Multivariate modeling and optimization of Cr(VI) adsorption onto carbonaceous material via response surface models assisted with multiple regression analysis and particle swarm embedded neural network

Heavy metals especially Hexavalent chromium commonly known for its toxic effects. Herein we explore the efficacy of carbonaceous material derived from the domestic firework place for the adsorption of Cr(VI). The effects of significant variables including adsorption time, pH, Cr(VI) solution temperature and Cr(VI) initial concentration on the adsorption process were studied by batch experimental tests. These variables were further utilized as input to analyze, predict, and optimize the Cr(VI) adsorption process using response surface models (RSM) assisted with multiple regression analysis (MRA) and particle swarm embedded neural network (ANN-PSO) models. Regression parameters and analysis of variance (ANOVA) confirmed the validity of both models, and conditions for maximum & accelerated adsorption of 1.37 mg g⁻¹ were predicted to be 94.7 min of adsorption time, Cr(VI) amount of 50.0 mg L⁻¹, temperature of 33.6 °C and pH of 2.0. The statistical accuracy and prediction capability of both models were assessed using validation experiments and in terms of nonlinear statistical metrics such as R², adju-R², mean absolute error (MAE), root mean squared error (RMSE), absolute averaged error (AAD), Marquardt’s percent standard deviation (MPSD), sum of squared error (SSE), hybrid fractional error function (HYBRID), and Pearson chi-square measure (χ²), which revealed the superiority of ANN over MRA. Furthermore, the kinetic data revealed that the uptake of Cr(VI) by carbonaceous material favored pseudo 2^nd  order equation and adsorption phenomenon is better defined by the Langmuir isotherm, while the physisorption was the dominant governing mechanism. The thermodynamic assessment confirmed that adsorption was favorable, spontaneous, and endothermic in nature.