Nonlinear kinetic parameter estimation using simulated annealing A. Eftaxias a , J. Font a , A. Fortuny b , A. Fabregat a , F. Stu ¨ber a, * a Departament d’Enginyeria Quı ´mica, ETSEQ, Universitat Rovira i Virgili, Paisos Catalans 26, 43007 Tarragona, Catalunya, Spain b Departament d’Enginyeria Quı ´mica, EUPVG, Universitat Polite `cnica de Catalunya, Av. Vı ´ctor Balaguer, s/n, 08800 Vilanova i la Geltru ´, Barcelona, Catalunya, Spain Received 4 December 2001; received in revised form 5 July 2002; accepted 5 July 2002 Abstract The performance of simulated annealing (S-A) in nonlinear kinetic parameter estimation was studied and compared with the classical Levenberg  /Marquardt (L  /M) algorithm. Both methods were tested in the estimation of kinetic parameters using a set of three kinetic models of progressively higher complexity. The models describe the catalytic wet air oxidation of phenol carried out in a small-scale trickle bed reactor. The first model only considered the phenol disappearance reaction, while the other two included oxidation intermediate compounds. The number of model parameters involved increased from 3 to 23 and 38, respectively, for the three models. Both algorithms gave good results for the first model, although the L  /M was superior in terms of computation time. In the second case the algorithms achieved convergence, but S-A resulted in a better criterion and kinetic parameters with physical meaning. In the more complex model, only S-A was capable of achieving convergence, whereas the L /M failed. For the second and third model the solution of S-A could be further improved, when used as an initial guess for the L  /M algorithm. # 2002 Elsevier Science Ltd. All rights reserved. Keywords: Simulated annealing; Nonlinear kinetic parameter estimation; Trickle bed reactor; Phenol; Catalytic wet air oxidation 1. Introduction Chemical process design and simulation requires precise knowledge of kinetic parameters. In general, they are obtained by fitting a set of experimental data obtained at specific operation conditions to a rate equation. The simplest approach to this problem, as presented by several classical chemical reactor design textbooks (Fogler, 1992; Froment & Bischoff, 1990), is based on an adequate reparameterisation of the kinetic equation, followed by a linear regression step. Cur- rently, this procedure is being severely criticised, as it may lead to erroneous results (Asprey & Naka, 1999; Buzzi-Ferraris, 1999). As computational power increases, nonlinear para- meter estimation methods have become an alternative to this procedure. According to these methods, the kinetic equations do not need to be linearised before fitting the experimental data. Among these methods, the Levenberg  /Marquardt (L /M) algorithm is the most commonly used. L /M is a gradient-based method, which exhibits rapid quadratic convergence. The major drawback of this algorithm is that convergence to local minima frequently occurs when poor initial guess values are provided. For example, several thermodynamic parameters included in the DECHEMA data bank, which were obtained via local optimisation methods, were proved not to correspond to the best fit (Gau & Stadtherr, 1999). A common practice to deal with the local convergence problem is to test different initial guess parameters. However, as the number of involved parameters increases, the probability to find an initial guess suitable for all parameters decreases. The following step in parameter estimation is the implementation of global optimisation methods. These methods permit to find the global minimum of the objective function, on cost of a significantly higher computational time. Global optimisation methods can be divided in deterministic (e.g. Schnepper & Stadtherr, 1996; Esposito & Floudas, 1998) and stochastic (e.g. Corana, Marchesi, Martini & Ridella, 1987; Cardoso, * Corresponding author. Tel.:  /34-977-559-671; fax:  /34-977-559- 667 E-mail address: fstuber@etseq.urv.es (F. Stu ¨ ber). Computers and Chemical Engineering 26 (2002) 1725  /1733 www.elsevier.com/locate/compchemeng 0098-1354/02/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved. PII:S0098-1354(02)00156-4