Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, Article ID 642101, 10 pages http://dx.doi.org/10.1155/2013/642101 Research Article Parametric Analysis of a Heavy Metal Sorption Isotherm Based on Fractional Calculus Enrico M. Gomes, 1 Rosana R. L. Araújo, 1 Marcelo K. Lenzi, 1 Fernanda R. G. B. Silva, 2 and Ervin K. Lenzi 2 1 Departamento de Engenharia Qu´ ımica, Universidade Federal do Paran´ a, Caixa Postal 19011, 81531-980 Curitiba, PR, Brazil 2 Departamento de F´ ısica, Universidade Estadual de Maring´ a, 87020-900 Maring´ a, PR, Brazil Correspondence should be addressed to Ervin K. Lenzi; ervinklenzi@gmail.com Received 5 October 2012; Revised 5 April 2013; Accepted 14 April 2013 Academic Editor: Jocelyn Sabatier Copyright © 2013 Enrico M. Gomes et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Heavy metals are widely recognized as being hazardous to human health and environmentally aggressive. Te literature reports diferent approaches for lead removal, for example, water hyacinths. Heavy metal sorption isotherm modeling represents an important tool towards the study of equilibrium conditions. Fractional calculus represents a novel approach and a growing research feld for process modeling, based on derivatives of arbitrary order. Recently, a novel isotherm based on fractional calculus was proposed for lead sorption using water hyacinth (Eichhornia crassipes). Tis paper reports a general procedure on error analysis and its infuence on parameter estimation. It was applied to mathematical models based on fractional diferential equations, focusing on a heavy metal novel isotherm sorption model. Parameter variance was calculated by using two diferent approaches (with the complete Hessian matrix and with a simplifed Hessian matrix), and joint parameter confdence regions were generated, being successfully able to show that the fractional nature of the model is statistically valid. 1. Introduction Heavy metals are widely recognized as being hazardous to human health and environmentally aggressive, being contin- uously generated by diferent chemical plants. Te use of lead in the battery industry [1] is an important example. Te liter- ature reports diferent approaches for heavy metals removal, such as chemical precipitation [2], ion exchange [3], and electrochemical [4] and water hyacinths [5]. Mathematical models represent an essential tool for in-depth process stud- ies, design, optimization, and control [6]. Terefore, heavy metal sorption isotherm modeling represents an important way towards the study of equilibrium conditions, which play a key role in sorption process design. Te most common approach for this task consists in the use of classical models [7], such as Langmuir, Freundlich, and Redlich-Peterson among others, followed by proper parameter estimation and model discrimination analysis. Fractional calculus represents a novel approach and a growing research feld for process modeling, being based on derivatives of arbitrary order [8–14]. Te literature reports a broad range of applications, concerning systems engineering [15], difusion processes [16], heat transfer [17], solid mixing [18], biological systems [19], and fuid mechanics [20] among others [21]. Recently, dos Santos et al. [22] proposed a novel isotherm based on fractional calculus for lead sorption using water hyacinths (Eichhornia crassipes). Te reported isotherm can successfully predict equilibrium concentrations of lead between the aqueous solution and the water hyacinth afer. Te model was validated using synthetic efuent [1]. It is important to highlight that the proposed model also leads to better performances when compared to classical models (Langmuir, Freundlich, and Redlich-Peterson), which were used for sake of comparison. Error analysis represents a crucial step in model valida- tion and further applications [23]. Recently, Joshi et al. [24] presented a detailed model analysis concerning classical sorp- tion models. Regarding fractional-calculus-based models, Gabano and Poinot [25], Khemane et al. [26], and Isfer et al. [15] report the calculation of parametric variance.