The cocoa husks, supplied by farmers in the Daloa area (Côte d’Ivoire), were prepared according to the protocol described in our previous work [14]. Briefly, after washing and air-drying at room temperature, the biomass was mechanically ground using a Retsch SK 100 knife mill, then sieved to obtain the desired particle size fractions. The powder was then activated by immersion in a NaOH or H₂SO₄ solution kept under magnetic stirring (350 rpm), at concentrations, durations, and temperatures that varied according to the trial. The activated materials were successively rinsed with tap water and distilled water, then dried before use.

Optimization of the biosorbent preparation conditions was carried out using experimental design methodology, a statistical approach that allows the simultaneous study of several parameters and of their interactions within a single series of runs, which the conventional one variable at a time strategy cannot achieve [15]. The phosphate ion removal rate (R, in %) was retained as the single response for evaluating the material’s efficiency. A preliminary screening study, carried out using a Plackett-Burman design (PBD), addressed seven preparation variables: biomass nature, particle size, activation time, activation temperature, solid/liquid ratio, type of activating agent, and concentration of the activating solution. Three parameters emerged as significant from this screening and were carried forward to the optimization step: activation temperature (X₁), activation time (X₂), and sodium hydroxide concentration (X₃) [14]. Their experimental domain is given in Table 1. These boundaries were defined based on the screening results, particularly the marked positive effect of the activating agent concentration. These three factors were subsequently studied by means of a full factorial design.

Table 1.Experimental domain of the full factorial design.

The design implemented is thus a duplicated three-factor full factorial design (2³), corresponding to 16 experimental runs.

The resulting first-degree polynomial model from this design is given by Equation (1).

Here, a i j denotes the coefficients of the two-factor interactions. Response surface methodology was then applied through a central composite design (CCD) in order to model the removal rate and locate the optimum. A CCD combines three groups of runs: the 2^k points of a two-level factorial design (k being the number of factors), the axial (star) points, and replicated center points. It leads to the second-order polynomial of Equation (2):

The a i i coefficients account for the curvature of the response (quadratic effects). Minitab 19 software was used to build the design matrices, estimate the regression coefficients and compute the statistical values for the different experimental designs.

Each coefficient was tested by multiple linear regression and the overall model by analysis of variance (ANOVA) at the 5% significance level; terms with p > 0.05 were discarded. A model was regarded as adequate when its residual sum of squares was lower than one third of the regression sum of squares [16]. The quality of fit was further judged from R² and adjusted R² [17] and from Fisher’s F-test [18]. Predictive accuracy was expressed through the coefficient of variation (CV), defined as:

where n is the number of runs, and R_exp and R_cal are the measured and predicted removal rates, respectively. A CV below 5% was taken as the acceptance criterion [19]. The optimum was located from the contour and three-dimensional response surface plots and then refined with the desirability function of Minitab 19.

The efficiency of each material was assessed through batch adsorption runs. In each run, 1 g of biosorbent was shaken with 25 mL of a 50 mg/L synthetic phosphate solution in a 100 mL Erlenmeyer flask. The pH was set at 4.7 with 1 N H₂SO₄ or 1 M NaOH, following [3]. The suspensions were stirred magnetically at 300 rpm for 2 h at room temperature and then centrifuged at 4000 rpm for 12 min. The residual phosphate concentration in the supernatant was measured by UV-Visible spectrophotometry at 880 nm, and the removal rate was calculated as follows:

R: phosphate ion removal rate (%)

C₀: initial phosphate ion concentration (mg/L)

C_e: residual phosphate ion concentration (mg/L)

Figure 1 reports the removal rate R (%) obtained for the eight factorial combinations, each run in duplicate.

Removal varied from 66.2% (run 1) to 85.0% (run 8). The lowest response is higher than the lowest one recorded during the screening stage, which indicates that the biosorbent was improved. The mean and median of the responses (76.8% and 79.3%, respectively) are close to each other, which points to a roughly symmetrical distribution of the data [20].

Figure 1.Results of the full factorial design.

Regression of the factorial data in Minitab gave the following linear model, where R₁ is the predicted removal rate:

The ANOVA of the factorial model is reported in Table 2.

Table 2.Analysis of variance for the full factorial design.

With a p-value of 0.021, below the 0.05 threshold, the ANOVA in Table 2 shows that the linear model is significant as a whole; in other words, the factors studied do affect the removal efficiency [15].

However, closer examination of the other statistical parameters reveals that the linear model cannot be validated, in accordance with the methodological criteria reported by [3][21]. Indeed, although the model sum of squares is relatively high (632.249), the error sum of squares remains substantial (150.845), reflecting non-negligible dispersion of the residuals [3]. This situation indicates that the model fails to satisfactorily explain the variability observed in the experimental data. Moreover, the presence of a low pure error (3.411) compared to the total error suggests that the observed discrepancy is mainly due to model inadequacy rather than random experimental error [21]. This confirms that the linear structure of the model does not correctly describe the actual behavior of the system studied [11]. Furthermore, the coefficients of determination shown in the table reinforce this conclusion. While R² is 80.74%, meaning that about four fifths of the total variability is accounted for by the model, the adjusted R² drops to 63.88%, revealing a significant loss of explanatory power once the number of factors and interactions is accounted for [11]. This marked difference between R² and adjusted R² reflects over-parameterization and insufficient predictive capacity of the model [22]. Thus, despite apparent statistical significance, the first-degree model shows major limitations in terms of adequacy and robustness. In line with recommendations from the literature [23], these results justify rejecting the linear model and highlight the need to resort to a second-order model [3] based on a response surface design, in order to better capture the joint and non-linear effects of the factors. Prior to building this second-order model, the relative weight of each factor and interaction was examined with a pareto chart (Figure 2), so as to single out the variables that actually govern removal efficiency.

Figure 2. Pareto chart of the standardized effects of the three factors and their interactions.

As can be seen in Figure 2, over 80% of the variability in the response is attributable to the activation time (X₂), the sodium hydroxide concentration (X₃), and the interaction between NaOH concentration-activation temperature-duration (X₁·X₂·X₃). The most influential factor is the activation time (X₂). Next came the normality and the interaction between NaOH concentration-activation temperature-duration (X₁·X₂·X₃), with respective contributions of 30.0% and 8.0% [11]. The contributions of temperature (X₁), the interaction between activation temperature and activation time (X₁·X₂), and the interaction between activation time and normality (X₂·X₃) each contribute less than 15%. The main effects plot (Figure 3) shows the direction and magnitude of the effect of each factor and indicates which level favors phosphate removal.

Figure 3.Main effects plot of the factors.

As shown in Figure 3, removal efficiency rises with normality (X₃): the mean removal rate goes from about 74% to 80% when the NaOH concentration is raised from 0.5 N to 2.5 N, i.e. a gain of about 6 percentage points. Extending the activation time (X₂) from 3 h to 6 h has an even stronger effect, the mean removal rate increasing from about 69% to 82%. Both parameters therefore improve the performance of the biosorbent. A longer and more concentrated alkaline treatment is likely to hydrolyze a larger fraction of lignin and hemicellulose, thereby opening up the porosity and generating oxygen-bearing functional groups that are known to participate in phosphate binding [14]. Similar trends have been reported for activated biosorbents [11] and for activated clays [3]. In contrast, the Pareto chart (Figure 2) shows that activation temperature (X₁) has no statistically significant effect, and Figure 3 even suggests a slight decrease in removal at its high level. Consequently, only activation time and NaOH concentration were kept for the next step. The second-order model used to describe and optimize the response is presented in the following section.

The CCD was built with two factors (k = 2), namely NaOH concentration (X₁) and activation time (X₂). It comprised 13 runs distributed as follows:

4 factorial runs (2²);

4 axial (star) runs (2k);

5 replicated runs at the center of the domain.

With α = (2k)^1/4 = 1.414, the coded levels of each factor range from −1.414 to +1.414, which corresponds to 1 - 3 N for NaOH concentration and 4 - 8 h for activation time. The measured responses R₂ (Exp), the values predicted by the model R₂ (Cal) and the corresponding coefficients of variation are listed in Table 3.

Table 3.Matrix, experimental design, removal rate, and coefficients of variation for the central composite design.

All CCD runs gave removal rates above 80% (Table 3), which confirms the improvement achieved after the factorial stage. Figure 4 presents the distribution of residuals as a function of fitted values. The residuals appear randomly distributed around zero, with no systematic trend.

No residual exceeds 2 in absolute value, showing close agreement between measured and predicted responses. Fitting the CCD data gave the second-order model of equation (6), in which R₂ is the predicted removal rate:

In this equation, the coefficients express the individual, interaction and quadratic effects of the factors. The ANOVA used to validate the model is summarized in Table 4.

Figure 4. Distribution of residuals versus fitted values.

Table 4.Analysis of variance for the central composite design.

In line with the criteria of [24], the residual sum of squares (6.800) is far below the regression sum of squares (232.664), and the model is highly significant (p < 0.001). The lack-of-fit test is not significant (p = 0.383 > 0.05), so the quadratic form describes the data adequately. Pure error (3.411) represents only 1.42% of the total sum of squares and can therefore be neglected [25]. R² (97.16%) and adjusted R² (95.13%) are both high and close to each other, which shows good agreement between measured and predicted values and rules out over-fitting. Finally, the mean CV (0.72%) is well below the 5% limit. Together, these results validate the proposed quadratic model [24]-[26].

The optimum was first sought graphically from the contour plot in Figure 5, which shows that removal rates above 95% can be achieved without leaving the experimental domain. The optimal region corresponds to combinations of NaOH concentration (X₁) and activation time (X₂) that place the response on or beyond the 99% contour (darkest blue zone). The problem therefore consists in finding the coded values of X₁ and X₂, within the design limits, that maximize R₂ as described by the second-degree polynomial.

Figure 5.Contour and response surface plots.

These coded values were then calculated with the desirability function of Minitab (Figure 6), the target being the maximum phosphate removal rate.

Figure 6.Desirability function.

According to Figure 6, a desirability of 1 is reached for a predicted removal rate of 99.67%. This optimum lies at −1.414 for NaOH concentration and +1.414 for activation time in coded units, which, after conversion to real units, gives 1 N NaOH and 8 h of activation. The 95% confidence interval of this prediction ranges from 94% to 100% (Table 5). Since the optimum lies at the boundary of the experimental domain, slightly better conditions outside the studied range cannot be excluded. Three additional (control) runs were performed under these conditions.

Table 6 gives the results of the confirmation runs.

Table 5.Response prediction.

Table 6.Control test results under optimal biosorbent preparation conditions.

The three control tests gave a mean removal rate of 98.23% (standard deviation 0.90). The difference from the predicted value (99.67%) is 1.44 percentage points, well within the 5% tolerance, which validates the model. This level of performance agrees with recent biosorption studies: comparable removal efficiencies, for various pollutants, have been reported for biosorbents prepared from Saba senegalensis hulls [11], argan shells [27], green walnut hulls [28], cocoa pod husks [29] and waste perlite [30], and they are of the same order as those of activated carbons [31][32]. This high efficiency results both from the statistical optimization of the preparation conditions [21][33][34] and from the NaOH activation step, which enhanced the adsorption capacity of the material.