# Neural Network Modeling for the Extraction of Rare Earth Elements from Eudialyte Concentrate by Dry Digestion and Leaching

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{4}(Ca, Ce, Fe)

_{2}ZrSi

_{6}O

_{17}(OH, Cl)

_{2}, but the component of the mineral may be sometimes different because of the zeolite crystal structure, which possesses ion-exchange properties of eudialyte [20,21].

_{2}SO

_{4}. It was found that dry digestion with high concentrated HCl benefits the extraction of REE and prevents gel formation. The recommended concentration of HCl to decompose the eudialyte concentrate is 10 mol/L [24].

## 2. Theoretical Background

#### 2.1. Multiple Linear Regression

#### 2.2. Stepwise Regression

#### 2.3. Design of Experiments

#### 2.4. Artificial Neural Network

## 3. Materials and Methods

#### 3.1. Material and Analysis

#### 3.2. Extraction Procedure

## 4. Experiments and Obtained Measurements of Process Outputs

## 5. Model Setup and Results

#### 5.1. Process Modeling in MLR Form

- (a)
- first order multiple linear regression$$y={b}_{0}+{\displaystyle \sum _{i=1}^{k}{b}_{i}}\xb7{x}_{i}$$
- (b)
- first order multiple linear regression with interaction effects$$y={b}_{0}+{\displaystyle \sum _{i=1}^{k}{b}_{i}}\xb7{x}_{i}+{\displaystyle \sum _{i=1}^{k-1}{\displaystyle \sum _{j=i+1}^{k}{b}_{ij}}\xb7{x}_{i}\xb7{x}_{j}+{\displaystyle \sum _{i=1}^{k-2}{\displaystyle \sum _{j=k+1}^{k-1}{\displaystyle \sum _{l=j+1}^{k}{b}_{ijl}\xb7}{x}_{i}\xb7{x}_{j}\xb7{x}_{k}}}}$$

_{2}O (variable $c{\mathrm{H}}_{2}\mathrm{O}$) rise, and that the maximum values of extraction are around 90%. Furthermore, TREE extraction efficiency increases with the extension of time from 20 min. (Figure 7a), to 60 min. (Figure 7b). Although the model is in the form of a hyperplane (7), i.e., (8), adequate (${F}_{rLF}=3.9<{F}_{t}=4$), it can be observed that the process is characterized by high levels of noise, which is indicated by the standard error ($s(y)=3.458072$). For this standard error value, it is relatively easy to perform proof of adequacy. In addition to the above, although the model (8) is adequate, its accuracy is very low, as can be seen from the value of its parameters: the coefficient of multiple determination (${R}^{2}=0.433$), the adjusted coefficient of multiple determination (${R}_{adj}^{2}=0.244$), and root mean squared error, which is calculated in total for all of the experiments, is $RMSE=4.668$.

#### 5.2. Modeling with Stepwise Regression

^{2}= 0.194481, 0.411

^{2}= 0.168921). This further extends the understanding of the significance of the observed regressors, and the sum of Part usually differs from ${R}_{adj}^{2}$ due to existence of joint effects.

#### 5.3. Modeling with ANN

#### 5.3.1. ANN Modeling of TREE Extraction Based on LOO CV

_{k}

_{=1}(4a) and Full ANN

_{k}

_{=2}(5a) were created on the basis of all 17 input/output pairs, where the 17th pair corresponds to the average measurement at the central point of the experimental plan (A17, Table 2). These models represent ANN models, which selected network architectures can generate. In addition, both network architectures (4) and (5) generate 16 ANN models (N − 1 = 16), based on the LOO CV strategy. Leave-one-out Cross validation is a model validation method, which is described in detail in [48]. For very sparse datasets, as in our research, LOO CV can be used in order to train the ANN on the largest possible number of examples. For a dataset with N examples, N ANN models are trained. For each model, N − 1 examples are used for learning (training) and the remaining example for ANN testing (Figure 11).

_{k}

_{=1}(4b) and LOO CV ANN

_{k}

_{=2}(5b) each generated 16 ANN models, as mentioned earlier. Therefore, for ANN LOO CV modeling, only one sample is in the test dataset, but its exclusion from the training dataset significantly influences the ANN model’s form. Due to space, as an example, Figure 12 shows only a few ANN models that were obtained during ANN LOO CV modeling.

#### 5.3.2. Simulation and Prediction Based on ANN Models

## 6. Validation of the Model and Scale-up

#### 6.1. Validation of REE Extraction Model and Obtained Optimal Process Regimes

_{ANN}= 1.65. The obtained results confirmed the adequacy of the adopted ANN model, which was the basis for selecting the optimal regime of the REE extraction technological procedure on the scale-up level.

#### 6.2. Dry Digestion and Leaching Process Scale-Up

## 7. Discussion

#### 7.1. REE Extraction Modeling and Optimal Regimes

#### 7.2. Phase Changes during REE Extraction

## 8. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 4.**QEMSCAN analysis of eudialyte concentrate ((

**left**) false color image after evaluation, (

**right**) initial BSE-image).

**Figure 6.**Proposed flowchart of the treatment of rare earth elements (REE) extraction from eudialyte concentrate.

**Figure 7.**TREE

_{eff.}= f(cHCl, cH

_{2}O) for (

**a**) T = const. = 20 °C, t = const. = 20 min.; and, (

**b**) T = const. = 20 °C, t = const. = 60 min.

**Figure 8.**(

**a**) Total rare earth elements (TREE) extraction model; and, (

**b**) Predicted by Observed plot for TREE

_{eff}.

**Figure 12.**Examples of ANN models obtained during the leave-one-out cross validation (LOO CV) process: (

**a**) ANN-4/3/2/1 model: excluded sample A4; (

**b**) ANN-4/3/2/1 model: excluded sample A5; (

**c**) ANN-4/3/2/1 model: excluded sample A10; (

**d**) ANN-4/10/1 model: excluded sample A2; (

**e**) ANN-4/10/1 model: excluded sample A10; and, (

**f**) ANN-4/10/1 model: excluded sample A16.

**Figure 13.**Final ANN data table numerical model for TREE extraction: (

**a**) T = 20 °C = const., t = 60 min. = const.; (

**b**) T = 20 °C = const., cH

_{2}O = 1.5 L/kg = const.

**Figure 14.**Treatment of eudialyte concentrate in two reactors at the Institute of Process Metallurgy and Metal Recycling (IME) demonstration plant, RWTH Aachen.

Element | Content (wt %) | Element | Content (wt %) |
---|---|---|---|

Al | 3.2 | Ce | 0.52 |

Ca | 5.7 | Pr | 532 mg/kg |

Fe | 6.04 | Nd | 0.21 |

Mn | 0.39 | Sm | 440 mg/kg |

Nb | 0.36 | Gd | 239 mg/kg |

Zr | 5.08 | Dy | 580 mg/kg |

Hf | 0.11 | Y | 0.33 |

Si | 23.1 | Yb | 368 mg/kg |

La | 0.25 | TREE | 1.52 |

No. | X1:HCl:Con. (L:kg) | X2:Water:Con. (L:kg) | X3:Leaching Temp. (°C) | X4:Leaching Time (min.) | Y:TREE Extr. (%) | ||||
---|---|---|---|---|---|---|---|---|---|

1:1, 1.25:1, 1.5:1 | 1:1, 2:1, 3:1 | 20, 50, 80 | 20, 40, 60 | [0–100] | |||||

c.v. | a.v. | c.v. | a.v. | c.v. | a.v. | c.v. | a.v. | ||

A1 | 1 | 1.5:1 | 1 | 3:1 | 1 | 20 | 1 | 60 | 93.475 |

A2 | −1 | 1:1 | 1 | 3:1 | 1 | 20 | 1 | 60 | 79.640 |

A3 | 1 | 1.5:1 | −1 | 1:1 | 1 | 20 | 1 | 60 | 82.010 |

A4 | −1 | 1:1 | −1 | 1:1 | 1 | 20 | 1 | 60 | 73.940 |

A5 | 1 | 1.5:1 | 1 | 3:1 | −1 | 20 | 1 | 60 | 86.465 |

A6 | −1 | 1:1 | 1 | 3:1 | −1 | 20 | 1 | 60 | 79.070 |

A7 | 1 | 1.5:1 | −1 | 1:1 | −1 | 20 | 1 | 60 | 81.170 |

A8 | −1 | 1:1 | −1 | 1:1 | −1 | 20 | 1 | 60 | 79.450 |

A9 | 1 | 1.5:1 | 1 | 3:1 | 1 | 20 | −1 | 20 | 85.350 |

A10 | −1 | 1:1 | 1 | 3:1 | 1 | 20 | −1 | 20 | 80.630 |

A11 | 1 | 1.5:1 | −1 | 1:1 | 1 | 20 | −1 | 20 | 76.940 |

A12 | −1 | 1:1 | −1 | 1:1 | 1 | 20 | −1 | 20 | 71.140 |

A13 | 1 | 1.5:1 | 1 | 3:1 | −1 | 20 | −1 | 20 | 77.650 |

A14 | −1 | 1:1 | 1 | 3:1 | −1 | 20 | −1 | 20 | 75.525 |

A15 | 1 | 1.5:1 | −1 | 1:1 | −1 | 20 | −1 | 20 | 88.600 |

A16 | −1 | 1:1 | −1 | 1:1 | −1 | 20 | −1 | 20 | 80.090 |

A17 * | 0 | 1.25:1 | 0 | 2:1 | 0 | 50 | 1 | 40 | 89.49, 87.99, 88.74, 90.85, 86.63, 94.35, 83.24 |

**Table 3.**Multiple Linear Regression (MLR) models, regression coefficients and their 95% confidence interval, F-test values, as well as the standard deviation of residuals σ

_{res}.

First-Order MLR Model: $\mathit{y}\mathbf{=}{\mathit{\beta}}_{\mathbf{0}}\mathbf{+}{\mathit{\beta}}_{\mathbf{1}}\mathbf{\xb7}{\mathit{x}}_{\mathbf{1}}\mathbf{+}{\mathit{\beta}}_{\mathbf{2}}\mathbf{\xb7}{\mathit{x}}_{\mathbf{2}}\mathbf{+}{\mathit{\beta}}_{\mathbf{3}}\mathbf{\xb7}{\mathit{x}}_{\mathbf{3}}\mathbf{+}{\mathit{\beta}}_{\mathbf{4}}\mathbf{\xb7}{\mathit{x}}_{\mathbf{4}}$ | |||
---|---|---|---|

${\beta}_{0}$ | $83.144565$ | $[81.378-84.911]$ | ${F}_{t}({f}_{R},{f}_{E},\alpha )={F}_{t}(12,6,0.05)=4$ ${F}_{rLF}=3.90$ ${s}^{2}(y)=11.958262;\text{}s(y)=3.458072$ ${{s}^{2}}_{LF}=46.61679$ ${F}_{rLF}<{F}_{t}\text{}is\text{}adequate$, ${\sigma}_{res}=6.8276488$ |

${\beta}_{1}$ | $3.2609375$ | $[1.143-5.379]$ | |

${\beta}_{2}$ | $1.5290625$ | $[-0.589-3.647]$ | |

${\beta}_{3}$ | $-0.3059375$ | $[-2.424-1.812]$ | |

${\beta}_{4}$ | $1.2059375$ | $[-0.912-3.324]$ |

Coefficients ^{a} | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Model | Unstandard. Coeff. | Standard. Coeff. | t | Sig. | 95.0% Confidence Interval for B | Correlations | Collinearity Statistics | ||||||

B | Std. Error | Beta | Lower Bound | Upper Bound | Zero-Order | Partial | Part | Tolerance | VIF | ||||

1 | (Constant) | 64.866 | 6.293 | 10.307 | 0.000 | 51.452 | 78.280 | ||||||

$c\mathrm{HCl}$ | 13.044 | 4.943 | 0.563 | 2.639 | 0.019 | 2.509 | 23.579 | 0.563 | 0.563 | 0.563 | 1.000 | 1.000 | |

2 | (Constant) | 64.866 | 5.651 | 11.478 | 0.000 | 52.746 | 76.986 | ||||||

$c\mathrm{HCl}$ | 10.554 | 4.587 | 0.456 | 2.301 | 0.037 | 0.715 | 20.393 | 0.563 | 0.524 | 0.441 | 0.936 | 1.068 | |

$c\mathrm{HCl}\xb7c{\mathrm{H}}_{2}\mathrm{O}\xb7t$ | 0.031 | 0.015 | 0.425 | 2.146 | 0.050 | 0.000 | 0.062 | 0.540 | 0.497 | 0.411 | 0.936 | 1.068 |

^{a}Dependent Variable: TREE

_{eff.}[%].

Parameter | Units | B1 | B2 | B3 | B4 |
---|---|---|---|---|---|

HCl:Concentrate | L:kg | 1.3:1 | 1.4:1 | 1.4:1 | 1.5:1 |

Water:Concentrate | L:kg | 2.25:1 | 2.25:1 | 2.5:1 | 3:1 |

Leaching time | min | 40 | 50 | 50 | 60 |

Predicted value | 85.35% | 87.90% | 89.40% | 89.40% | |

Actual value | 87.65% | 89.40% | 89.70% | 91.20% |

Element | Concentration (g/L) | Element | Concentration (g/L) |
---|---|---|---|

Al | 0.56 | Ce | 0.951 |

Ca | 10.49 | Pr | 0.105 |

Fe | 2.02 | Nd | 0.370 |

Mn | 0.269 | Sm | 0.094 |

Nb | <0.0001 | Gd | 0.075 |

Zr | 0.02 | Dy | 0.105 |

Hf | 0.010 | Y | 0.601 |

Si | 0.099 | Yb | 0.069 |

La | 0.425 | TREE | 2.80 |

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**MDPI and ACS Style**

Ma, Y.; Stopic, S.; Gronen, L.; Milivojevic, M.; Obradovic, S.; Friedrich, B.
Neural Network Modeling for the Extraction of Rare Earth Elements from Eudialyte Concentrate by Dry Digestion and Leaching. *Metals* **2018**, *8*, 267.
https://doi.org/10.3390/met8040267

**AMA Style**

Ma Y, Stopic S, Gronen L, Milivojevic M, Obradovic S, Friedrich B.
Neural Network Modeling for the Extraction of Rare Earth Elements from Eudialyte Concentrate by Dry Digestion and Leaching. *Metals*. 2018; 8(4):267.
https://doi.org/10.3390/met8040267

**Chicago/Turabian Style**

Ma, Yiqian, Srecko Stopic, Lars Gronen, Milovan Milivojevic, Srdjan Obradovic, and Bernd Friedrich.
2018. "Neural Network Modeling for the Extraction of Rare Earth Elements from Eudialyte Concentrate by Dry Digestion and Leaching" *Metals* 8, no. 4: 267.
https://doi.org/10.3390/met8040267