Experimental design and data prediction by Bayesian statistics for adsorption of tetracycline in a GAC fixed-bed column

Analytical Perspective and Modeling of Breakthrough Curves for a Fixed-Bed Column of Modified Dolomite by Calcination and Ultrasound for Phosphate Immobilization: Experimental Design, Bayesian Statistics, and Data Prediction

“…For the selection of the models, for all three evaluated solids, it can be noticed that the classical statistical metrics ( R 2 and R ajstd 2 ) were not able to select the best adjustment of each adsorption process since several models had the same values (e.g., for the RD: Yoon–Nelson, Clark, and Gompertz models; for the CD: Yan, Yoon–Nelson, Clark, Gompertz, and Log-Gompertz models; and for the UD: Yoon–Nelson, Clark, and Gompertz models), proving that the Bayesian statistical metrics are indispensable for adsorption modeling and through them the models were selected. The importance of the Bayesian statistical metrics was also observed by de Oliveira et al , In that context, for the raw dolomites and dolomites modified with an ultrasound bath, the Yoon–Nelson model better represents the mechanisms of the breakthrough curves (RD: AIC = 27.10, AIC C = 28.60, and BIC = 27.90; UD: AIC = −17.65, AIC C = −15.93, and BIC = −17.04), while for the calcinated dolomite, the Clark model has a better adjustment (AIC = 27.90, AIC C = 31.90, and BIC = 28.80).…”

“…According to the Gompertz and Log-Gompertz models used for practical approaches, ,,, it is seen from the results obtained in Tables – that the Gompertz model had better adjustments for all three solids. This was also observed by de Oliveira et al…”

“…Table lists the models that describe the dynamics of the breakthrough curves, their mechanisms, and the parameters that need to be calculated. The methodology applied in this work was proposed by de Oliveira et al, which consists of using a nonlinear analysis through Bayesian statistics, implemented by the Monte Carlo Markov Chain method (MCMC) with the Metropolis–Hastings (MH) algorithm, in order to model and predict the breakthrough curves and estimate the parameters present in the models, leaving the maximum adsorption capacity deterministically, which is based on the use of the experimental value of the solid’s adsorption capacity; this value can be obtained by applying eq . Two types of statistical metrics were assessed in order to choose the appropriate adjustment for each process: the traditional metrics, which are the coefficient of determination, frequently used in adsorption research ( R 2 ), and the adjusted coefficient of determination ( R ajstd 2 ); and the Bayesian, which is the Akaike information criterion (AIC), the corrected Akaike information criterion (AIC C ), and the Bayesian information criterion (BIC). q max = C i Q 1000 W ∫ 0 t sat ( 1 − C sat C normali ) normald t where C i is the initial phosphate concentration (mg L –1 ); q max is the maximum adsorption capacity (mg g –1 ); k th is the Thomas constant (L –1 min –1 mg –1 ); k YN (min –1 ) is the Yoon–Nelson kinetic constant; τ (min –1 ) is the time required to reach 50% of rupture; n is the Freundlich isotherm constant; A and r (min –1 ) are constants of the Clark model; a Y is the dimensionless constant of the Yan model; α G and α LG are the dimensionless Gompertz and Log-Gompertz constants, respectively; β G and β LG (min –1 ) are the constants of the Gompertz and Log-Gompertz models, respectively; t (min) is the time process; W (g) is the mass of adsorbent in the bed; and Q (mL min –1 ) is the volumetric flow rate applied in the fixed-bed column.…”

“…Continuous adsorption experiments are generally performed to investigate the effects of certain parameters on the process of pollutant removal, such as flow rate, bed depth, and initial concentration. − Continuous adsorption has several advantages over batch adsorption, these being: (I) can treat a constant flow of water, allowing for a higher treatment capacity; (II) the adsorbent can be used multiple times, making it more efficient than batch adsorption; (III) it allows for more consistent treatment quality, as the process can be optimized and monitored continuously; (IV) it can be faster than batch adsorption, as the contact time between the adsorbent and the water can be adjusted by varying the flow rate and the length of the column; and (V) it can be easily scaled up to treat larger volumes. − However, continuous adsorption has disadvantages, such as the optimization to use on a pilot scale and its operational conditions, which need many experiments in order to be implemented; therefore, the operational cost is relatively high . Thus, to solve this problem, the Bayesian technique provides a statistical analysis of the parameters, with a more reliable estimation, and can generate a prediction of the breakthrough curves, making the system scale-up more affordable …”

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

“…32 Habineza et al (2017), 79 Medina et al (2021), 34 and Anastopoulos et al (2020) 35 concluded that the value of q max directly depends on the operational conditions applied in the process, such as the pH, time, temperature, and initial pollutant and solid concentrations, and the adsorbent physical properties, e.g., diameter particle size (d p ), often not informed in papers. In addition, it is worth mentioning that only Danish (2020) 72 52,81 also observed that leaving this parameter as a random variable usually causes its under-or overestimation. By a comparison of the experimental value obtained in the present work (24.85 mg•g −1 ) with those obtained by Danish (2020) 72 (11.34 mg•g −1 ) and Melo et al (2020) 46 (13.50 mg•g −1 ) using modified ACs, q max of this research is quite promising.…”