Equations vapor-liquid equilibrium calculation

These are general equations that do not depend on the particular mixing rules adopted for the composition dependence of a and b. The mixing rules given by Eqs. (4-221) and (4-222) can certainly be employed with these equations. However, for purposes of vapor/liquid equilibrium calculations, a special pair of mixing rules is far more appropriate, and will be introduced when these calculations are treated. Solution of Eq. (4-232) for fugacity coefficient at given T and P reqmres prior solution of Eq. (4-231) for V, from which is found Z = PV/RT. 

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... 

The vapor-liquid x-y diagram in Figures 4.6c and d can be calculated by setting a liquid composition and calculating the corresponding vapor composition in a bubble point calculation. Alternatively, vapor composition can be set and the liquid composition determined by a dew point calculation. If the mixture forms two-liquid phases, the vapor-liquid equilibrium calculation predicts a maximum in the x-y diagram, as shown in Figures 4.6c and d. Note that such a maximum cannot appear with the Wilson equation. 

Equation 12.22 relates the compositions of vapor and liquid streams passing each other in the stripping section of a column. Equation 12.22 can be used together with vapor-liquid equilibrium calculations to calculate a composition profile in the stripping section of the column, similar to that of the rectifying section of the column as described above. The calculation is started with an assumed bottoms composition and Equation 12.22 applied repeatedly with vapor-liquid equilibrium calculations working up the column. 

Either Equation 12.28 or Equation 12.32 can be used in conjunction with vapor-liquid equilibrium calculations to calculate the section profile for the middle section of the two-feed column. 

Flash Calculations. The ability to carry out vapor-liquid equilibrium calculations under various specifications (constant temperature, pressure constant enthalpy, pressure etc.) has long been recognized as one of the most important capabilities of a simulation system. Boston and Britt ( 6) reformulated the independent variables in the basic flash equations to make them weakly coupled. The authors claim their method works well for both wide and narrow boiling mixtures, and this has a distinct advantage over traditional algorithms ( 7). 

Many modifications of the original Redlich/Kwohg equation that appear in the literature are intended for special-purpose applications. The SRJt equation, developed for vapor/liquid equilibrium calculations, is designed specifically to yield reasonable vapor pressures for pure fluids. Thus, there is no assurance that molar volumes calculated by the SRK equation are more accurate than values given by the original Redlich/Kwong equation. 

Physical Properties The on equilibrium-stage simulation are i and enthalpies these same properties are needed for nonequilibrium models as well. Enthalpies are required for the energy balance equations vapor-liquid equilibrium ratios are needed for the calculation of driving forces for mass and heat transfer. The need for mass- (and heat-) transfer coefficients means that nonequilibrium models are rather more demanding of physical property data than are equilibrium-stage models. These coefficients may depend on a number of other physical properties, as summarized in Table 13-12. 

The Soave/Redlich/Kwong and the Peng/Robinson equations were developed specifically for vapor/liquid equilibrium calculations (Sec. 14.2). 

The gas phase of the system will mainly consist of H20(g), NHj(g), and C02(g). In order to perform vapor-liquid equilibrium calculations with an activity coefficient model at pressures higher than ambient pressure, the activity coefficient model can be combined with an equation of state for the gases. Usually, there is no need for binary interaction parameters in the equation of state as gas phase fiigacities are only slightly dependent on these binary interaction parameters. For the equilibrium between ammonia in the liquid phase and ammonia in the vapor phase. Equation (13) gives ... 

Stryjek, R., and Vera, J. H., 1986b. PRSV2 A cubic equation of state for accurate vapor-liquid equilibrium calculations. Can. J. Chem. Eng. 64 820-826. 

Equilibrium compositions of liquid phases at equilibrium are calculated by equating the component fugacities, similar to vapor-liquid equilibrium calculations, described in more detail in Chapter 2. The activity coefficients may be calculated by equations presented in Section 1.3.3, in particular the UNIQUAC and NRTL equations. The composition dependence of these equations is developed to the point where the same equation with the same constants can predict activity coefficients over wide ranges of composition, thus allowing it to predict two immiscible liquid phases at equilibrium. 

The column is solved by computer simulation with vapor-liquid equilibrium calculations based on the van Laar liquid activity coefficient equation (Chapter 1). Initially the column is solved without a side draw to determine the composition profiles in the column. The column is solved at a reflux ratio of 1 and a bottoms rate of 800 kmol/h. The composition profiles with no side draw are shown in Figure 9.7. The trays are numbered from the top down, with the condenser as number one. A total of 20 trays are shown, which include the condenser and the reboiler. 

For vapor-liquid equilibrium calculations up to moderate pressures, the B equation is suitable and convenient for the vapor phase for its applicability and simple form. Formulas have been derived from statistical theory for the calculation of virial coefficients, including B, from intermo-lecular potential energy functions, but intermolecular energy functions are hardly known quantitatively for real molecules. B is found for practical calculations by correlating experimental B values. Pitzer [1] correlated B of normal flnids in a generalized form with acentric factor to as the third parameter. 

A common logarithm is used in this equation. Both Equation (4.456) and Equation (4.458) are useful for making an initial estimate of either T or p in iterative computer calculations for vapor-liquid equilibrium calculations. 

While the same eos-derived ([) equation is used for both ( ),l and ( ) the input variables to the equation are different for (() l, the liquid mol fractions X and liquid molar volume are the input for (j) the gas mol fractions and gas molar volume Vy are the input. Find the liquid volume Vl by solving the eos at specified [X p,] and pick the small root find the vapor volume Vy by solving the eos at specified, [y p,] and pick the large root. The vapor-liquid equilibrium calculations using K values are much the same as in the use of y-cj) K values. 

Note the difference between this method of calculation and the one used in the previous illustration. There we did vapor-liquid equilibrium calculations only for the conditions needed, and then solved the mass balance equations analytically. In this illustration we first had to do vapor-liquid equilibrium calculations for all compositions (to construct the. t- v diagram), and then for this binary mixture we were able to do all further calculations graphically. As shown in the following discussion, this makes it easier to consider other reflux ratios than the one u.sed in this illustration. 

In the study of the solubility of a gas in a liquid one is interested in the equilibrium when the mixture temperature T is greater than the critical temperature of at least one of the components in the mixture, the gas. If the mixture can be described by an equation of state, no special difficulties are involved, and the calculations proceed as described in Sec. 10.3. Indeed, a number of cases encountered in Sec. 10.3 were of this type (e.g., ethane in the ethane-propylene mixture at 344.3 K). Consequently, it is not necessary to consider the equation-of-state description of gas solubility, as it is another type of equation-of-state vapor-liquid equilibrium calculation, and the methods described in Sec. 10.3 can be used. 

In a significant departure from conventional practice, Chueh and Prausnitz (11,12) proposed that the critical constraints on the RK equation be relaxed, and that parameters b and c be treated as empirical constants, determined separately for the liquid phase and for the vapor phase of a given substance. The conventional RK expression for (T) was retained the application was to vapor-liquid equilibrium calculations, in which the vapor-phase version of the equation was used for computation of vapor-phase fugacity coefficients, but in which the liquid-phase version was used only for Poynting corrections. Thus, they proposed that... 

In adapting the RK equation for vapor-liquid equilibrium calculations, pure-component parameters are adjusted to match vapor and liquid fugacity along the vapor pressure locus. In the Soave modification only the RK parameter a is temperature dependent, while for the JZ modification both a and b are temperature dependent. 

Since ethylene is above its critical temperature, its liquid-phase vapor-pressure will have to be estimated if we are to do the vapor-liquid equilibrium calculation. However, since we need only a moderate extrapolation (from 7 = 9.2°C to T = 50°C), we will do an extrapolation of the vapor-pressure data, and not use Shair s correlation. Using vapor-pressure equation in the Handbook of Chemistry and Physics, NQfmA P P(50°C) 1223 bar for ethylene. 

Chemical and industrial engineering both provide a series of examples involving nonideal vapor-liquid equilibrium calculation. For example, the equations governing a flash drum separator with molar fractions Zi, Z2, , Zc, flow rate F, and enthalpy H are... 

In order to determine the solubility parameter (8j and molar volume VJ for solvent-oil mixtures at a given pressure and temperature conditions, vapor-liquid equilibrium calculations are performed using Peng-Robinson Equation of State. 

An improvement can be achieved with the volume translation concept introduced by Peneloux et al [55]. The idea is that the specific volume calculated by the equation of state is corrected by addition of a constant parameter c. The volume translation has no effect on the vapor-liquid equilibrium calculation, as both the liquid and the vapor volume are simultaneously translated by a constant value. The procedure has also little effect on the calculated vapor volumes, as c is in the order of magnitude of a liquid volume far away from the critical temperature. 

The development of the equation was targeted primarily at improving the accuracy of vapor-liquid equilibrium calculations. The underlying reasoning in developing the equations was that a necessary condition for an equation of state to predict mixture vapor-liquid equilibrium properties was that it accurately predict pure component vapor-liquid equilibrium properties, namely pure component... 

Flash Units. In simulators, the term flash refers to the module that performs a single-stage vapor-liquid equilibrium calculation. Material, energy, and phase equilibrium equations are solved for a variety of input parameter specifications. In order to specify completely the condition of the two output streams (liquid and vapor), two parameters must be input. Many combinations are possible—for exanple, temperature and pressure, temperature and heat load, or pressure and mole ratio of vapor to liquid in exit streams. Often, the flash module is a combination of two pieces of physical equipment, that is, a phase separator and a heat exchanger. These should appear as separate equipment on the PFD. Note that a flash unit can also be specified for batch operation, in which case the unit can serve as a surge or storage vessel. 

Volumetric Properties From an Equation of State. In general, most equations of state give relatively good predictions of volumetric properties at high temperature and low pressure. However, near the saturation envelope and especially in the critical region, volumetric predictions based on equations of state are poor, particularly for the saturated liquid. Therefore, with the exception of vapor-liquid equilibrium calculations, where internally consistent liquid densities are needed to calculate the liquid fugaclty, empirical liquid density correlations are normally used in industrial design calculations. 

This paper has dealt with the characteristics of equations of state required by industry, has discussed a number of equations that are used in Industrial vapor-liquid equilibrium calculations, and has covered a number of everyday and sometimes unusual practical applications of equations of state. In all three areas an attempt was made to analyze the shortcomings, deficiencies, and handicaps of specific equations of state as well as equations of state in general, from an industrial viewpoint. It is hoped that some of the material discussed in this paper will prove advantageous in future equation of state development work. 

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