Estimation of Vapor-Liquid Equilibrium

As expected, often not all the necessary for design purposes VLE data are avulable. Consider, for example, the case where vapor-liquid equilibrium information is needed for the design of a fractionation system to recover component A firom a mixture containing, in addition, components B and C. 

If experimental VLE data for the three binary systems A-B, B-C and C-A are available, they can be used - as we have seen - for the prediction of the ternary behavior. Reliable results should be expected, and the availability of compilations of a large number of binary data (Hala et al  

Hirata et al and especially the series of volumes by Gmehling and coworkers) makes this approach very attractive. 

In the typical case, however, some - or even all - of the necessary binary data may not be available. In such cases we resort to estimation techniques. 

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Estimation of Vapor Liquid Equilibrium of Binary Systems Tert-Butanol+2-Ethyl-1-HexanolandN-Butanol+2-Ethyl-1-Hexanol Using Artificial Neural Network... 

Estimation of Vapor Liquid Equilibrium of Binary Systems... 

Q Development of Artificial Neural Network (ANN) Model for Estimation of Vapor Liquid Equilibrium (VLE) Data... 

The most reliable estimates of the parameters are obtained from multiple measurements, usually a series of vapor-liquid equilibrium data (T, P, x and y). Because the number of data points exceeds the number of parameters to be estimated, the equilibrium equations are not exactly satisfied for all experimental measurements. Exact agreement between the model and experiment is not achieved due to random and systematic errors in the data and due to inadequacies of the model. The optimum parameters should, therefore, be found by satisfaction of some selected statistical criterion, as discussed in Chapter 6. However, regardless of statistical sophistication, there is no substitute for reliable experimental data. 

The computer subroutines for calculation of vapor-liquid equilibrium separations, including determination of bubble-point and dew-point temperatures and pressures, are described and listed in this Appendix. These are source routines written in American National Standard FORTRAN (FORTRAN IV), ANSI X3.9-1978, and, as such, should be compatible with most computer systems with FORTRAN IV compilers. Approximate storage requirements for these subroutines are given in Appendix J their execution times are strongly dependent on the separations being calculated but can be estimated (CDC 6400) from the times given for the thermodynamic subroutines they call (essentially all computation effort is in these thermodynamic subroutines). 

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. 

This would be an exact equality if both temperature and pressure were constant, but that would be inconsistent with the concept of vapor-liquid equilibrium.) In other words, the net area under the curve In (Yi /yi) versus x should be zero. This means that the area above and below the x axis must be equal. Since real data entail changes in system pressure and are subject to experimental errors and errors in estimating / " and , the preceding requirement cannot be expected to be exactly satisfied. However, the deviation should not be more than a few percent of the total absolute area. 

In using simulation software, it is important to keep in mind that the quality of the results is highly dependent upon the quahty of the liquid-liquid equilibrium (LLE) model programmed into the simulation. In most cases, an experimentally vmidated model will be needed because UNIFAC and other estimation methods are not sufficiently accurate. It also is important to recognize, as mentioned in earlier discussions, that binary interaction parameters determined by regression of vapor-liquid equilibrium (VLE) data cannot be rehed upon to accurately model the LLE behavior for the same system. On the other hand, a set of binary interaction parameters that model LLE behavior properly often will provide a reasonable VLE fit for the same system—because pure-component vapor pressures often dominate the calculation of VLE. 

For the analysis of distillation and other vapor-liquid separation processes one must estimate the compositions of the vapor and liquid in equilibrium. This topic is considered in detail in this chapter with particular reference to the preparation of mixture vapor-liquid equilibrium (VLE) phase diagrams, partial vaporization and condensation. calculations, and the use of vapor-liquid equilibrium ippasurements to,obtain infonnac-. 

To evaluate the potential use of carbon dioxide in tertiary oil recovery, it is necessary to estimate the vapor-liquid equilibrium between carbon dioxide and reservoir petroleum, which we will take to be n-hexane, at oil well conditions, typically 140 bar and 75°C. Mdke this estimate as best you can. 

To illustrate the criterion for parameter estimation, let 1, 2, and 3 represent the three components in a mixture. Components 1 and 2 are only partially miscible components 1 and 3, as well as components 2 and 3 are totally miscible. The two binary parameters for the 1-2 binary are determined from mutual-solubility data and remain fixed. Initial estimates of the four binary parameters for the two completely miscible binaries, 1-3 and 2-3, are determined from sets of binary vapor-liquid equilibrium (VLE) data. The final values of these parameters are then obtained by fitting both sets of binary vapor-liquid equilibrium data simultaneously with the limited ternary tie-line data. 

In Equation (24), a is the estimated standard deviation for each of the measured variables, i.e. pressure, temperature, and liquid-phase and vapor-phase compositions. The values assigned to a determine the relative weighting between the tieline data and the vapor-liquid equilibrium data this weighting determines how well the ternary system is represented. This weighting depends first, on the estimated accuracy of the ternary data, relative to that of the binary vapor-liquid data and second, on how remote the temperature of the binary data is from that of the ternary data and finally, on how important in a design the liquid-liquid equilibria are relative to the vapor-liquid equilibria. Typical values which we use in data reduction are Op = 1 mm Hg, = 0.05°C, = 0.001, and = 0.003... 

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. 

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 14.3 Vapor-liquid equilibrium data and calculated values for the carbon dioxide-n-hexane system. Calculations were done using interaction parameters from implicit and constrained least squares (LS) estimation, x and y are the mote fractions in the liquid and vapor phase respectively [reprinted from the Canadian Journal of Chemical Engineering with permission]...

Patino-Leal, H., and P.M. Reilly, "Statistical Estimation of Parameters in Vapor-Liquid Equilibrium", AIChEJ., 28(4), 580-587 (1982). 

These models are semiempirical and are based on the concept that intermolecular forces will cause nonrandom arrangement of molecules in the mixture. The models account for the arrangement of molecules of different sizes and the preferred orientation of molecules. In each case, the models are fitted to experimental binary vapor-liquid equilibrium data. This gives binary interaction parameters that can be used to predict multicomponent vapor-liquid equilibrium. In the case of the UNIQUAC equation, if experimentally determined vapor-liquid equilibrium data are not available, the Universal Quasi-chemical Functional Group Activity Coefficients (UNIFAC) method can be used to estimate UNIQUAC parameters from the molecular structures of the components in the mixture3. 

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. 

Air needs to be dissolved in water under pressure at 20°C for use in a dissolved-air flotation process (see Chapter 8). The vapor-liquid equilibrium between air and water can be predicted by Henry s Law with a constant of 6.7 x 104 bar. Estimate the mole fraction of air that can be dissolved at 20°C, at a pressure of 10 bar. 

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