Simplified Constrained Least Squares Estimation

Solution of the above constrained least squares problem requires the repeated computation of the equilibrium surface at each iteration of the parameter search. This can be avoided by using the equilibrium surface defined by the experimental VLE data points rather than the EoS computed ones in the calculation of the stability function. The above minimization problem can be further simplified by satisfying the constraint only at the given experimental data points (Englezos et al. 1989). In this case, the constraint (Equation 14.25) is replaced by 

In Equation 14.27, cT, oP and ax are the standard deviations of the measurements of T, P and x respectively. All the derivatives are evaluated at the point where the stability function cp has its lowest value. We call the minimization of Equation 14.24 subject to the above constraint simplified Constrained Least Squares (simplified CLS) estimation. 

2 A Potential Problem with Sparse or Not Well Distributed Data 

The problem with the above simplified procedure is that it may yield parameters that result in erroneous phase separation at conditions other than the given experimental ones. This problem arises when the given data are sparse or not well distributed. Therefore, we need a procedure that extends the region over which the stability constraint is satisfied. 

The objective here is to construct the equilibrium surface in the T-P-x space from a set of available experimental VLE data. In general, this can be accomplished by using a suitable three-dimensional interpolation method. However, if a sufficient number of well distributed data is not available, this interpolation should be avoided as it may misrepresent the real phase behavior of the system. 

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Copp and Everet (1953) have presented 33 experimental VLE data points at three temperatures. The diethylamine-water system demonstrates the problem that may arise when using the simplified constrained least squares estimation due to inadequate number of data. In such case there is a need to interpolate the data points and to perform the minimization subject to constraint of Equation 14.28 instead of Equation 14.26 (Englezos and Kalogerakis, 1993). First, unconstrained LS estimation was performed by using the objective function defined by Equation 14.23. The parameter values together with their standard deviations that were obtained are shown in Table 14.5. The covariances are also given in the table. The other parameter values are zero. 

This system illustrates the use of simplified constrained least squares (CLS) estimation. In Figure 14.3, the experimental data by Li et al. (1981) together with the calculated phase diagram for the system carbon dioxide-n-hexane are shown. The calculations were done by using the best set of interaction parameter values obtained by implicit LS estimation. These parameter values together with standard deviations are given in Table 14.3. The values of the other parameters (k, kj) were equal to zero. As seen from Figure... 

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