# Grey Box Modelling of Decanter Centrifuges by Coupling a Numerical Process Model with a Neural Network

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory

#### 2.1. Grey Box Modelling

#### 2.2. Artificial Neural Networks

#### 2.3. Decanter Centrifuge Numerical Model and Experimental Procedure

## 3. Methodology

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ANN | Artificial Neural Network |

CFD | Computational Fluid Dynamics |

MSE | Mean of the sum of squared errors |

MSEREG | Regularised mean of the sum of squared errors |

MSW | Mean of the sum of squared weights |

RMSE | Root mean of the sum of squared errors |

## References

- Merkl, R.; Steiger, W. Properties of decanter centrifuges in the mining industry. Min. Metall. Explor.
**2012**, 29, 6–12. [Google Scholar] [CrossRef] - Stahl, W.H. Fest-Flüssig-Trennung. 2, Industrie-Zentrifugen. Maschinen-& Verfahrenstechnik; DrM Press: CH-Maennedorf, Switzerland, 2004. [Google Scholar]
- Venkatasubramanian, V. The promise of artificial intelligence in chemical engineering: Is it here, finally? AIChE J.
**2019**, 65, 466–478. [Google Scholar] [CrossRef] - McCoy, J.; Auret, L. Machine learning applications in minerals processing: A review. Miner. Eng.
**2019**, 132, 95–109. [Google Scholar] [CrossRef] - Duch, W.; Diercksen, G.H.F. Neural networks as tools to solve problems in physics and chemistry. Comput. Phys. Commun.
**1994**, 82, 91–103. [Google Scholar] [CrossRef] - Willis, M.J.; Di Massimo, C.; Montague, G.A.; Tham, M.T.; Morris, A.J. Artificial neural networks in process engineering. IEE Proc. D
**1991**, 138, 256–266. [Google Scholar] [CrossRef] - Gonzalez-Fernandez, I.; Iglesias-Otero, M.A.; Esteki, M.; Moldes, O.A.; Mejuto, J.C.; Simal-Gandara, J. A critical review on the use of artificial neural networks in olive oil production, characterization and authentication. Crit. Rev. Food Sci. Nutr.
**2019**, 59, 1913–1926. [Google Scholar] [CrossRef] - Funes, E.; Allouche, Y.; Beltrán, G.; Aguliera, M.P.; Jiménez, A. Predictive ANN models for the optimization of extra virgin olive oil clarification by means of vertical centrifugation. J. Food Process Eng.
**2018**, 41, e12593. [Google Scholar] [CrossRef] - Jiménez, A.; Beltrán, G.; Aguilera, M.P.; Uceda, M. A sensor-software based on artificial neural network for the optimization of olive oil elaboration process. Sens. Actuators B Chem.
**2008**, 129, 985–990. [Google Scholar] [CrossRef] - Jiménez Marquez, A.; Aguilera Herrera, M.; Uceda Ojeda, M.; Beltrán Maza, G. Neural network as tool for virgin olive oil elaboration process optimization. J. Food Eng.
**2009**, 95, 135–141. [Google Scholar] [CrossRef] - Thompson, M.L.; Kramer, M.A. Modeling chemical processes using prior knowledge and neural networks. AIChE J.
**1994**, 40, 1328–1340. [Google Scholar] [CrossRef] - Menesklou, P.; Nirschl, H.; Gleiss, M. Dewatering of finely dispersed calcium carbonate-water slurries in decanter centrifuges: About modelling of a dynamic simulation tool. Sep. Purif. Technol.
**2020**, 251, 117287. [Google Scholar] [CrossRef] - Gleiss, M.; Hammerich, S.; Kespe, M.; Nirschl, H. Development of a dynamic process model for the mechanical fluid separation in decanter centrifuges. Chem. Eng. Technol.
**2018**, 41, 19–26. [Google Scholar] [CrossRef] [Green Version] - Menesklou, P.; Sinn, T.; Nirschl, H.; Gleiss, M. Scale-Up of Decanter Centrifuges for The Particle Separation and Mechanical Dewatering in The Minerals Processing Industry by Means of A Numerical Process Model. Minerals
**2021**, 11, 229. [Google Scholar] [CrossRef] - Hammerich, S.; Gleiß, M.; Nirschl, H. Modeling and Simulation of Solid-Bowl Centrifuges as an Aspect of the Advancing Digitization in Solid-Liquid Separation. ChemBioEng Rev.
**2019**, 6, 108–118. [Google Scholar] [CrossRef] - Psichogios, D.C.; Ungar, L.H. Direct and indirect model based control using artificial neural networks. Ind. Eng. Chem. Res.
**1991**, 30, 2564–2573. [Google Scholar] [CrossRef] - Pitarch, J.L.; Sala, A.; de Prada, C. A Systematic Grey-Box Modeling Methodology via Data Reconciliation and SOS Constrained Regression. Processes
**2019**, 7, 170. [Google Scholar] [CrossRef] [Green Version] - Henrique, H.M.; Lima, E.L.; Seborg, D.E. Model structure determination in neural network models. Chem. Eng. Sci.
**2000**, 55, 5457–5469. [Google Scholar] [CrossRef] - Hornik, K.; Stinchcombe, M.; White, H. Multilayer feedforward networks are universal approximators. Neural Netw.
**1989**, 2, 359–366. [Google Scholar] [CrossRef] - Leung, W.W.F. Industrial Centrifugation Technology; McGraw-Hill: New York, NY, USA, 1998. [Google Scholar]
- Wakeman, R.J.; Tarleton, S. Solid/Liquid Separation: Scale-Up of Industrial Equipment, 1st ed.; OCLC: 254305213; Elsevier: Oxford, UK, 2005. [Google Scholar]
- Records, A.; Sutherland, K. Decanter Centrifuge Handbook; Elsevier: Oxford, UK, 2001. [Google Scholar] [CrossRef]
- Hagan, M.; Menhaj, M. Training feedforward networks with the Marquardt algorithm. IEEE Trans. Neural Netw.
**1994**, 5, 989–993. [Google Scholar] [CrossRef] - Foresee, F.D.; Hagan, M. Gauss-Newton approximation to Bayesian learning. In Proceedings of the International Conference on Neural Networks (ICNN’97), Houston, TX, USA, 12 June 1997; Volume 3, pp. 1930–1935. [Google Scholar] [CrossRef]
- MacKay, D.J.C. Bayesian Interpolation. Neural Comput.
**1992**, 4, 415–447. [Google Scholar] [CrossRef] - Snoek, J.; Larochelle, H.; Adams, R.P. Practical Bayesian Optimization of Machine Learning Algorithms. arXiv
**2012**, arXiv:1206.2944. [Google Scholar] - Leung, W.W.F. Inferring in-situ floc size, predicting solids recovery, and scaling-up using the Leung number in separating flocculated suspension in decanter centrifuges. Sep. Purif. Technol.
**2016**, 171, 69–79. [Google Scholar] [CrossRef]

**Figure 4.**Solids mass fraction of the cake for different rotational speeds, and variation of cake trust interval at a constant volumetric flow rate of ${\dot{V}}_{\mathrm{in}}=2000\mathrm{L}{\mathrm{h}}^{-1}$ for CC1.

**Figure 5.**Solids mass fraction of the centrate for different volumetric flow rates and variation of the centrate trust interval at a constant rotational speed of ${n}_{\mathrm{rot}}=2950\mathrm{rpm}$ for CC2.

**Figure 6.**Variation number of neurons for the relative correction parameter of the centrate at a constant volumetric flow rate of ${\dot{V}}_{\mathrm{in}}=2000\mathrm{L}{\mathrm{h}}^{-1}$ for the product CC1.

**Figure 7.**Solids mass fraction of the cake at a constant volumetric flow rate of ${\dot{V}}_{\mathrm{in}}=2000\mathrm{L}{\mathrm{h}}^{-1}$ for the product CC1: (

**a**) relativised output of the black box model, (

**b**) comparison of the results from experiment, white, and grey box model.

**Figure 8.**Solids mass fraction of the centrate at a constant volumetric flow rate of ${\dot{V}}_{\mathrm{in}}=2000\mathrm{L}{\mathrm{h}}^{-1}$ for the product CC1: (

**a**) relativised output of the black box model, (

**b**) comparison of the results from experiment, white, and grey box model.

**Figure 9.**Solids mass fraction of the cake at a constant rotational speed of ${n}_{\mathrm{rot}}=2950\mathrm{rpm}$ for the product CC2: (

**a**) relativised output of the black box model, (

**b**) comparison of the results from experiment, white, and grey box model.

**Figure 10.**Solids mass fraction of the centrate at a constant rotational speed of ${n}_{\mathrm{rot}}=2950\mathrm{rpm}$ for the product CC2: (

**a**) relativised output of the black box model, (

**b**) comparison of the results from experiment, white, and grey box model.

Variable | Value Range | Order of Magnitude |
---|---|---|

Rotational speed | 1000 $\mathrm{rpm}$ to 3000 $\mathrm{rpm}$ | 10^{3} |

Differential speed | 15 $\mathrm{rpm}$ to 36 $\mathrm{rpm}$ |
10^{1} |

Volumetric flow rate | 500 $\mathrm{L}$ ${\mathrm{h}}^{-1}$ to 3000 $\mathrm{L}$ ${\mathrm{h}}^{-1}$ |
10^{2} to 10^{3} |

Solids content inlet | 20 $\mathrm{wt}.\%$ to 30 $\mathrm{wt}.\%$ |
10^{1} |

Pool depth | 0.054 $\mathrm{m}$ to 0.094 $\mathrm{m}$ |
10^{−2} |

Solids content centrate | 0 $\mathrm{wt}.\%$ to 30 $\mathrm{wt}.\%$ |
10^{0} to 10^{1} |

Solids content cake | 50 $\mathrm{wt}.\%$ to 75 $\mathrm{wt}.\%$ |
10^{1} |

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**MDPI and ACS Style**

Menesklou, P.; Sinn, T.; Nirschl, H.; Gleiss, M.
Grey Box Modelling of Decanter Centrifuges by Coupling a Numerical Process Model with a Neural Network. *Minerals* **2021**, *11*, 755.
https://doi.org/10.3390/min11070755

**AMA Style**

Menesklou P, Sinn T, Nirschl H, Gleiss M.
Grey Box Modelling of Decanter Centrifuges by Coupling a Numerical Process Model with a Neural Network. *Minerals*. 2021; 11(7):755.
https://doi.org/10.3390/min11070755

**Chicago/Turabian Style**

Menesklou, Philipp, Tabea Sinn, Hermann Nirschl, and Marco Gleiss.
2021. "Grey Box Modelling of Decanter Centrifuges by Coupling a Numerical Process Model with a Neural Network" *Minerals* 11, no. 7: 755.
https://doi.org/10.3390/min11070755