Constrained Least Squares Estimation

It is well known that cubic equations of state may predict erroneous binary vapor liquid equilibria when using interaction parameter estimates from an unconstrained regression of binary VLE data (Schwartzentruber et al.. 1987 Englezos et al. 1989). In other words, the liquid phase stability criterion is violated. Modell and Reid (1983) discuss extensively the phase stability criteria. A general method to alleviate the problem is to perform the least squares estimation subject to satisfying the liquid phase stability criterion. In other 

Given a set of N binary VLE (T-P-x-y) data and an EoS, an efficient method to estimate the EoS interaction parameters subject to the liquid phase stability criterion is accomplished by solving the following problem 

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Let us consider constrained least squares estimation of unknown parameters in algebraic equation models first. The problem can be formulated as follows ... 

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. 

Englezos, P. and N. Kalogerakis, "Constrained Least Squares Estimation of Binary Interaction Parameters in Equations of State", Computers Chem. Eng.. 17. 117-121 (1993). 

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. 

If incorrect phase behavior is predicted by the EOS then constrained least squares (CLS) estimation should be performed and new parameter estimates be obtained. Subsequently, the phase behavior should be computed again and if the fit is found to be acceptable for the intended applications, then the CLS estimates should suffice. This was found to be the case for the carbon dioxide-n-hexane system presented later in this chapter. 

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]...

The adjustment of measurements to compensate for random errors involves the resolution of a constrained minimization problem, usually one of constrained least squares. Balance equations are included in the constraints these may be linear but are generally nonlinear. The objective function is usually quadratic with respect to the adjustment of measurements, and it has the covariance matrix of measurements errors as weights. Thus, this matrix is essential in the obtaining of reliable process knowledge. Some efforts have been made to estimate it from measurements (Almasy and Mah, 1984 Darouach et al., 1989 Keller et al., 1992 Chen et al., 1997). The difficulty in the estimation of this matrix is associated with the analysis of the serial and cross correlation of the data. 

The data reconciliation problem can be generally stated as the following constrained weighted least-squares estimation problem ... 

Using the Q-R orthogonal factorization method described in Chapter 4, the constrained weighted least-squares estimation problem (5.4) is transformed into an unconstrained one. The following steps are required ... 

It is possible to arrive at a better estimate for the known components using a constrained least squares approach. In the case where the data can be modeled by Equation 1, and only one component is a known, the following relationship is certainly valid ... 

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... 

Apart from the original method mentioned above, Morrison and eo-workers [143,144] formulated a new iterative teehnique ealled CAEDMON (Computed Adsorption Energy Distribution in the Monolayer) for the evaluation of the energy distribution from adsorption data without any a priori assumption about the shape of this function. In this case, the local adsorption is calculated numerically from the two-dimensional virial equation. The problem is to find a discrete distribution function that gives the best agreement between the experimental data and calculated isotherms. In this order, the optimization procedure devised for the solution of non-negative constrained least-squares problems is used [145]. The CAEDMON algorithm was applied to evaluate x(fi) for several adsorption systems [137,140,146,147]. Wesson et al. [147] used this procedure to estimate the specific surface area of adsorbents. 

Besides, the Automeasure software includes a non-negatively constrained least squares technique, which estimates the intensity-weighed particle size distribution. 

Although satisfactory criteria for deciding whether data are better analyzed by distributions or multiexponential sums have yet to established, several methods for determining distributions have been developed. For pulse fluorometry, James and Ware(n) have introduced an exponential series method. Here, data are first analyzed as a sum of up to four exponential terms with variable lifetimes and preexponential weights. This analysis serves to establish estimates for the range of the preexponential and lifetime parameters used in the next step. Next, a probe function is developed with fixed lifetime values and equal preexponential factors. An iterative Marquardt(18) least-squares analysis is undertaken with the lifetimes remaining fixed and the preexponential constrained to remain positive. When the preexponential... 

In this model we are constraining the coefficient of log(T) to be 1 and log (rjB) to be -1. Again these are both linear models where the coefficients can be estimated using least squares. 

If the yield stress of a sample is known from an independent experiment, ATh and can be determined from linear regression of log a — ctoh versus log()>) as the intercept and slope, respectively. Alternatively, nonlinear regression technique was used to estimate ctoh> and h (Rao and Cooley, 1983). However, estimated values of yield stress and other rheological parameters should be used only when experimentally determined values are not available. In addition, unless values of the parameters are constrained a priori, nonlinear regression provides values that are the best in a least squares sense and may not reflect the true nature of the test sample. 

The online solution of this constrained estimation problem, known as full information estimator because we consider all the available measurements, is formulated as an optimization problem - typically posed as a least squares mathematical program-subject to the model constraints and inequality constraints that represents bounds on variables or equations. 

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