# A Parametric Model of the LARCODEMS Heavy Media Separator by Means of Multivariate Adaptive Regression Splines

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## Abstract

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## 1. Introduction

#### The LARCODEMS

## 2. Materials and Methods

#### 2.1. Description of LARCODEMS Test Procedure

- Twelve, 21.4, 32.9 and 45 cm long separation cylinder versions with and without a vortex extractor of the 110 mm LARCODEMS.
- Fifty-two, 72, and 172 mm diameter versions of the scaled 45 cm long, 110 mm LARCODEMS.
- Variations in diameter of the feed DA of the 110 mm LARCODEMS.
- Variations in FPA of the 110 mm LARCODEMS.
- Variations in SPA of the 52, 72, 110, and 172 mm LARCODEMS.
- Three variations in total media feed flow rate for all of the above variables when separations of plastic particles that had received a treatment (TM) to reduce hydrophobic effects were conducted without a VE.
- Five variations in total media feed flow rate for all of the above variables when separations were conducted without a VE.

- The 32.9 cm-long and the 45 cm-long separation cylinder versions with and without a vortex extractor of the 110 mm LARCODEMS.
- Variations in diameter of the float port of the 110 mm LARCODEMS.
- Three variations in total media feed flow rate for all of the above variables when separations of plastic particles that had received a treatment (TM) to reduce hydrophobic effects were conducted without a vortex extractor.
- Five variations in total media feed flow rate for all of the above variables when separations were conducted without a vortex extractor.

#### 2.2. The Multivariate Adaptive Regression Splines

#### 2.3. Model Performance Measurement

^{2}[18,19] of the model are measures that can be employed with this aim.

#### 2.4. Model Training and Validation

^{2}were calculated. This permitted the determination of the minimum, average, and maximum values of those parameters in each one of the 1000 models replications.

## 3. Results

^{2}for the models of the LARCODEMS tests performed were as follows: model one with media: 0.6254; into the vortex: 0.9553; model two with media: 0.4820; and into the vortex: 0.8630. This shows a high correlation coefficient of real and predicted efficiencies. In the case of model one, the RMSE, for the test performed with feed into the vortex has a value of 1.9894; while for those test with media feed it is 2.3741. For model two, RMSE values are 2.421 with the media, and 4.8996 into the vortex. In the case of the MAE, for model one with media, the value is 1.7404, and into the vortex, 1.3456. Finally, for model two, the MAE value is 1.8419 with media, and 3.7928 into the vortex. The R

^{2}values obtained show a high correlation for the case of the LARCODEMS test performed with media. In the case of the model for the test performed into the vortex, although the correlation is significant, it is not as high. As far as it is known to the authors, there are no other studies that would allow us to compare the RMSE and MAE obtained in this work.

^{2}, RMSE, and MAE of the 1000 different five-fold cross-validation sets created are summarized in Table 9 for both models one and two with media and into the vortex. The average value of R

^{2}is better in the case of both models with into the vortex feed when compared with the model with media feed. In the case of both RMSE and MAE, the average value is larger in the models trained and validated with the into the vortex media test.

## 4. Discussion and Conclusions

- Separation efficiencies increase with both the length and size of the separation cylinder. This is reflected by the relation between the separation efficiency and media flow rates in the sink and float ports and by the relation between the cylinder length to the cylinder area.
- For a given separation cylinder size there is an optimum length above which the separation efficiency does not increase.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Schematic diagram of the large coal density media separator (LARCODEMS) (courtesy of JMC Engineering Ltd.).

**Figure 2.**Graphical representation of the basis functions of MARS model number one and their coefficients for feed with media.

**Figure 3.**Graphical representation of the basis functions of MARS model number one and their coefficients for feed into the vortex.

**Figure 4.**Graphical representation of the basis functions of MARS model number two and their coefficients for feed with media.

**Figure 5.**Graphical representation of the basis functions of the MARS model number one and their coefficients for feed into the vortex the ratio CL to CA relative to separation efficiency.

**Figure 6.**Real value of efficacy versus the calculated by MARS model one for into the vortex and with media feed tests.

**Figure 7.**Real value of efficacy versus the calculated by MARS model two for into the vortex and with media feed tests.

**Table 1.**List of basis functions of multivariate adaptive regression splines (MARS) model number one and their coefficients for feed with media.

B_{i} | Definition | C_{i} |
---|---|---|

B_{1} | Constant | 87.045425 |

B_{2} | Cylinder length | 0.263318 |

B_{3} | Float port area | 0.292959 |

B_{4} | Diaphragm area | –0.198685 |

B_{5} | h(Sink flow − 0.677) | –164.231069 |

B_{6} | h(Sink flow − 0.677) | 193.832769 |

**Table 2.**List of basis functions of MARS model number one and their coefficients for feed into the vortex.

B_{i} | Definition | C_{i} |
---|---|---|

B_{1} | Constant | 88.024967 |

B_{2} | h(21.4 − Cylinder length) | –3.149586 |

B_{3} | h(Cylinder length − 21.4) | 0.491264 |

B_{4} | h(Float flow − 0.243) | –22.817374 |

B_{5} | h(Sink flow − 0.588) | 19.200122 |

**Table 3.**List of basis functions of MARS model number two and their coefficients for feed with media.

B_{i} | Definition | C_{i} |
---|---|---|

B_{1} | Constant | 81.97549 |

B_{2} | Cylinder length/Cylinder Area | 21.50694 |

B_{3} | Float port area/Total Area | 41.73611 |

**Table 4.**List of basis functions of the MARS model number two and their coefficients for feed into the vortex.

B_{i} | Definition | C_{i} |
---|---|---|

B_{1} | constant | 99.00408 |

B_{2} | h(0.35 − Cylinder length/Cylinder Area) | –157.73737 |

B_{3} | h(Cylinder length/Cylinder Area − 0.35) | –13.58808 |

**Table 5.**Relative importance of variables in MARS model number one and their coefficients for feed with media.

Variable | Nsubsets | GCV | RSS |
---|---|---|---|

Diaphragm area | 5 | 100 | 100 |

Cylinder length | 4 | 69.9 | 79.3 |

Sink flow | 3 | 41.8 | 61.7 |

Float port area | 1 | 13.2 | 32.3 |

**Table 6.**Relative importance of variables in MARS model number one and their coefficients for feed into the vortex.

Variable | Nsubsets | GCV | RSS |
---|---|---|---|

Cylinder length | 4 | 100 | 100 |

Float flow | 1 | 17.2 | 16.4 |

Sink flow | 1 | 17.2 | 16.4 |

**Table 7.**List of basis functions of MARS model number two and their coefficients for feed with media.

Variable | Nsubsets | GCV | RSS |
---|---|---|---|

Float A/Total A | 2 | 100 | 100 |

Cyl L/Cyl A | 1 | 40.1 | 55.2 |

**Table 8.**List of basis functions of MARS model number two and their coefficients for feed into the vortex.

Variable | Nsubsets | GCV | RSS |
---|---|---|---|

Cyl L/Cyl A | 2 | 100 | 100 |

**Table 9.**Performance measurements of models one and two trained with the media feed and the feed into the vortex.

Variable | With Media | Into the Vortex | ||||
---|---|---|---|---|---|---|

MODEL 1 | Min. | Avg. | Max. | Min. | Avg. | Max. |

R^{2} | 0.3543 | 0.5080 | 0.6776 | 0.6303 | 0.7001 | 0.8078 |

RMSE | 2.1672 | 2.9757 | 6.3582 | 7.4346 | 9.2680 | 22.1758 |

MAE | 1.8560 | 2.3265 | 4.8040 | 6.2205 | 7.9052 | 15.6619 |

MODEL 2 | Min. | Avg. | Max. | Min. | Avg. | Max. |

R^{2} | 0.3355 | 0.4333 | 0.5519 | 0.6838 | 0.7785 | 0.8601 |

RMSE | 2.6046 | 3.3521 | 4.6182 | 5.5193 | 7.4982 | 13.0074 |

MAE | 2.1771 | 2.7911 | 4.0124 | 4.3370 | 6.2126 | 10.9078 |

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**MDPI and ACS Style**

Álvarez, M.M.; Sierra, H.M.; Lasheras, F.S.; Juez, F.J.d.C.
A Parametric Model of the LARCODEMS Heavy Media Separator by Means of Multivariate Adaptive Regression Splines. *Materials* **2017**, *10*, 729.
https://doi.org/10.3390/ma10070729

**AMA Style**

Álvarez MM, Sierra HM, Lasheras FS, Juez FJdC.
A Parametric Model of the LARCODEMS Heavy Media Separator by Means of Multivariate Adaptive Regression Splines. *Materials*. 2017; 10(7):729.
https://doi.org/10.3390/ma10070729

**Chicago/Turabian Style**

Álvarez, Mario Menéndez, Héctor Muñiz Sierra, Fernando Sánchez Lasheras, and Francisco Javier de Cos Juez.
2017. "A Parametric Model of the LARCODEMS Heavy Media Separator by Means of Multivariate Adaptive Regression Splines" *Materials* 10, no. 7: 729.
https://doi.org/10.3390/ma10070729