3.1. FCC Optimization Results
The experimental data for both responses (shown in
Table 2) were statistically analyzed by analysis of variance, and the significance of the independent variables was evaluated based on their
p-value. As shown from ANOVA results (
Table 3), in the case of Y
1− Conversion time, the independent variables X
A-Pressure, X
B-Fe(II), and X
C-Ni(II), along with the interactions X
AX
B, X
AX
D, X
CX
D, and the quadratic term X
C2, were highly significant, with
p-values < 0.001. Additionally, most of the other interactions and quadratic terms demonstrated statistical significance (
p < 0.05), except the concentration of Co, which was not statistically significant. Conversely, for Y
2, the independent variables X
A-Pressure, X
B-Fe(II), and X
D-Co(II), together with the interactions X
AB and X
AD, were significant. Furthermore, the interaction X
BD and the quadratic terms A
2 and B
2 were also statistically significant (
p < 0.05), while the concentration of Ni was not statistically significant.
A second-order quadratic model was developed to examine the correlation between the responses and the independent variables, including trace element concentrations and applied pressure. The model’s acceptability was evaluated using the
p-value and F-test criteria [
15], and both adequate precision and the Coefficient of Variation were within acceptable limits. The resulting Multiple Linear Regression (MLR) models, presented below, provide predictive capability for the response values based on the specified independent variables.
For Y
1 (conversion time), the data were transformed using a power algorithm to ensure the validity of the ANOVA assumptions, including normality and independence of the data distribution. The developed regression model exhibited an F-test value of 75.23 and a
p-value < 0.0001, indicating statistical significance. The quadratic model’s R² value (0.9458) fell within the desirable range (0.8–1), confirming the model’s strong predictive accuracy [
39]. Additionally, the difference between the adjusted R² (0.9458) and predicted R² (0.9364) values was <0.2, ensuring consistency in predictive capability. Other metrics, such as an adequate precision value of 31.78 and a Coefficient of Variation of 8.39%, further validated the model’s reliability.
For Y
2 (conversion rate), the quadratic model demonstrated the best fit, as indicated with a
p-value < 0.0001 and an F-test value of 59.97. The model’s adjusted R² (0.9204) and predicted R² (0.8993) values showed a variance <0.2, signifying robust predictive performance. The adequate precision value (27.78) and the Coefficient of Variation (2.10%) further confirmed the model’s adequacy and reliability [
40]. These findings highlight the suitability of the quadratic model for investigating the correlation between conversion rate, trace element concentrations, and applied pressure.
The predicted values generated by the model exhibited a strong correlation with the actual experimental data, as demonstrated in the accompanying graphs in
Figure S1. This high degree of agreement underscores the reliability and robustness of the developed model in predicting system responses under the tested conditions. To evaluate the validity of the model assumptions, normal probability plots of the internally studentized residuals were generated (
Figure S2). Normal probability plots serve as essential diagnostic tools for assessing whether residuals adhere to a normal distribution. These plots compare the observed data with theoretical normal distribution values by plotting data points against their expected positions. Ideally, when errors follow a normal distribution, the data points align closely along a straight line. Significant deviations from this line indicate potential departures from normality, suggesting issues such as skewness, non-independence, or heteroscedasticity. In this study, the normal probability plots (
Figure S2) revealed minimal deviation from the straight line, confirming that the residuals were normally distributed. Furthermore, the residuals exhibited independence and homoscedasticity, as no discernible patterns or outliers were observed. These results validate the underlying assumptions of the model, ensuring its statistical rigor and applicability for predicting conversion rates and times within the defined experimental range. The alignment between predicted and experimental values, coupled with the robust diagnostic evaluation of residuals, highlights the accuracy and reliability of the modeling approach. This outcome confirms the suitability of the model for process optimization and predictive performance in biological methanation systems [
30].
The perturbation plots in
Figure 2 provide valuable insights into the influence of independent variables on the observed responses. In these plots, variables exhibiting steeper slopes or more pronounced curves have the most significant impact on the responses. In
Figure 2a, pressure (X
A, green line) demonstrated the most substantial influence on conversion time. This observation highlights the pivotal role of pressure in accelerating the biomethanation process. Increasing pressure enhances the gas–liquid mass transfer of H
2, which significantly reduces the reaction time by improving substrate availability at the microbial interface. This phenomenon has been corroborated in previous studies, which also reported that elevated pressure facilitates the conversion of CO
2 and H
2 into CH
4 through improved gas dissolution and diffusion [
41]. Furthermore, there is an additional benefit of Fe(II) and Ni(II) in the biomethanation process, since the increase in their concentration decreases the required time of the reaction.
Figure 2a illustrates the positive contributions of iron (X
B) and nickel (X
C) concentrations in the biomethanation process. An increase in Fe(II) and Ni(II) concentrations was found to decrease the reaction time, underscoring their critical role as essential trace elements in methane formation pathways. Iron, as a cofactor for key enzymes such as hydrogenases, facilitates electron transfer processes, while nickel serves as a fundamental component in the active sites of enzymes involved in methanogenesis, such as methyl-coenzyme M reductase [
42,
43,
44,
45].
Figure 2b further demonstrates the synergistic effect of pressure (X
A, green line) on the conversion rate, where the increase in pressure notably enhanced conversion rate. This observation aligns with the well-documented relationship between higher gas–liquid mass transfer rates and improved methanation efficiency under elevated pressure conditions. Furthermore, the perturbation plot reveals that nickel concentration (X
C, blue line) exerts a significant influence on conversion rate, as evidenced by the steep curvature of the response. Nickel’s indispensable role in methanogenic pathways, particularly in methane synthesis, has been extensively recognized in the literature [
42,
43,
44,
45]. Cobalt concentration (X
D, orange line) exhibited a moderate influence on the conversion rate, reinforcing its known role as a cofactor in enzyme activity, particularly in energy metabolism and electron transfer reactions. In contrast, iron concentration (X
B, red line) showed a relatively smaller effect on both conversion time and rate, which contrasts with studies highlighting its positive contributions to methanogenesis [
7,
46,
47]. This discrepancy may be attributed to the experimental conditions, including nutrient balance and microbial adaptation, which could influence the extent of iron’s contribution to methanation.
Overall, the perturbation plots demonstrate the dominant role of pressure and nickel concentration in enhancing biomethanation efficiency, while also highlighting the supportive, yet comparatively smaller, contributions of iron and cobalt. These findings underscore the importance of optimizing trace metal supplementation alongside pressure conditions to achieve maximum methane production efficiency.
3.2. Response Surface Analysis
Response surface plots were employed to evaluate the effects of the interaction between the independent variables (X
A, X
B, X
C, and X
D) on the response variables. These relationships were visualized through three-dimensional response surface plots and two-dimensional contour plots. As shown in
Figure 3a, conversion time (Y
1) decreases notably when the iron concentration exceeds 43 mg/L, while pressure is maintained within a range of 1.0 to 1.8 bar. This trend highlights the synergistic effect of elevated iron concentrations and moderate pressures on reducing reaction time, likely due to enhanced enzyme activity and improved microbial metabolism. The interaction effects between cobalt and iron (
Figure S3c) and between cobalt and nickel (
Figure S3d) further corroborate these findings. Specifically, the data indicate that conversion time is minimized at iron concentrations above 30 mg/L and nickel concentrations within the range of 0.15 to 0.40 mg/L, regardless of the cobalt concentration. These observations underscore the limited influence of cobalt compared to iron and nickel in this experimental setup. In summary, the response surface analysis highlights the critical importance of optimizing iron and nickel concentrations, along with applied pressure, to minimize conversion time. These findings underscore the critical importance of optimizing iron and nickel concentrations to enhance biomethanation yield.
In contrast,
Figure 3b highlights a significant interaction between pressure (X
A) and Fe(II) concentration (X
B) on the conversion rate (Y
2). The conversion rate reaches its maximum value when pressure exceeds 1.5 bar and iron concentration is maintained above 20 mg/L. The maximum conversion rate was achieved at high pressure (above 1.5 bar) and high Fe(II) concentration (over 29 mg/L), underscoring the synergistic effects of these factors, as observed for each factor individually in previous studies [
29,
48,
49,
50]. This interaction underscores the importance of simultaneous optimization of pressure and iron levels to enhance substrate utilization efficiency and methane production.
Figure S4a,b further illustrate the effects of interactions between pressure and nickel (X
C) or cobalt (X
D) concentrations on Y
2. These results indicate that higher conversion rates can be achieved consistently when pressure is maintained above 1.5 bar, irrespective of variations in Ni(II) and Co(II) concentrations. This observation suggests that pressure plays a more dominant role in influencing conversion rates compared to nickel and cobalt under the tested conditions. Finally,
Figure S4c examines the interaction between cobalt and iron on conversion rate. The results reveal that optimal conversion rates are achieved when iron concentration exceeds 30 mg/L and cobalt concentration is maintained above 0.05 mg/L. These findings highlight the supporting role of cobalt in enhancing iron’s catalytic effect, which aligns with its recognized function as a cofactor in critical methanogenic enzymes. In conclusion, the response surface analysis emphasizes that pressure and iron concentration are the most influential factors for optimizing the conversion rate. While nickel and cobalt contribute to the process, their effects are secondary compared to the dominant influence of pressure and iron.
3.3. Optimization of the Independent Variables by RSM
A Desired Space (DSp) was developed using Design Expert
® by overlaying the contour plots to identify an optimal set of biomethanation pressure and nutrient concentrations that simultaneously maximize the conversion rate and minimize the conversion time. Low-limit criteria were established for both responses to determine these optimal conditions. Specifically, the target values were set as follows: 13.5 h ≤ Y
1 ≤ 24 h and 95% ≤ Y
2 ≤ 100%. These values were selected to achieve biomethane production under sustainable conditions, such as shorter operating times and moderate pressure levels [
10,
51,
52]. The overlay plot (
Figure 4) highlights a bright yellow region where all the imposed criteria were satisfied. Results indicated a Desired Space located within a moderate pressure range (X
A: 0.5–1.5 bar) and an iron concentration (X
B) of 1–25 mg/L, with additional constraints for nickel (X
C: 0.01–0.25 mg/L) and cobalt (X
D: 0.01–0.05 mg/L).
To evaluate the accuracy and predictive performance of the proposed model, a validation experiment was conducted using the same inoculum within this DSp. The optimal conditions selected to ensure process viability were a pressure (XA) of 1.25 bar, an iron concentration (XB) of 30.00 mg/L, a nickel concentration (XC) of 0.100 mg/L, and a cobalt concentration (XD) of 0.045 mg/L. Under these optimal conditions, the model predictions for the responses were: Y1 corresponding to 16.9 h and Y2 achieving 97.41%. Validation experiments conducted under the same conditions achieved a conversion rate of 95.10% and a conversion time of 17.3 h, yielding a prediction error of less than 5% for both responses. This high level of accuracy underscores the effectiveness of RSM in optimizing biomethanation parameters; it highlights its potential to streamline the process, delivering faster reaction times and improved efficiency in methane production under controlled conditions.
3.4. Development of ANN Model
Based on the FCC dataset (
Table 2), feed-forward Artificial Neural Networks (ANNs) with varying numbers of hidden neurons (ranging from 1 to 20) were constructed and trained using MATLAB 2024b to predict conversion rate and time as output variables, with pressure, Fe(II), Ni(II), and Co(II) concentrations as the four input variables. The training results (
Table S2) indicated that an ANN with seven neurons in the hidden layer achieved the lowest mean squared error (MSE) value (0.0006) and the highest R² value (0.9968) among all tested topologies. The optimal ANN architecture is depicted in
Figure 5a. The input layer consists of four neurons, each representing one of the independent variables investigated in the study: pressure and Fe(II), Ni(II), and Co(II) concentrations. These neurons act as channels to transfer the raw input data into the network for processing. The hidden layer contains seven neurons, each receiving weighted inputs from the input layer, and processes them using a nonlinear activation function (hyperbolic tangent sigmoid). In addition to the weighted inputs, each neuron incorporates a bias term, which serves as an adjustable parameter to enhance the network’s ability to fit the data by shifting the activation function. The hidden layer plays a crucial role in capturing complex, nonlinear interactions between the input and output variables. The output layer comprises two neurons corresponding to the response variables: conversion time and rate. These outputs are derived from the processed data in the hidden layer and are computed using a linear transfer function, ensuring accurate predictions of the target responses. For this optimal ANN (4–7–2), the MSE for the training, validation, and testing datasets was minimized at epoch 41, as shown in
Figure S5.
The regression performance for predicting conversion time and rate using normalized values is illustrated in
Figure 5b. The R² values for the training, validation, testing, and total datasets were 0.99835, 0.99785, 0.99691, and 0.99783, respectively, all of which were close to 1. These results demonstrate the high predictive accuracy of the developed ANN model.
The histogram presented in
Figure 6 illustrates the error distribution (i.e., the difference between target and predicted values) for the Training, Validation, and Test datasets used in the development of the Artificial Neural Network (ANN) model. The highest frequency of errors is concentrated near the zero-error region, reflecting the model’s high accuracy in most predictions. The tails of the histogram represent instances with larger prediction errors, which occur less frequently. The overall error distribution is relatively smooth and symmetrically centered around zero, indicating minimal bias in the model’s predictions. This balanced distribution further confirms the ANN model’s robustness and reliability in capturing the underlying patterns within the dataset.
Consequently, the optimized ANN (4-7-2) architecture, after rigorous training, validation, and testing, proved effective as a predictive function for modeling and forecasting the conversion time and rate of the biomethanation process. This model reliably incorporates the effects of pressure and trace metal concentrations as input variables, making it a valuable tool for process optimization and operational decision-making in biomethanation systems.
3.5. Optimization of the Independent Variables by ANN Coupled with GA
The best-developed ANN (4-7-2) architecture was employed as the fitness function within a genetic algorithm (GA) framework to identify the optimal combination of input variables for minimizing conversion time and maximizing conversion rate. The ANN fitness function provided values for Y
1 (conversion time) and Y
2 (conversion rate) based on iteratively adjusted upper and lower parameter boundaries.
Figure 7 depicts the evolution of fitness values across generations, demonstrating that the fitness value decreased steadily from generation G1 to G16 before plateauing for over 70 generations. This plateau indicated a lack of further crossover or mutation among variables (genes) capable of influencing the fitness value of the two responses. As illustrated in
Figure 7, the optimal conditions determined by the ANN-GA integration yielded a minimum conversion time of 15.8 h and a maximum conversion rate of 98.9%. These values were achieved at an optimal pressure of 1.5 bar and trace metal concentrations of Fe(II) = 25.00 mg/L, Ni(II) = 0.20 mg/L, and Co(II) = 0.02 mg/L, as summarized in
Table 4. Notably, the optimized value for Y
2 (conversion rate) was slightly higher than the predictions generated by the FCC-RSM model but required approximately 1 h less time, indicating a more efficient biomethanation process.
The integration of ANN as a fitness function within the GA framework, as implemented in this study, represents a novel approach for optimizing the biomethanation process. This method effectively combines the predictive capabilities of artificial neural networks with the optimization efficiency of genetic algorithms. It is the first application of its kind for optimizing biomethanation under elevated pressure conditions, alongside optimization of micro-nutrient (metal) concentrations. However, further experimental validation of the ANN-GA-recommended optimum conditions is necessary to assess the accuracy and reliability of the integrated model. Comparing the predicted values with the respective experimental data will provide critical insights into the practical applicability and robustness of this advanced optimization technique.
3.6. Comparison and Validation of RSM and ANN
In this study, both Response Surface Methodology (RSM) and Artificial Neural Network-Genetic Algorithm (ANN-GA) approaches were employed for modeling and optimization of conversion time and rate in the biomethanation process. These experiments were conducted under elevated pressure and controlled nutrient concentrations, using data derived from FCC design. RSM, a mathematical and graphical modeling technique, involves fitting a polynomial equation to the experimental data, while ANN utilizes interconnected processing nodes to model complex, nonlinear relationships between inputs and outputs.
As summarized in
Table S3, FCC-RSM produced two second-order quadratic models (one for each response, Equations (2) and (3), while the ANN generated a network with four input neurons, seven hidden neurons, and two output neurons to model and predict the two studied responses. Both the RSM regression models and the trained ANN demonstrated strong predictive accuracy. The fitting performance of these models, evaluated using statistical parameters, is compared in
Table S3. In general, the ANN model had significant better results as evidenced by its higher R² value and lower MSE and Root Mean Squared Error (RMSE) values compared to the respective values of the RSM model, suggesting slightly better overall prediction performance.
Figure 8 compares the predicted values from the FCC-RSM and ANN models with the actual experimental data, demonstrating the superior predictive performance of the ANN model. The deviations between the predicted and actual values were significantly smaller for both responses when using the ANN model compared to the FCC-RSM model. This finding aligns with conclusions reported in previous studies [
15,
53], which demonstrated that the ANN model consistently outperforms the RSM model in predicting biogas or methane production in batch anaerobic digestion experiments. In engineering and scientific applications, both artificial neural networks and response surface methodology are widely employed for modeling the relationships between input variables and output responses. Each approach offers distinct advantages and limitations, depending on the complexity of the problem under investigation. RSM is a robust and frequently used method for elucidating relationships between variables and identifying optimal conditions with a relatively small number of experiments. However, it is inherently limited to quadratic nonlinear coefficients and may not adequately represent more complex systems [
16,
21]. Conversely, ANN excels in capturing intricate, nonlinear relationships between parameters, offering superior predictive accuracy for complex datasets. Despite its advantages, ANN models often lack interpretability and typically require computational tools for their development and analysis [
13]. In conclusion, RSM is well suited for systems with straightforward mathematical relationships, while ANN is more appropriate for scenarios involving complex, nonlinear interactions. The selection of the appropriate modeling approach should thus be guided by the specific characteristics of the system under study and the research objectives.
Regarding the optimization results, RSM predicted a maximum conversion rate of 97.41% at a conversion time of 16.9 h. In contrast, ANN-GA predicted a slightly higher conversion rate of 98.9%, but at a significantly reduced conversion time of 15.8 h, under the optimal conditions of pressure = 1.5 bar, Fe(II) = 25.00 mg/L, Ni(II) = 0.200 mg/L, and Co(II) = 0.020 mg/L. Validation experiments conducted in triplicate for both optimization approaches confirmed these results, as shown in
Table 4. The prediction errors for RSM and ANN-GA were 2.31% and 0.63% for the time and −2.43% and −1.02% for the conversion rate, respectively, with negative values indicating that the actual measured results were lower than the model predictions. Both models exhibited low prediction errors (<2.5%), but the ANN-GA predictions were slightly more accurate overall, as indicated by the smaller prediction error (absolute value). This finding underscores the distinct characteristics and mechanisms of the two modeling approaches.
The validated minimum conversion time obtained using ANN-GA was 15.9 h, which is approximately 34% less than the average conversion time of the full dataset, indicating a significant enhancement in the efficiency of the biomethanation process. This improvement is attributed to the interaction between the optimized independent variables, including pressure and trace metal concentrations. The mechanism of the improvement is related to the synergistic effects of elevated pressure on enhancing hydrogen gas–liquid mass transfer, coupled with the catalytic role of trace metals such as iron and nickel in facilitating enzymatic reactions within the methanogenic pathway. Elevated pressure increases the concentration gradient between the gaseous and liquid phases, accelerating substrate availability for hydrogenotrophic methanogens. Concurrently, trace metals such as Fe(II) and Ni(II) serve as cofactors in key enzymes, including hydrogenases and methyl-coenzyme M reductase, which are essential for efficient methane synthesis. This dual effect highlights the importance of optimizing both physical and biochemical parameters to achieve superior process performance.
These results highlight the potential of advanced computational modeling and optimization techniques to significantly enhance biomethanation efficiency, paving the way for improved methane production processes.
