Estimation of Parameters and Selection of Models Applied in Adsorption

Abstract

The modeling of complex phenomena such as adsorption often requires the determination of parameters that cannot be directly measured and, therefore, must be estimated. An important point is related to the analysis of the inverse problem as a method of estimating parameters and selecting models. In view of this, this work aims to apply the Monte Carlo method via Markov Chains (MCMC) as a technique for solving the inverse problem of estimating fixed-bed adsorption parameters using analytical models proposed in the literature. In addition, this work aims to select the best model through the statistical metrics Akaike, corrected Akaike and Bayesian Information Criterion. The use of the Bayesian approach allowed the analysis of the convergence of the chains, as well as selected the best model to represent the experimental data obtained from the literature.

Bayesian Computational Technique for Modeling Caffeine Adsorption in a Fixed-Bed Column: Use of the Maximum Adsorption Capacity Deterministically and Experimental Design

“…The breakthrough curves are defined as the ratio between the adsorbate concentration at the exit of the column I, from a certain length ( z ), and the initial adsorbate concentration ( C 0 ) as a function of time or volume of treated effluent. The adsorption occurs in the mass transfer zone (MTZ), ,, a region inside the column where the concentration of the adsorbate varies, considered by many authors, from 95 to 5% of the feed value. ,− The mathematical models of the breakthrough curves used in this work are described in Sections , 2.2, 2.3, 2.4, and 2.5, and the equation to obtain the maximum adsorption capacity experimentally is described in Section .…”

“…The maximum adsorption capacity is the most important parameter in adsorption processes, and it is present in the breakthrough curve models, being essential for the column design, both pilot and industrial scales . This parameter is extensively estimated along with the other parameters of the model; ,, however, it can be calculated experimentally providing the real value of the process under study, which often, when the parameter is estimated, leaving it as a random variable, can lead to its under or overestimation, not representing the process itself. Studies of caffeine removal by adsorption have gained attention of researchers around the world due to its high consumption and for its resistance in conventional treatments. − …”

Applying the Bayesian Technique, Statistical Analysis, and the Maximum Adsorption Capacity in a Deterministic Way for Caffeine Removal by Adsorption: Kinetic and Isotherm Modeling

“…Both analyses were performed on a pore size and area analyzer (Quantachrome, model NOVA 4200e). The zero-charge-point analysis (pH PCZ ) was determined using the methodology of the 11-point experiment …”

“…The zero-charge-point analysis (pH PCZ ) was determined using the methodology of the 11-point experiment. 44 The solid was characterized by X-ray diffraction (XRD) using a Phillips diffractometer, model X'Pert MDP, with angular variation from 5°to 75°(2θ), Cu Kα radiation, and an exposure speed of 0.05°•s −1 . Fourier transform infrared (FTIR) spectroscopy of the samples was obtained on a PerkinElmer spectrophotometer, model Frontier, in the wavelength range of 4000 to 400 cm −1 , with a resolution of 4 cm −1 , in 16 scans, at room temperature.…”

“…The mechanisms and characteristics governing the adsorption process are informed through mathematical and/or empirical models. − Those models have parameters that are not possible to obtain by direct measurements, being characterized as inverse problems. , Inverse problems can be defined as those that aim to determine causes from experimental measures of effects; thus, its solution, being the estimation of the parameters, is usually done through the minimum squares method. ,,, Although, this approach can modify the intrinsic interpretation of the parameters, violating the assumptions of the method, because a priori information on the parameters and uncertainties of the measurements are not implemented in its resolution. ,− …”