Fig. 5

Analysis of residual graphs, 2-D and 3-D response surface plots, and desirability ramp for optimizing copper removal parameters. (a) Diagnostic residual plots: (1) Normal probability plot confirmed the normality assumption; (2) Residuals vs. predicted values showed a random scatter, indicating homoscedasticity and model adequacy; (3) Residuals vs. run order confirmed the independence of residuals over time; (4) Cook’s distance plot revealed no influential outliers; and (5) Box-Cox plot indicated that no data transformation was required. These diagnostics collectively confirm the adequacy and validity of the fitted model. (b) 2D and 3D response surface plots. In all plots, the red color indicates the highest response, while dark blue represents the lowest response. For each 3D plot, the third parameter is fixed at its central value. (1) The A-axis denotes pH and the B-axis shows inoculation percentage. (2) The A-axis indicates pH and the B-axis shows the initial copper concentration. (3) The A-axis represents inoculation percentage and the B-axis shows the copper concentration. (c) Desirability ramp charts for the optimization of initial solution pH, copper concentration, and inoculation percentage to maximize copper removal efficiency. The optimal values for each parameter and the predicted maximum removal percentage are indicated. This figure was created by using Design-Expert software (version 12.0.3.0, Stat-Ease Inc., Minneapolis, USA).