#
Development of a Mathematical Model Based on an Artificial Neural Network (ANN) to Predict Nickel Uptake Data by a Natural Zeolite^{ †}

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

_{4}and AlO

_{4}, and the isomorphic substitution of Si

^{+4}by Al

^{+3}provides a net negative charge on the framework surface, which is balanced by alkaline and alkali-earth metals, such as Na

^{+}, Ca

^{+}, K

^{+}, and Mg

^{+2}[4,7].

## 2. Materials and Methods

#### 2.1. Data Collection

#### 2.2. ANN Model

_{j}is the output variable, f is the transfer function, B

_{j}is the bias in the hidden layer, n is the number of neurons in the hidden layer, w

_{ji}is the connection weights between the input and hidden layers, and Xi is the input variable. To avoid overfitting or underfitting, the data were normalized in the scaled range of −1 to 1, using Equation (2) [11]:

_{nor}is the normalized data, and M

_{max}and M

_{min}are the maximum and minimum values of the scaling range, respectively. y

_{i}is the actual data. Max(y

_{i}) and Min(y

_{i}) are the maximum and minimum values of the actual data, respectively.

#### 2.3. ANN Optimization

## 3. Results

#### 3.1. ANN Performance

^{2}values for both the training and validation sets are above 90%, which indicates the high accuracy of the ANN model [14].

#### 3.2. Mathematical Model Development

_{0}is the bias in the output layer, n is the number of neurons in the hidden layer, w

_{k}is the connection weights between the hidden and output layers, f

_{sig}is the transfer function, b

_{nk}is the bias at each neuron in the hidden layer, m is the number of neurons in the input layer, w

_{ik}is the connection weights between the input and hidden layers, X

_{i}is the normalized input data, and y is the normalized output data.

_{n}is unknown and can be calculated using Equation (5):

_{n}is also unknown and can be calculated using Equation (6):

_{n-in p}and w

_{n-outp}are the connection weights in the input and output layers, respectively. The final equation used for predicting the nickel removal after de-normalizing the data is presented as follows:

## 4. Discussion

#### 4.1. ANN Validation

^{2}value of 0.98, indicating the validation and high accuracy of the model. Furthermore, the error between the output and the target is very low. Therefore, the model can be chosen as appropriate for predicting future data.

#### 4.2. Isotherm Prediction

^{2}of 0.993, and the lowest ARE and RMSE values. These results are very consistent with the experimental results. In addition, the predicted maximal adsorption capacity (Qm = 28.92) was very close to the experimental value (Qm = 28.79), with a standard deviation of−0.13. As a result, the developed ANN model was a valid and appropriate model for nickel-adsorption data prediction.

## 5. Conclusions

^{2}of 0.98, which indicates the high accuracy of the model. In addition, the model was tested for isotherm data prediction, where the prediction data were in agreement with the experimental data. The developed ANN model was accurate and appropriate for nickel-adsorption data prediction.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 7.**Isotherm plot with different models: (

**a**) isotherm plot of the original data (modified after [4]); (

**b**) isotherm plot of predicted data.

**Table 1.**Data ranges used for ANN model development (modified after [4]).

Input Data | Range | Output | ||
---|---|---|---|---|

Min | Max | Mean | Nickel removal (R %) | |

Initial concentration (mg/L) | 10 | 150 | 80 | |

Adsorbent dosage (mg/g) | 0.1 | 0.5 | 0.3 | |

pH | 3 | 6 | 4.5 |

n ^{1} | Weights | Biases | ||||
---|---|---|---|---|---|---|

Ic ^{2} | Ad ^{3} | pH | R (%) | b_{nk} | b_{0} | |

n = 1 | 2.53 | 1.63 | 0.87 | 0.09 | −3.26 | −0.18 |

n = 2 | 2.40 | 1.44 | −1.56 | −0.05 | −2.61 | |

n = 3 | 1.73 | 1.06 | −2.48 | −0.79 | −2.03 | |

n = 4 | 2.59 | 1.10 | −1.49 | 0.47 | −1.52 | |

n = 5 | −0.17 | −2.19 | −2.29 | 0.65 | 0.89 | |

n = 6 | 2.16 | −0.13 | 2.36 | −0.38 | −0.28 | |

n = 7 | −0.89 | 2.17 | 2.41 | 0.72 | −0.33 | |

n = 8 | −1.52 | −1.39 | −2.43 | −0.17 | −0.94 | |

n = 9 | −0.01 | −3.05 | 0.98 | −0.50 | −1.43 | |

n = 10 | −2.21 | −1.02 | −2.10 | 0.20 | −2.00 | |

n = 11 | 2.51 | 2.01 | −0.56 | −0.78 | 2.54 | |

n = 12 | −2.56 | −1.85 | 0.23 | −0.33 | −3.23 |

^{1}Neurons number.

^{2}Initial concentration.

^{3}Adsorbent dosage.

Isotherm Model | Isotherm Parameters | R^{2} | adjR^{2} | RMSE | ARE | ||
---|---|---|---|---|---|---|---|

Langmuir isotherm | Qm | K1 | 0.993 | 0.991 | 0.85 | 0.54 | |

28.92 (mg/g) | 0.08 (L/mg) | ||||||

Freundlich isotherm | Kf | nf | 0.94 | 0.92 | 2.47 | 0.81 | |

5.74 | 2.95 | ||||||

Redlich–Peterson isotherm | arp | krp | β | 0.996 | 0.993 | 0.74 | 0.50 |

0.15 | 3.05 | 0.91 |

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**MDPI and ACS Style**

Mehdi, B.; Brahmi-Ingrachen, D.; Belkacemi, H.; Muhr, L.
Development of a Mathematical Model Based on an Artificial Neural Network (ANN) to Predict Nickel Uptake Data by a Natural Zeolite. *Phys. Sci. Forum* **2023**, *6*, 4.
https://doi.org/10.3390/psf2023006004

**AMA Style**

Mehdi B, Brahmi-Ingrachen D, Belkacemi H, Muhr L.
Development of a Mathematical Model Based on an Artificial Neural Network (ANN) to Predict Nickel Uptake Data by a Natural Zeolite. *Physical Sciences Forum*. 2023; 6(1):4.
https://doi.org/10.3390/psf2023006004

**Chicago/Turabian Style**

Mehdi, Boukhari, Daouia Brahmi-Ingrachen, Hayet Belkacemi, and Laurence Muhr.
2023. "Development of a Mathematical Model Based on an Artificial Neural Network (ANN) to Predict Nickel Uptake Data by a Natural Zeolite" *Physical Sciences Forum* 6, no. 1: 4.
https://doi.org/10.3390/psf2023006004