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Brazilian Journal of Chemical Engineering

Print version ISSN 0104-6632On-line version ISSN 1678-4383

Braz. J. Chem. Eng. vol.17 n.3 São Paulo Sept. 2000 



A. L. H. Costa, F. P. T. da Silva and F. L. P. Pessoa
Department of Chemical Engineering, Federal University of Rio de Janeiro (UFRJ),
Escola de Química, Bloco E, Centro de Tecnologia, Cidade Universitária, CEP 21949-900,
Rio de Janeiro - RJ, Brazil, Phone: 21-590-3192,


(Received: February 10, 2000 ; Accepted: April 14, 2000)



Abstract - This paper presents the utilization of a stochastic global optimization method for the problem of parameter estimation in thermodynamic models. The method is based on an adaptation of the simulated annealing method to continuous variables. The proposed approach is applied to a vapor-solid equilibrium with supercritical carbon dioxide as solvent. Numerical results indicate that the simulated annealing method calculated a set of parameter values associated with considerably smaller errors as compared with a traditional method of local optimization.
Keywords: parameter estimation; global optimization; supercritical fluid.




The design, simulation and optimization of a process demand a mathematical model that must be able to predict the behavior of the system. An important component of chemical engineering models is the set of equations that describes the phase equilibrium phenomenon. In supercritical fluid extraction processes, the thermodynamic model is a fundamental aspect. The accuracy of the results depends on model parameters that must be estimated from experimental data.

A classical procedure employed for parameter estimation is the method of least squares. It consists in the minimization of the sum of the squared differences of the experimental values and the values predicted by the model. The resultant mathematical problem corresponds to a nonlinear programming problem (NLP). Conventional approaches to resolution of this kind of problem utilize local optimization methods. However, due to the mathematical structure of thermodynamic models, more than one optimum can occur in the problem of optimization of parameter estimation. Previous studies of the use of global optimization methods for parameter estimation can be found in Park and Froment (1998), Esposito and Floudas (1998) and Kleiber and Axmann (1998).

This work aims to investigate the utilization of a stochastic method of global optimization for parameter estimation in thermodynamic models to predict the behavior of supercritical systems. The stochastic method employed is based on the simulated annealing method adapted to a continuous space. The proposed parameter estimation approach is applied to a vapor-solid equilibrium problem involving supercritical carbon dioxide. The efficiency of the utilization of simulated annealing in parameter estimation is demonstrated by a comparison with Powell’s method.



The method of least squares determines the model parameters by solution of an optimization problem. Consider a model with a set of independent variables, x, and only one dependent variable, y. It is assumed that there is no error in the independent variables or that the error is much lower than the error in the dependent variable. Vector q represents the set of model parameters. In each experiment, k = 1,..., N, the measured values of the independent and dependent variables are represented by xk and , respectively. The model is represented by the function ycalc = ycalc (q, x). The method of least squares presents the following structure (Edgar and Himmelblau, 1988):


The usual approach to solving this problem is to use local optimization methods, e.g., Newton, gradient, Powell, etc. However, due to the mathematical form of the model, the objective function may have a nonconvex behavior. In this case, the optimization problem can present multiple local optima. Thus, utilization of a global optimization method can achieve better results.



Simulated annealing (SA) is a global stochastic optimization method that originated in the computational reproduction of the thermal process of annealing, where a material is heated and cooled slowly in order to reach a minimum energy state. In the SA method, starting from an initial configuration, a new configuration is generated randomly. If this new configuration has a smaller value of objective function (in a minimization context), then this new configuration will become the current configuration. Otherwise, a stochastic test is applied to indicate whether or not the new configuration will be accepted. This process of movement-acceptation is repeated, and as the number of analyzed alternatives increases, the acceptance probability of the worse configurations is gradually reduced. Due to the possibility of carrying out "wrong way" movements, the search can move from a local optimum toward the global optimum to avoid being trapped in a poor local solution.

The SA method is largely employed for the resolution of combinatorial problems, but it has also been adapted to optimization with continuous decision variables (Bohachevsky et al., 1986). This paper employs the continuous version of the SA method described in Platt et al. (1999), but without utilizing the concept of nonequilibrium (Cardoso et al., 1994).



Mixtures involving supercritical conditions originate complex phase equilibrium problems. In this context, utilization of the SA approach is suggested to estimate parameters in a thermodynamic model of a high-pressure system. In this paper, solid-vapor equilibrium will be addressed.

Thermodynamic Model

Consider the equilibrium of a pure solid (species 1) with a binary vapor mixture containing species 1 and the solvent (species 2), which is insoluble in the solid phase. The mole fraction of species 1 in the vapor phase, y1, is its solubility in the solvent.

As species 2 is not distributed between the two phases, the phase equilibrium equation for this system can be written as


With the fugacity definition, we can write, for the vapor phase,


where is the fugacity coefficient of species 1 in the solution. For the solid phase, which is pure species 1,


where is the solid/vapor saturation pressure at temperature T, is the molar volume of the solid and is the fugacity coefficient of species 1 at the saturated conditions. Combining Equations 3 and 4 and then solving for y1, yields the solubility of species 1 in the vapor phase, as follows:


The fugacity coefficient evaluated by the Peng-Robinson cubic equation of state (Peng and Robinson, 1976) with Van der Waals mixing rule (Smith et al., 1996) is given by


where z is the compressibility factor, a and b are equation of state parameters, , , , , and .

Subscript "m" represents the mixture, "i" and "j" represent a defined chemical species and the summations include all species present in the mixture. Parameter aij is given by the following combination rule:


and parameter bij is given by the expression


where Kij and lij are the binary interaction parameters with kij = kji, lij = lji, and kii = kjj = lii = ljj = 0. In this example, the dependent variable is solubility and the parameters to be estimated are the binary interaction parameters.

Numerical Results

The mixture analyzed is composed of naphtalene and carbon dioxide. The binary interaction parameters were estimated using the experimental data of solubility at 35 ° C presented in McHugh and Paulatis (1980). The physical properties of the substances were obtained in Daubert and Danner (1991).

Due to the stochastic nature of SA, ten runs were conducted to provide a more complete description of the method performance. The efficiency of the estimation procedure is measured by the average relative error of the model prediction in the set of experimental data ():


Starting each run with a configuration with kij = lij = 0, the values of the parameters obtained by estimation in the best run are presented in Table 1.



The value of associated with the best run and the average value of in the run set are shown in Table 2.



Table 2 indicates that the proposed parameter estimation approach provided a good fit with a small value of . It is important to note that the best value of and the average value of are close, i.e., despite the stochastic nature of SA, the difference in performance between distinct runs is small.

This problem of parameter estimation was also solved using Powell’s method (Press et al., 1989), a conventional local optimization method. Three different initial guesses were applied. Comparison with the SA results is shown in Table 3.



According to Table 3, utilization of the SA method can give better results when compared to Powell’s method. Each run using Powell’s method achieved a solution with different parameter values, thus suggesting the occurrence of several local optima in the parameter estimation problem.



This paper presents the utilization of the SA method, adapted to a continuous space for the problem of parameter estimation. Vapor-solid equilibrium with supercritical carbon dioxide as solvent was studied. The proposed method found parameter values associated with considerably smaller errors when compared to Powell’s method, a conventional local optimization method.



The authors are grateful to the Research Support Foundation of the State of Rio de Janeiro (FAPERJ) and the Brazilian National Council for Scientific and Technological Development (CNPq) for their financial support.



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