Chemical Engineering Science 63 (2008) 4631--4635 Contents lists available at ScienceDirect Chemical Engineering Science journal homepage: www.elsevier.com/locate/ces Optimum reparameterization of power function models Marcio Schwaab, José Carlos Pinto ∗ Programa de Engenharia Química/COPPE, Universidade Federal do Rio de Janeiro, Cidade Universitária - CP: 68502, Rio de Janeiro, RJ 21941-972, Brazil ARTICLE INFO ABSTRACT Article history: Received 29 February 2008 Received in revised form 28 June 2008 Accepted 1 July 2008 Available online 4 July 2008 Keywords: Parameter correlation Reparameterization Parameter estimation Kinetics Adsorption Mathematical modeling Power function models are frequently used to describe rates of adsorption (as in the common Freundlich model) and chemical reaction (for estimation of reaction orders). When power function models are used to fit available experimental data, correlations among obtained parameter estimates are normally very high, which may cause significant numerical problems during the estimation of the model parameters and lead to misinterpretation of the statistical significance of final results. In this work, a reparameterization technique is presented to allow for reduction of parameter correlations in power function models. After- wards, the two-step parameter estimation procedure [Schwaab, M., Pinto, J.C., 2007. Optimum reference temperature for reparameterization of the Arrhenius equation. Part 1: problems involving one kinetic constant. Chemical Engineering Science 62, 2750–2764; Schwaab, M., Lemos, L.P., Pinto, J.C., 2008b. Opti- mum reference temperature for reparameterization of the Arrhenius equation. Part 2: problems involving multiple reparameterizations. Chemical Engineering Science 63, 2895–2906.] is used for optimum repa- rameterization and estimation of uncorrelated model parameters in power function models. © 2008 Elsevier Ltd. All rights reserved. 1. Introduction Existence of high correlations among the parameter estimates of a mathematical model can cause significant numerical problems during the estimation of model parameters, as the minimization of the objective function may become inefficient (Espie and Macchietto, 1988) and the statistical characterization of the final parameter es- timates may lack significance (Watts, 1994). Some of these dif- ficulties can be observed during the estimation of parameters of power function models because of the usually very high correlations among the obtained parameter estimates. Power function models are commonly used in the chemical engineering field to describe rate expressions and equilibrium conditions. Some examples are the Freundlich equation, used to describe adsorption of gases and liquids onto solid surfaces (Guo et al., 2006), and the well-known nth-order reaction rate models. Schwaab and Pinto (2007) and Schwaab et al. (2008b) have re- cently proposed an algorithm for optimum reparameterization of the Arrhenius equation (frequently used in kinetic models) and reduc- tion of parameter correlations in parameter estimation problems. It was shown that the explicit introduction of a reference tempera- ture into the standard Arrhenius equation and the proper selection of reference temperature values can allow for minimization (and ∗ Corresponding author. Tel.: +55 21 25628337; fax: +55 21 25628300. E-mail address: pinto@peq.coppe.ufrj.br (J.C. Pinto). 0009-2509/$ - see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2008.07.005 sometimes elimination) of correlations among parameter estimates and simultaneous minimization of the relative standard errors of the parameter estimates. In this work, a reparameterization technique is proposed in order to allow for minimization of correlations among parameter estimates during estimation of the parameters of power function models. The proposed reparameterization technique is based on the definition of a reference value, which is used to renormalize the original model equation. First, the proposed technique is applied in a simple prob- lem, where it is possible to derive an analytical solution for the reference value that leads to null parameter correlations (and mini- mum relative error content of final parameter estimates). Then, the numerical optimization procedures proposed by Schwaab and Pinto (2007) and Schwaab et al. (2008a,b) are used to provide the opti- mum reference values (and, consequently, uncorrelated parameter estimates) in a more complex numerical parameter estimation problem, where an analytical solution cannot be derived. 2. Theoretical framework A simple power function model can be defined as y = kx n (1) where x and y are the independent and the dependent measured variables and k and n are two model parameters. The correlation between the parameter estimates for k and n are usually very high. In order to minimize the correlation between the two parameter