Adsorption equilibrium models: Computation of confidence regions of parameter estimates

Abstract

Estimation of parameter values of adsorption equilibrium models from available experimental data is a common practice, but the evaluation of parameter uncertainties is not usually performed. Furthermore, adsorption equilibrium models generally present two or more parameters and the uncertainty analysis must be performed with help of confidence regions, allowing for evaluation of parameter correlations. In the present work confidence regions of estimated parameters were determined for Langmuir, Freundlich, Sips and Redlich–Peterson isotherms through the elliptical approximation and the likelihood ratio method, using experimental data related to dyes adsorption from liquid solutions. The effects caused by the number of measurements, the experimental data region and the definition of experimental measurement variance were also evaluated. The results showed that changes of the experimental region can exert large influence on the precision of the estimated parameters when experimental variances are estimated with the minimum value of the least squares objective function, since in this case model performance affects directly the estimated parameter uncertainties. On the other hand, when experimental variances are obtained independently, parameter uncertainties become less dependent on model performance, as expected. It was also observed that the elliptical and likelihood ratio approaches for determination of confidence regions lead to similar results when uncertainties of the parameter estimates are low, even when nonlinear models are considered. Finally, it was shown that the proper evaluation of the experimental uncertainty exerts a major effect on the obtained confidence regions, so that the use of replicates in the experimental grid must certainly be encouraged.