Marquardt's Percent Standard Deviation Research Articles

Isotherms and thermodynamics by linear and non-linear regression analysis for the sorption of methylene blue onto activated carbon: Comparison of various error functions

A comparison of linear and non-linear regression method in selecting the optimum isotherm was made to the experimental equilibrium data of methylene blue sorption by activated carbon. The r 2 was used to select the best fit linear theoretical isotherm. In the case of non-linear regression method, six error functions, namely coefficient of determination ( r 2), hybrid fractional error function (HYBRID), Marquardt's percent standard deviation (MPSD), average relative error (ARE), sum of the errors squared (ERRSQ) and sum of the absolute errors (EABS) were used to predict the parameters involved in the two and three parameter isotherms and also to predict the optimum isotherm. For two parameter isotherm, MPSD was found to be the best error function in minimizing the error distribution between the experimental equilibrium data and predicted isotherms. In the case of three parameter isotherm, r 2 was found to be the best error function to minimize the error distribution structure between experimental equilibrium data and theoretical isotherms. The present study showed that the size of the error function alone is not a deciding factor to choose the optimum isotherm. In addition to the size of error function, the theory behind the predicted isotherm should be verified with the help of experimental data while selecting the optimum isotherm. A coefficient of non-determination, K 2 was explained and was found to be very useful in identifying the best error function while selecting the optimum isotherm.

Modelling Individual and Competitive Adsorption of Cadmium(II) and Zinc(II) Metal Ions from Aqueous Solution onto Bagasse Fly Ash

The present study deals with the competitive adsorption of cadmium (Cd(II)) and zinc (Zn(II)) ions onto bagasse fly ash (BFA) from binary systems. BFA is a waste obtained from the bagasse‐fired boilers of sugar mills. The initial pH≈6.0 is found to be the optimum for the individual removal of Cd(II) and Zn(II) ions by BFA. The equilibrium adsorption data were obtained at different initial concentrations (C 0 = 10–100 mg/l), 5 h contact time, 30°C temperature, BFA dosage of 10 mg/l at pH 0 = 6. The Redlich–Peterson (R–P) and the Freundlich models represent the single ion equilibrium adsorption data better than the Langmuir model. The adsorption capacities in the binary‐metal mixtures are in the order Zn(II)>Cd(II) and is in agreement with the single‐component adsorption data. The equilibrium metal removal decreases with increasing concentrations of the other metal ion and the combined action of Cd(II) and Zn(II) ions on BFA is found to be antagonistic. Equilibrium isotherms for the binary adsorption of Cd(II) and Zn(II) ions on BFA have been analyzed by non‐modified Langmuir, modified Langmuir, extended‐Langmuir, Sheindorf–Rebuhn–Sheintuch (SRS), non‐modified R–P and modified R–P adsorption models. The isotherm model fitting has been done by minimizing the Marquardt's percent standard deviation (MPSD) error function using MS Excel. The SRS model satisfactory fits for most of the adsorption equilibrium data of Cd(II) and Zn(II) ions onto BFA.

Arsenic adsorption onto iron oxide-coated cement (IOCC): Regression analysis of equilibrium data with several isotherm models and their optimization

Abstract The present work examines the equilibrium sorption of arsenic (III and V) from aqueous environment onto iron oxide-coated cement (IOCC) at 288, 298 and 308 K, and determines the equilibrium sorption isotherms. The equilibrium for both As(III) and As(V) was achieved in 2 h. The experimental isotherm data were analysed using Langmuir, Freundlich, Dubinin–Radushkevich (D–R), Toth and Temkin isotherm equations. In order to determine the best-fit isotherm for each system, both linear and non-linear regressions were carried out. For this, six error analysis methods were used to evaluate the data: the linear coefficient of determination, the sum of the squares of errors, the sum of absolute errors, the average relative error, the hybrid fractional error function and the Marquardt's percent standard deviation. The error values indicated that the Freundlich isotherm was able to provide the best quality of fit for all the experimental data, over the concentration range studied, for both As(III) and As(V) at 288, 298 and 308 K. The values of the parameter sets of the isotherms also indicated that the adsorption of arsenic onto IOCC is a temperature dependent phenomena with As(III) and As(V) exhibiting exothermic and endothermic nature of adsorption, respectively.

Use of optimization to develop a correlation model for predicting residual NAPL saturation

Predicting the residual saturation of a trapped non-aqueous phase liquid (NAPL) contaminant is critical to estimate the region of contamination, the design of remediation strategies, and risk assessment. Models were developed to predict residual NAPL saturation utilizing optimization and non-linear functions, consequently allowing for a broader mathematical approach to model development. The input parameters evaluated represent soil and fluid properties; the uniformity coefficient (C u), the coefficient of gradation, the capillary number (Nc), the bond number (Nc), and the total trapping number (Nt). Overall, the model that performed the best was based on a second-order equation with the independent variables C u and Nt1 using the sum of the squares of the errors. The non-linear error function based on a derivative of Marquardt's percent standard deviation performed the best for three other cases.

EQUILIBRIUM ISOTHERM STUDIES FOR THE SORPTION OF DIVALENT METAL IONS ONTO PEAT: COPPER, NICKEL AND LEAD SINGLE COMPONENT SYSTEMS

The sorption of three divalent metal ions — copper, nickel and lead — from aqueous solution onto peat in single component systems has been studied and the equilibrium isotherms determined. The experimental data have been analysed using the Langmuir, Freundlich, Redlich-Peterson, Toth, Temkin, Dubinin-Radushkevich and Sips isotherm models. In order to determine the best fit isotherm for each system, six error analysis methods were used to evaluate the data: the coefficient of determination, the sum of the errors squared, a hybrid error function, Marquardt's percent standard deviation, the average relative error and the sum of absolute errors. The error values demonstrated that the Sips equation provided the best model for the three sets of experimental data overall.