Statistical Analysis of Linear and Non-Linear Regression for the Estimation of Adsorption Isotherm Parameters

Abstract

A very common practice during the parameter estimation of adsorption isotherms, including the well-known Langmuir and Freundlich isotherms, consists in manipulating the isotherm equation to obtain a linear equation and estimate the model parameters using a linear least squares method. This procedure is also usually used for estimating the thermodynamic adsorption parameters, despite the fact that personal computers and software are available for prompt implementation of non-linear solutions of the original parameterestimation problem. For this reason, the main purpose of this work is to show that the use of linear least-squares methods for estimating adsorption isotherm parameters leads to some serious drawbacks, which can be readily avoided through proper use of non-linear procedures and posterior statistical analyses of the parameter-estimation results, enhancing the quality of the obtained results.

Adsorptive characterization of porous solids: Error analysis guides the way

“…Since the direct fitting method results in a smaller variation in BET area and confidence interval for data inside the recommended pressure window, it is recommended to use the direct fitting procedure. This conclusion is in line with recent work of Osmari et al, who show that Langmuir parameters from different adsorption measurements determined using the nonlinear (direct) fitting approach yields better parameters and lower uncertainties than using different linearization schemes [71]. Furthermore they concluded that the direct method is much more robust, as it is less influenced by the in-or exclusion of a specific measured point, which can also be seen from the results presented here (see Fig.…”

“…The linear fitting method may be preferred over directly fitting C and q m because of visual tractability and simplicity of fitting, not because of profound physical insights or statistical benefits. Regarding the latter, the error distribution is changed by linearization [71], similarly as in the determination of (bio)catalytic reaction kinetic parameters (Hougen-Watson, Lineweaver-Burk approach) [72][73][74].…”

Statistical Evaluation of Non-Linear Parameter Estimation Procedures for Adsorption Equilibrium Models

“…Traditionally, minimization of deviations is performed through minimization of the least-squares function, as presented in equation (1): (1) where S is the objective function value, and are the experimental and model calculated values of the equilibrium concentration in the solid phase, is the experimental value of the equilibrium concentration in the bulk phase and q is the vector of adsorption equilibrium parameters, whose values must be optimized during the parameter estimation procedure to achieve the best adsorption model fit. This is the commonest non-linear procedure used to determine the adsorption equilibrium parameters (Kumar andSivanesan 2005, 2006;Kumar 2006;Bolster and Hornberger 2007;Parimal et al 2010;Osmari et al 2013).…”

“…Recently, Osmari et al (2013) performed the comparison of parameter estimates obtained from linear and non-linear regression procedures based on sound statistical arguments, giving particular attention to the estimated parameter values and respective uncertainties, and showed that non-linear regression techniques present more robust performances and can provide more significant parameter estimates.…”

Estudo De Regiões De Confiança De Parâmetros De Isotermas Em Sistema De Adsorção De Corante Em Carvão Ativado

“…Para evitar o uso de procedimentos iterativos, alguns modelos podem ser manipulados matematicamente gerando um modelo linear em um novo conjunto de variáveis dependentes e independentes; porém tais transformações modificam as distribuições de erros das variáveis dependentes e independentes, invalidando hipóteses implícitas do procedimento de estimação, o que compromete a interpretação da significância estatística dos resultados. (Osmari et al, 2013, Cassol et al, 2014.…”