Modeling Heavy Metal Sorption Kinetics Using Fractional Calculus

In this article is introduced a new kinetic semi‐empirical model for drying. The model was developed by arbitrary‐order generalization of Lewis's kinetic equation that was obtained using the Laplace transform and Laplace's Inverse Transform. Kinetic data on soybean drying at 50, 60, 70, and 80 °C were retrieved to test the model which was compared to first‐order Lewis's model and to Page's model by quantitative criteria. Results show that the process is best described by the fractional‐order model and that arbitrary‐order equation may be employed to adjust experimental data on drying, with better results among other models analyzed. Practical applications Drying is one of the most complex and energy‐consuming chemical unit operations. The modeling of drying kinetics can be applied in the project of drying equipment and to reduce costs of energy consumption. In general, the approach used in drying kinetics modeling is based on differential balances to represent the temporal variability of moisture content. In some cases, the Mathematical models cannot fit good the experimental data by to the fact that the models obtained by differential balances are exponential and the experimental data nonexponential. The fractional calculus can be an alternative to generalize the order of differential equations and fit process. The modeling procedure presented in this article presents the use of fractional calculus to generalize the order of a classic differential model. The model obtained by the fractional calculus approach provides best fits and more reliable adjusts than the classic model, the results indicate an anomalous diffusion.

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The fractional calculus in studies on drying: A new kinetic semi‐empirical model for drying

Matias

Bissaro

Jorge

et al. 2018

“…For kinetic models, for example, this mathematical technique can be carried out considering a derivative of fractional order to represent the variation of the property to be described. In recent studies on the adsorption kinetics of heavy metals, Friesen, Leitoles, Gonçalves, Lenzi, and Lenzi (2015) and Gomes, Ara ujo, Lenzi, Silva, and Lenzi (2013) proposed lumped parameters models of fractional order and obtained satisfactory results in the description of the experimental data analyzed.…”

Section: Introductionmentioning

confidence: 96%

Generalization of a lumped parameters model using fractional derivatives applied to rice hydration

Nicolin

Balbinoti

Jorge

et al. 2017

This study presents a generalization of a lumped parameters model through the use of fractional derivatives of arbitrary order. The conventional model has an analytical solution in terms of exponential function. However, when applied to kinetic data on moisture absorption in rice grains, its performance was worse when compared to the proposed generalized model. The results were analyzed for four hydration temperatures. Despite the similarity of the mass transfer coefficients for both models, the proposed generalization provided a statistically significant improvement in prediction of moisture values in relation to time and process temperature. Practical applications The model proposed in this paper is applicable in describing the hydration kinetics of rice which is an important step in parboiling process, for instance. The simplicity of the analytical solution of the model and the improvement of its prediction capability makes the presented approach useful to project of equipment intended to process rice.

“…The fractional calculus is able to capture different characteristics, often not covered by models based on conventional calculus (Machado, Galhano, & Trujillo, 2014), so that this calculation can be applied to the modeling of processes such as drying, once it allows a generalization of the conventional calculation, providing a description of the process by differential equations of arbitrary order (Oldham & Spanier, 1974;Podlubny, 1999). Heavy metal adsorption studies have found highly satisfactory results based on generalized models of fractional order, which were originally designed for first-order derivatives (Friesen, Leitoles, Gonçalves, Lenzi, & Lenzi, 2015;Gomes, Ara ujo, Lenzi, Silva, & Lenzi, 2013). Other authors who have also applied the fractional calculus in their studies were Machado et al (2011) and Sabatier et al (2007).…”

Section: Introductionmentioning

confidence: 99%

Mathematical modeling of soybean drying by a fractional‐order kinetic model

Nicolin

Defendi

Rossoni

et al. 2017

In this article, a mathematical model based on the generalization of first‐order kinetic model, to model soybean drying kinetic, is proposed by taking into account that the humidity variation rate is given by a derivative of arbitrary order. The Bootstrap technique was used to study the minimum required number of terms of the analytic solution to fit the parameters with stable variability. Results were highly successful for the estimated moisture curves. These results were compared with the first‐order kinetic model and with the classic Page model. Series solution minimum number of terms varied both with respect to the model parameters and temperature. Practical applications The modeling procedure presented in this article presents the use of fractional calculus as a potential tool to generalize mathematical models based on differential equations. The proposed model proved to be statistically better than its version which was based on conventional calculus. Thus, the model proposed is suitable for modeling kinetic processes in general and can be used to project drying equipment. Due to the fact that drying of food could present nonexponential behavior, the fractional model proposed would be an adequate option to capture the characteristics of the kinetic curve which the conventional calculus cannot. In addition, this article presents a statistical approach to evaluate the correct number of terms to be used in an equation based on an infinite series in order to result the least variability possible. Such approach provides more reliable results when the model is applied to experimental data.