Estimation of Interaction Parameters

As we mentioned earlier, a volumetric EoS expresses the relationship among pressure, P, molar volume, v, temperature, T and composition z for a fluid mixture. This relationship for a pressure-explicit EoS is of the form 

Given an EoS, the objective of the parameter estimation problem is to compute optimal values for the interaction parameter vector, k, in a statistically correct and computationally efficient manner. Those values are expected to enhance the correlational ability of the EoS without compromising its ability to predict the correct phase behavior. 

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Englezos, P., N. Kalogerakis and P.R. Bishnoi, "A Systematic Approach for the Efficient Estimation of Interaction Parameters in Equations of State Using Binary VLE Data", Can. J. Chem. Eng., 71,322-326 (1993). 

Rodriguez, A. Canosa, J. Dominguez, A. Tojo, J. Vapour-hquid equilibria of dimethyl carbonate with hnear alcohols and estimation of interaction parameters for the UNIFAC and ASOG method Fluid Phase Equilib. 2002,201,187-201... 

Table 2. Solubility Parameters 6 for Estimation of Interaction Parameters x (<5, — 6j) ...

At the end of this section, we mention the estimation of interaction parameters between polymer and solvent, or (more generally) between two species A and B in a binary mixture. The simplest possibility is to use the standard Lorentz-Berthelot combining mles [210] ... 

Leo, A. J., Hoekman, D., Calculating log Poet with no missing fragments the problem of estimating new interaction parameters. Perspect. Drug Discov. Des. 2000, 18,19-38. 

The implicit LS, ML and Constrained LS (CLS) estimation methods are now used to synthesize a systematic approach for the parameter estimation problem when no prior knowledge regarding the adequacy of the thermodynamic model is available. Given the availability of methods to estimate the interaction parameters in equations of state there is a need to follow a systematic and computationally efficient approach to deal with all possible cases that could be encountered during the regression of binary VLE data. The following step by step systematic approach is proposed (Englezos et al. 1993)... 

The first task of the estimation procedure is to quickly and efficiently screen all possible sets of interaction parameters that could be used. For example if the Trebble-Bishnoi EoS were to be employed which can utilize up to four binary interaction parameters, the number of possible combinations that should be examined is 15. The implicit LS estimation procedure provides the most efficient means to determine the best set of interaction parameters. The best set is the one that results in the smallest value of the LS objective function after convergence of the minimization algorithm has been achieved. One should not readily accept a set that... 

The methane-methanol binary is another system where the EoS is also capable of matching the experimental data very well and hence, use of ML estimation to obtain the statistically best estimates of the parameters is justified. Data for this system are available from Hong et al. (1987). Using these data, the binary interaction parameters were estimated and together with their standard deviations are shown in Table 14.1. The values of the parameters not shown in the table (i.e., ka, kb, kc) are zero. 

Consider each type of data separately and estimate the best set of interaction parameters by Least Squares. 

If the estimated best set of interaction parameters is found to be different for each type of data then it is rather meaningless to correlate the entire database simultaneously. One may proceed, however, to find the parameter set that correlates the maximum number of data types. 

Prior work on the use of critical point data to estimate binary interaction parameters employed the minimization of a summation of squared differences between experimental and calculated critical temperature and/or pressure (Equation 14.39). During that minimization the EoS uses the current parameter estimates in order to compute the critical pressure and/or the critical temperature. However, the initial estimates are often away from the optimum and as a consequence, such iterative computations are difficult to converge and the overall computational requirements are significant. 

The above equation following from relations (2.3.1)-(2.3.4) reduces, with the values listed in Table 2.2, to known expressions for dispersion80 and repulsion78 interactions and proves helpful in the experiment-based estimation of the parameters entering into it. 

Empirical and semi-empiriad approaches. The problem of making dieoretical estimates for the interaction coefficients for the liquid phase has been tackled in different ways by various authors. Kaufman and Bernstein (1970) considered that the liquid state would exhibit the lowest repulsive forces of all the states of condensed matter and that a description of the interaction parameters for the liquid state would be the best basis for die prediction of interaction parameters for various solid phases. 

A useful compendium of formulas for estimating the interaction parameters e and cr in the Lennard-Jones or Stockmayer potentials has been presented by Svehla [389]. His work... 

The VDW interactions seem to have little effect on the rate of aggregation of small vesicles in a primary minimum. However, this statement may be made only because the magnitudes of Hamaker coefficients are less than 10 13 erg (5 X 10 14 erg), in contrast to much higher values frequently used in treatments in colloid science (3). Our estimates of VDW parameters for phospholipid vesicles are based on the analysis of a significant amount of recent data (33,42). 

The average separation distance and its root-mean-square fluctuation were recently experimentally determined as functions of the applied pressure, for lipid bilayers/water multilayers,10 and a disagreement between the existing theories and experiment was noted.12 No comparable experimental results are yet available for two lipid bilayers, and it is difficult to extend the present theory to multilayers. For this reason, we will employ the present procedure to estimate the interaction parameters of eq 10 from the experimental results for multilayers.10... 

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