Central composite design (CCD)

The design expert software (11.1.0.1) was used to generate the statistical design of experiments and the data analysis. The CCD is one of the RSM design methods that makes use of a second-order model for improved optimization process. CCD offers high quality significant predictions of linear and quadratic interactive effects of the operating parameters influencing the process. It is a full factorial design consisting of two levels, a centre point that refers to the middle level of the factors and two axial points.

In this study, four operating parameters were selected for the bio-sorption of Cu(II) and Pb(II) ions using banana peels, namely; initial metal concentration (X₁), pH (X₂), adsorbent dosage (X₃) and particle size (X₄). The factor levels of the independent variables were coded as −1 (low), 0 (centre point) and + 1 (high) while the % removal of Cu (II) and Pb(II) ions were the dependent variables. A total of 30 experimental runs were generated from the design matrix. The experimental range and levels of the independent variables in coded form is depicted in Table ​Table1.1. In the optimization process, the second order polynomial equation was used to explain the effects of the independent variables and their interactions. The quadratic model used to optimize the variables is as shown in Eq. 7,

Experimental range and level of independent variables

*Actual value used in the experiment was 250 μm due to available sieve sizes

Where, Y is the response predicted, X_i and X_j are the independent variables, β_o, β_i, β_ii and β_ij are the regression coefficient and Ɛ is the residual error. The significant variables and the interpretation of the experimental results are explained using a set of mathematical functions called analysis of variance (ANOVA). The analysis of variance ANOVA is a statistical technique used to determine the significance of a factor in a multi-significant model. ANOVA helps to identify the most important factors in a model as well as the meaning of the experimental results. The F-value and p value are important coefficients for determining the fitness of the model and the significant of each factor represented in the model equation. The coefficient F-value is obtained as a quotient of the residual mean square and the mean square [15].