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MSC: 65L11 65L20 65L60 65L70 Keywords:Singularly perturbed problem Two small parameters Galerkin finite element method Bakhvalov-type mesh Clément quasi-interpolant a b s t r a c t A singularly perturbed problem with two small parameters is considered. On a Bakhvalovtype mesh we prove uniform convergence of a Galerkin finite element method with piecewise linear functions. Arguments in the error analysis include interpolation error bounds for a Clément quasi-interpolant as well as discretization error estimates in an energy norm. Numerical experiments support theoretical findings.

The aim of this study was the optimization of supercritical fluid extraction (SFE) of wheat germ oil obtained as by‐product from industrial mill. Extraction kinetics modeling and response surface methodology (RSM) were used for that purpose. SFE was performed with broadening Box‐Behnken experimental design, where pressure (250–350 bar), temperature (40–60°C), and CO2 flow rate (0.2–0.4 kg/hr) were used as independent variables. Five empirical kinetic equations were successfully utilized for modeling of SFE. Model IV (Kandiah and Spiro model) provided the best fit with experimental data, according to statistical parameters (R2, sum of squared errors and average absolute relative deviation). Furthermore, initial slope calculated from this model was used as a response variable for RSM optimization. It could be concluded that SFE should be performed at elevated pressure (350 bar) and CO2 flow rate (0.4 kg/hr), while temperature should be held at a lower level (40°C) in order to achieve a maximal initial slope. Practical applications Optimization and modeling of industrial processes are crucial factors that will determine its efficiency and profitability. This research provided information about modeling of green and environmentally friendly extraction technique, that is, SFE, which could be successfully utilized for recovery of valuable oil from food industry by‐product, that is, wheat germ. Commonly used empirical models were successfully applied for modeling of extraction process and influence of SFE factors on model parameters was determined providing information about mass transfer phenomena during extraction. Further investigation provided determination of the highest initial extraction rate constant which could be applicable for optimization of industrial scale processes.

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

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