those that relate certain measurements to state variables). This system is made up of material and energy balances, liquid-vapor equilibrium relationships, pressure equality equations, link equations, etc. 

If a liquid is placed into a sealed container, molecules will evaporate from the surface of the liquid and will eventually establish a gas phase over the liquid that is in equilibrium with the liquid phase. This is the vapor pressure of the liquid. This vapor pressure is temperature dependent, the higher the temperature the higher the vapor pressure. If a solution is prepared, then the solvent contribution to the vapor pressure of the solution depends upon the vapor pressure of the pure solvent, P°soivenb and the mole fraction of the solvent. We can find the contribution of solvent to the vapor pressure of the solution by the following relationship ... 

Finally, we must be certain we are observing vapor-liquid equilibrium in the column not vapor-solid equilibrium. Braun and Guillet (1976) reported that a discontinuity in a plot of Jin (V°) vs 1/T (where V° is the retention volume at 273.15 K) indicated a phase transition. We calculated V° from the following relationship... 

Example 2.7. To show what form the energy equation takes for a two-phase system, consider the CSTR process shown in Fig. 2.6. Both a liquid product stream f and a vapor product stream F (volumetric flow) are withdrawn from the vessel. The pressure in the reactor is P. Vapor and liquid volumes are and V. The density and temperature of the vapor phase are and L. The mole fraction of A in the vapor is y. If the phases are in thermal equilibrium, the vapor and liquid temperatures are equal (T = T ). If the phases are in phase equilihrium, the liquid and vapor compositions are related by Raoult s law, a relative volatility relationship or some other vapor-liquid equilibrium relationship (see Sec. 2.2.6). The enthalpy of the vapor phase H (Btu/lb or cal/g) is a function of composition y, temperature T , and pressure P. Neglecting kinetic-energy and potential-energy terms and the work term,... 

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... 

Dewpoint calculations must be made when we know the composition of the vapor yj and P (or T) and want to find the liquid composition Xj and T (or P). Flash calculations must be made when we know neither Xj nor yj and must combine phase equilibrium relationships, component balance equations, and an energy balance to solve for all the unknowns. 

We will assume a binary system (two components) with constant relative volatihty throughout the column and theoretical (100 percent efficient) trays, i.e., the vapor leaving the tray is in equilibrium with the liquid on the tray. This means the simple vapor-hquid equihbrium relationship can be used... 

An appropriate vapor-liquid equilibrium relationship, as discussed in Sec. 2.2.6, must be used to find y j. Then Eq. (3.96) ean be used to calculate the y j for the inefficient tray. The yj- i, y would be calculated Irom the two vapors entering the tray Fj( i and V . ... 

The digital simulation of a distillation column is fairly straightforward. The main complication is the large number of ODEs and algebraic equations that must be solved. We will illustrate the procedure first with the simplified binary distillation column for which we developed the equations in Chap. 3 (Sec. 3.11). Equimolal overflow, constant relative volatility, and theoretical plates have been assumed. There are two ODEs per tray (a total continuity equation and a light component continuity equation) and two algebraic equations per tray (a vapor-liquid phase equilibrium relationship and a liquid-hydraulic relationship). 

Suppose we apply these relationships to the equilibrium of a liquid mixture and its vapor. At equilibrium, fx, must have the same value for each component in both the liquid and vapor phases. Therefore... 

A method of prediction of the salt effect of vapor-liquid equilibrium relationships in the methanol-ethyl acetate-calcium chloride system at atmospheric pressure is described. From the determined solubilities it is assumed that methanol forms a preferential solvate of CaCl296CH OH. The preferential solvation number was calculated from the observed values of the salt effect in 14 systems, as a result of which the solvation number showed a linear relationship with respect to the concentration of solvent. With the use of the linear relation the salt effect can be determined from the solvation number of pure solvent and the vapor-liquid equilibrium relations obtained without adding a salt. 

Deviations from ideality often occur, and the Kt value depends not only on temperature and pressure but also on the composition of the other components of the mixture. A more detailed discussion of vapor-liquid equilibrium relationships for nonideal mixtures is outside the scope of this article. 

If a fluid composed of more than one component (e.g., a solution of ethanol and water, or a crude oil) partially or totally changes phase, the required heat is a combination of sensible and latent heat and must be calculated using more complex thermodynamic relationships, including vapor-liquid equilibrium calculations that reflect the changing compositions as well as mass fractions of the two phases. 

The basic equations describing a single stage in a fractionator in which chemical reaction may occur include component material balances, vapor-liquid equilibrium relationships, and energy balance, and restrictions on the liquid vapor phase mol fractions. The model equations for stage j may be expressed as follows ... 

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