On the Operational Solution of a System of Fractional Differential Equations

MSC 2010: 26A33, 44A45, 44A40, 65J10We consider a linear system of differential equations with fractional derivatives, and its corresponding system in the field of Mikusiński operators, written in a matrix form, by using the connection between the fractional and the Mikusiński calculus. The exact and the approximate operational solution of the corresponding matrix equations, with operator entries are determined, and their characters are analyzed. By using the packages Scientific Work place and GeoGebra, the exact and the approximate solution of the given numerical example are constructed, and their dependence on the initial condition and the fractional derivatives is shown graphically

The impact of calculation on the value of thermodynamic parameters of copper adsorption on poplar sawdust

The adsorption of Cu(II) onto poplar sawdust as an adsorbent is analyzed. The experimental data were fitted by the Langmuir isotherm using four linearized forms at the isotherm along with the original one. The least squares regression method was applied. Using the obtained Langmuir constants by each at methods, the enthalpy, entropy and Gibbs free energy at adsoption were calculated. A comparison of the used linear and non-linear regression methods in view at the goodness of the fit is presented. The coefficient of correlation was adopted as a criterionn to select the best method. The impact of the choice at regression model on the resulting estimates of the thermodynamic parameters is discussed. The best fit of the experimental data is obtained by the nonlinear regression. Thus, it is recommended to use the Langmuir parameters calculated by the nonlinear regression for estimating the thermodynamic parameters of adsorptin. The differences in the values obtained by different models are not so large to change the basic conclusion that the adsorption of copper ions on poplar sawdust is a spontaneous endothermic process i.e. that tested adsorbent has an affinity for copper ions

Isotherms for the adsorption of Cu(II) onto lignin -comparison of linear and non-linear methods

Abstract Equilibrium studies were carried out for the adsorption of Cu(II) onto Kraft lignin as an adsorbent. The experimental data were fitted to the Freundlich, Langmuir and Redlich-Peterson isotherms by linear and non-linear methods. Comparison of linear and non-linear regression methods was given in selecting the optimum isotherm for the experimental data. The coefficient of correlation r 2 and chi-square test χ 2 were used to select the best linear theoretical isotherm. The best linear model is Redlich-Peterson isotherm model, where r 2 = 0.985 and χ 2 = 0.02. In order to predict the error ERRSQ, HYBRD, MPSD, ARE and EABS were used. Moreover, by minimizing these error functions the optimal values of parameters and also the optimum isotherm was found. The Redlich-Peterson isotherm was found to be the best representative for adsorption of Cu(II) on the adsorbent in the cases when ERRSQ, HYBRD and MPSD functions were used. Their coefficients of determination were 0.986, 0.985 and 0.984, respectively and chi-square was 0.02 in all cases. Freundlich isotherms obtained by minimization of the ERRSQ, HYBRD, MPSD, ARE and EABS function showed very good agreement with experimental data. In all cases the coefficients of determination were greater than 0.91. Besides, it was observed that non-linear isotherm models were better for representation of equilibrium data than linearized models. There is increasing interest in the development of economically viable new applications of lignin, which can be found as waste from cellulose and paper industry. This is evidenced by the existence of the International Lignin Institute, which promotes new technologies with lignin as the main component. Among others, the possibility of using lignin as an adsorbent for different pollutants from water has been considered Equilibrium relationships between adsorbent and adsorbate are described by adsorption isotherms and characterized by certain constants, the values of which provide information on adsorbent capacity and affinity for some kind of adsorbate. In addition, the quality of the fit of experimental results indicates whether ad- sorption is monolayer or multilayer, on a homogeneous or heterogeneous surface. Isotherm equations do not consider which adsorption mechanisms, such as external mass transfer of solute, intraparticle diffusion or adsorption at sites, are decisive in the adsorption process, but can be used for design of batch adsorption systems. Some of the isotherm equations that can explain solid-liquid adsorption systems are: Langmuir, Freundlcih, Temkin, Sips, Toth, Redlich-Peterson isotherm equations Linear regression has been frequently used to determine the most fitted models and the method of least squares has been frequently used for finding the parameters of the models In this study, binding characteristics of Cu(II) on Kraft lignin were investigated and feasible application of different adsorption models for description of its adsorption on this low-cost and environment-friendly adsorbent was described. For that purpose, linear and non-linear equations of Freundlich, Langmuir and Red

Influence of impurities and hull content in material for pressing on sensory quality cold-pressed sunflower oil

This paper analyzes influence of different percentage of impurities and hull in the material for pressing on sensory quality (appearance, smell, taste, aroma and colour) of edible nonrefined sunflower oil. It has been concluded that simultaneous presence of impurities and hull in the starting material, especially presence of bigger quantities of impurities, has a rather negative effect on sensory quality of sunflower oil, made by cold pressing on a screw press. Apart from sensory analysis, the effect of impurities and hull on oil colour has also been identified by transparency determining, i.e. content of total pigments - carotenoids and chlorophyll. It has been concluded that presence of bigger quantities of impurities and hull cause the increase of content of both carotenoids and chlorophyll, and at the same time, decrease of transparency value in cold-pressed sunflower oil

Isotherms for the adsorption of Cu(II) onto lignin: Comparison of linear and non-linear methods

Equilibrium studies were carried out for the adsorption of Cu(II) onto Kraft lignin as an adsorbent. The experimental data were fitted to the Freundlich, Langmuir and Redlich-Peterson isotherms by linear and non-linear method. Comparison of linear and non-linear regression method was given in selecting the optimum isotherm for the experimental data. The coefficient of correlation r2 and Chi-square test χ2 was used to select the best linear theoretical isotherm. The best linear model is Redlich-Peterson isotherm model, where r2=0,985 and χ2=0,02. In order to predict the error ERRSQ, HYBRD, MPSD, ARE and EABS were used. Moreover, by minimizing these error functions the optimal values of parameters and also the optimum isotherm was found. The Redlich-Peterson isotherm was found to be the best representative for adsorption of Cu(II) on the adsorbent in the cases when ERRSQ, HYBRD, MPSD functions were used. There coefficients of determination are 0.986, 0.985, 0.984, respectively and Chi-square is 0.02 in all cases. Freundlich isotherms which were obtained by minimization of the ERRSQ, HYBRD, MPSD, ARE and EABS function showed very good agreement with experimental data. In all cases the coefficients of determination are greater than 0.91. Besides, it was observed that non-linear isotherm models were better for representation of equilibrium data than linearized models

Effect of seed quality on oxidative stability of cold-pressed sunflower oil

The objective of this study was to investigate the impact of the storage time, content of the husk and impurities in the seed mass on the oxidative stability of the cold pressed sunflower oil. The oxidative stability of oil was analyzed by the RSM (Response Surface Methodology), where the response value (output) of the model was the content of oxidation products measured via specific absorbance at 232 nm for conjugated dienes and 270 nm for conjugated trienes. Analyses were performed on the fresh oil samples and oil samples tempered under the Schaal-Oven’s test conditions (96 h at 63±2ºC). It was concluded that the seed storage time had significant impact on the primary (A1%232nm) and secondary (A1%270nm) oil oxidation products, both before (b1-p<0,001; b1-p=0,021) and after (b1-p=0,048; b1-p=0,033) the heating of oil. The content of impurities and husks in the seeds used for pressing also had an influence of the conjugated dienes and trienes contents in both, fresh and heated oil samples. However, their impact was not statistically significant. Interaction of seed storage time, the content of impurities and husks also exert some effect on the values of these quality parameters in the fresh oil as well as in the oil after the heat treatment, even though their influence is not significant. The obtained results have shown that the cold pressed sunflower oil had a good oxidative stability at temperatures up to 63±2ºC