A comparison between linear least-squares method and non-linear regression method of the widely used equilibrium isotherms (Langmuir, Freundlich and Redlich-Peterson) for the sorption of Cr(VI) onto Cystoseira indica, which was chemically-modified by cross-linking with epichlorohydrin (CB1, CB2), or oxidized by potassium permanganate (CB3), or only washed by distilled water (RB) was examined. The biosorption equilibriums were established in about 2 h and the maximum removal was observed at pH 3.0 with solid to liquid ratio of 0.05 g/l. The four different linearized forms of Langmuir isotherms are also discussed. Langmuir isotherm parameters obtained from the four Langmuir linear equations are different but they are the same by using non-linear Langmuir equation. The best-fitting isotherms are Langmuir and Redlich-Peterson. The present investigation showed that the non-linear method is the more appropriate method to determine the isotherm parameters. A design procedure was proposed using the Langmuir isotherm to design a two stage sorption system to minimize the amount of biomass required for the treatment of Cr(VI) solution using Cystoseira indica. A two stage sorption system reduced the amount of biomass required by 51.2, 50.7, 51.1 and 51.3%, for CB1, CB2, CB3 and RB, respectively, to achieve the required amount of Cr(VI) removal for any solution volume.
A two-stage non-linear program for optimal electrical grid power balance under uncertainty
