Low-Pressure VLE Calculations

BoiUng point temperature of one species at this pressure  

FIGURE 8.15 S ymmetry between region where one cannot use Raoult s law and must use Henry s law, and region where one cannot use the L-R rule with pure species fugacity, because the pure species 

Henry s law makes a strange standard state, because its f° = Hi does not correspond to the behavior of pure i but rather to the behavior of i in the solution as x,- — 0.00. This is one of the reasons it says in Chapter 7 that is sometimes chosen to correspond to the fugacity of pure i at this temperature and pressure and sometimes not. For dissolved oxygen in liquid water at 68°F there is no pure state of pure liquid oxygen, because oxygen cannot exist as a liquid at this temperature. The fact that Henry s law leads to this strange value off seems to have no practical consequences, and is pointed out here only to remind the reader that we make several choices for f°, some of them not very intuitive. (Some authors make much of this unimportant distinction, under the name unsymmetrical standard states.) 

We made several VLE calculations in Chapter 3, one in Chapter 7, and several in this chapter. Most of those were simple enough that we used shortcut, manual methods. In this section we consider the six standard types of VLE calculation, more formally than in previous sections. 

For the most low-pressure VLE (up to several hundred psia) we don t worry much about nonideality in the gas phase if we use the L-R rule we will have reasonable confidence in our estimates in the gas phase. All of the examples before Example 8.8 assumed that the gas phase was practically an ideal gas. However, those examples showed that the liquid phase was often quite nonideal. Our attempts to correlate and predict the VLE have mostly been attempts to correlate and predict liquid-phase activity coefficients. Many mixtures with widely different chemical structures and widely different vapor pressures can be represented reasonably well by Eq. 8.5 and fairly simple equations or prediction methods for liquid-phase activity coefficients. The next chapter shows how that is done. Before we begin that, we will borrow a result from that chapter, and show how the results are used. 

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The general low-pressure VLE calculation procedure for mixtures with more than two species is the same as for two species we attempt to find equations for the fugacities in gas and liquid phases that present/] as a function of T, P, and the mol fractions in that phase. As the number of species goes up, the mathematical complexity increases and the intuitive... 

Most low-pressure VLE calculations use the Raoult s law type of formulation, which uses the pure-species vapor pressures. A widely used alternative approach that works much better at high pressmes computes both liquid- and gas-phase fugacities from the same EOS, most often using the SRK or PR EOSs, discussed in Chapter 10 and Appendix F. 

For low pressure VLE calculations involving a completely miscible system, give the range of typical % errors resulting from the assumption of ideality in the ... 

These data can be reasonably well correlated by an equation of the form GB/RT = Bx,x2. Making the usual assumptions for low-pressure VLE, determine a suitable value for B and calculate values of the deviations dy, and SP between values calculated from the correlation and experimental values, basing the correlation on ... 

We could of course calculate f values by Eq. (11.67) for conditions of low-pressure VLE and combine them with experimental values of P, Tf Xj, and y, for the evaluation of activity coefficients by Eq. (11.66). However, at. low pressures (up to at least 1 bar), vapor phases usually approximate ideal gases, for which < , = 7 = 1, and the Poynting factor (represented by the exponential) differs from unity by only a few parts per thousand. Moreover, values of 4>, and tf>f differ significantly less from each other than from unity, and their influence in Eq. (11.67) tends to cancel. Thus the assumption that = 1 introduces little error for low-pressure VLE, and it reduces Eq. (11.66) to... 

This simple equation is adequate to our present purpose, allowing easy calculation of activity coefficients from experimental low-pressure VLE data. For comparison, when a system obeys Raoult s law, y(P = P at, and yf = 1. 

In Sec. 10.5 we treated dew- and bubble-point calculations for multicomponent systems that obey Raoult s law [Eq. (10.16)], an equation valid for low-pressure VLE when an ideal-liquid solution is in equilibrium with an ideal gas. Calculations for the general case are carried out in exactly the same way as for Raoult s law,... 

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 ... 

Thus it is apparent that real systems exhibit a diversity of LLE behavior. The thermodynamic basis for calculation or correlation of LLE is an expression for G /RT, from which activity coefficients are derived. The suitability of a particular expression is determined by its ability to accommodate the various features illustrated by Fig. 14.12. This is a severe test, because, unlike their role in low-pressure VLE where they represent corrections to Raoult s law, the activity coefficients here are the only thermodynamic contribution to an LLE calculation. 

Once the interaction energies were obtained, they were used to calculate the parameters in the UNIQUAC and Wilson models given by Eq. (24). To test the validity of the method, low-pressure vapor-liquid equilibrium (VLE) predictions were made for several binary aqueous systems. The calculations were done using the usual method assuming an ideal vapor phase (Sandler, 1999). Figures 7 and 8 show the low-pressure VLE diagrams for the binary aqueous mixtures of ethanol and acetone [see Sum and Sandler (1999a,b) for results for additional systems and values of the... 

It should be emphasized that this expression is only valid for gases at low pressure (below several atmospheres) and provided that the compounds do not associate. Hydrogen fluoride and acetic acid are two examples of species that associate in the vapor phase and for which eqn. (2.3.5) would not be correct, even at low pressure. To calculate VLE at low pressures with the y-(j> approach, we need to solve the equation... 

At low pressures, such calculations can be performed for ideal gases or as ideal mixture for real gases without Poynting correction. Equation (22.54) was used for all VLE calculations with the UNIFAC method for 9-17 model species (see mixtures A-C in Table 22.12) ... 

In most low-pressure VLE hand calculations, we assume that the vapor phase is an ideal solution of ideal gases. Most computer VLE programs include vapor-phase nonideahty in their calculations. One easily understood approach uses the two-term, pressure-explicit form of the virial EOS, for which molecular theory provides the basis for estimating the values of the constants for various molecular interactions. For low pressures the result is practically the same as the L-R rule, because the computed deviations from ideal solution behavior of the mixture are much smaller than the calculated deviations from ideal gas behavior of the pme species. 

Chapters 8 and 9 showed the common observations and calculation methods for low-pressure VLE. However, the observations at high pressures are different, and those calculation methods run into trouble at high pressures. Figure 10.1 shows why. 

In low-pressure VLE (see (Chapters 8 and 9) we normally begin with experimental data, calculate liquid-phase activity coefficients, use those to estimate the appropriate constants in a suitable liquid-phase activity coefficient equation, and then use that plus a suitable estimate of the vapor-phase nonideality (often the ideal gas law or the L-R rule for low-pressure VLE) to calculate equihbrium phase concentrations. In LLE we most often begin with some kind of liquid-phase activity coefficient equation, use it to calculate the composition of the equilibrium phases (without going through the intermediate step of calculating activity coefficients), and then compare the predicted to the experimental equilibrium concentrations, adjusting our equations as needed to get agreement. Then we use the equation to estimate other data points, the values at other temperatures, and so on. 

The presence, thus, of nonideality in both phases characterizes high pressure vapor-liquid equilibrium as compared to that at low pressure, where the main source of nonideality is in the liquid phase. And as a result, high pressure VLE calculations can be more complex than those at low pressures. 

High pressure VLE calculations are carried out using K values obtained with the aforementioned methods, in a fashion similar to that used for low pressure ones. 

The result of a calculation can be quite sensitive to the values for the k. Although these quantities can be correlated at times against combinations of properties for pure species i and / (e.g., critical-volume ratios), they are best treated as purely empirical parameters, values of which are (ideally) backed out of good experimental mixture data for the type of property which is to be represented. Thus, if accurate calculation of low-to-moderate-pressure volumetric properties is required, then the kif could be estimated from available data on mixture second virial coefficients for the constituent binaries. Alternatively, if application to multicomponent VLE calculations is envisioned, then the ki would be best estimated from available VLE data on the constituent binaries. (It... 

Setting Yi = 1 (ideal solution) this reverts to Raoult s law. We may view then the activity coefficient as a correction to Raoult s law that accounts for deviations from ideal-solution behavior. Equation (12.24) is the basis of VLE calculations, provided that pressure is sufficiently low. It is also used to extract activity coefficients from experimental data, as we will see with examples below. 

Equations (1.5-12)-(1.5-15) together constitute the most common formulation for predicting or correlating subcritical VLE at low to moderate pressures. When using the formulation for VLE predictions, one requites data or correlations for pure-component vapor pressures (e.g., Antoine equations), for the activity coefRcients (e.g., the UNIQUAC equation or the UNIFAC correlation), for the second virial coefficients (e.g., one of the correlations referent in Section 1.3-2), and for the molar volumes of the saturated liquid (e.g., the Rackett equation - for v ). The actual VLE calculations are iterative and require the use of a computer, details are given in the monograph by Ptausnitz et al. ... 

The final type of hybridization is the use of different models for different unit operations. Although this appears to be inconsistent at first, it is reality that thermodynamic models are not perfect and that some work much better for LLE than for VLE, some work better for low pressures and others for high pressures, and some work for hydrocarbons but not for aqueous phases. Furthermore, simulators perform calculations for individual units and then pass only component flowrates, temperature, and pressure to the next unit. Thus, consistency is not a problem. Therefore, one should always consider the possibility of using different models for different unit operations. All the simulators allow this, and it is essential for a complex flowsheet. An activity-coefficient model can be used for the liquid-liquid extractor and an equation of state for the flash unit. This hybridization can be extremely important when, for exanple, some units contain mainly complex organics and other units contain light hydrocarbons and nitrogen. 

Fiend s Constant. Henry s law for dilute concentrations of contaminants ia water is often appropriate for modeling vapor—Hquid equiHbrium (VLE) behavior (47). At very low concentrations, a chemical s Henry s constant is equal to the product of its activity coefficient and vapor pressure (3,10,48). Activity coefficient models can provide estimated values of infinite dilution activity coefficients for calculating Henry s constants as a function of temperature (35—39,49). 

In most industrial processes coexisting phases are vapor and liquid, although liquid/liquid, vapor/solid, and liquid/solid systems are also encountered. In this chapter we present a general qualitative discussion of vapor/liquid phase behavior (Sec. 12.3) and describe the calculation of temperatures, pressures, and phase compositions for systems in vapor/liquid equilibrium (VLE) at low to moderate pressures (Sec. 12.4).t Comprehensive expositions are given of dew-point, bubble-point, and P, T-flash calculations. 

Gas solubilities can be calculated using Henry coefficients, //,j In contrast to VLE, the value of the activity coefficient y approaches unity, when x, becomes zero. For small solubilities and low gas pressure, the activity coefficient and the fugacity coefficient can be neglected and Equation (7) can be simplified ... 

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