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

Abstract

Adsorption equilibrium is a fundamental concept in the adsorption science and relates the equilibrium between the quantity of the adsorbed material and its concentration in the bulk phase. Several models have been proposed for prediction of adsorption equilibrium and all models depend on parameters whose values must be estimated from available experimental data. Although linear parameter estimation procedures can be used for model fitting, through transformation of available experimental data and model parameters, non-linear parameter estimation procedures lead to more reliable results and allow for direct comparison of results obtained with different adsorption equilibrium models. The main objective of this work is to present and compare different non-linear procedures for parameter estimation of adsorption equilibrium models, based on theoretical arguments and also on the numerical estimation of adsorption equilibrium parameters, using available experimental data for adsorption of methylene blue onto activated carbon. The results obtained indicate that the best parameter estimation procedure is the one that relies on available equilibrium concentrations in the bulk phase as a function of the fluid volume, adsorbent mass and initial concentrations in the bulk phase, without transformation of measured experimental values and model parameters. Besides, it is shown that parameter estimates should be obtained through proper minimization of weighted least-squares objective function, in accordance with maximum likelihood procedures.

Modeling of Basil Essential Oil Extraction With Variable Diffusion Coefficient

“…According to Cassol et al (2014), from the assumed assumptions for measurement errors, the covariance matrix of the parameter estimates, , was given by [eq. (15)].…”

Is the Frumkin (Fowler–Guggenheim) adsorption isotherm a two- or three-parameter equation?

“…( 7) for nonlinear least-squares fitting of explicitly defined isotherm models such as the Langmuir and Freundlich equations. The parameter estimation procedure requires that the values of ci,fit be obtained from the solution of the mass balance equation for a batch adsorber [41]. In this treatment, c becomes the dependent variable and the other measured quantities in the mass balance equation (initial concentration, adsorbent mass, solution volume) become the independent variables.…”

“…The ODR method for fitting explicit isotherm models to experimental equilibrium data has been implemented in a freely available Excel spreadsheet [45]. It should be mentioned that ODR does not address the statistical problem of correlation that exists between c and q [41,46].…”

Evaluation of Cd(II) Ion Removal from Aqueous Solution by a Low-Cost Adsorbent Prepared from White Yam (Dioscorea rotundata) Waste Using Batch Sorption

“…However, this approach has been observed to be limited as it has an inherent bias resulting from the linearization, as such, data transformations implicitly alters the error structure and may result in a violation of the error equality of variance and normality hypotheses for standard least squares [61,[64][65][66]. The use of non-linear optimization has been reported as a better approach for the determination of model fitting to experimental data, and of isotherm parameter values as it most commonly uses algorithms for the determination of the parameters [61,64,[67][68][69]. The utilization of the non-linear approach requires the definition of error function to enable the optimization process determine and evaluate the fitting of the models to the experimental data [61].…”