# Monitoring Liquid-Liquid Mixtures Using Fractional Calculus and Image Analysis

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{0}and $\frac{d}{d\left(mf\right)}B\left(mf=0\right)=0$:

_{0}), which basically concerns an exponential variation of the color component, i.e.:

_{i}(mf) is close to zero, the parametric variance matrix is commonly simplified [28] to the following equation:

**G**, present in Equation (12), regarding the fractional order model can be written as:

_{OBJ}(k

_{estimated}, α

_{estimated}) is the final value of the objective function.

## 3. Results

^{−5}as the convergence criteria for all parameters. Table 2 presents the parameter estimation results.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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Sample Purpose | Olive Oil Mass Fraction | Color Component B |
---|---|---|

Parameter Estimation | 0 | 175 |

0.1 | 144 | |

0.2 | 116 | |

0.3 | 83 | |

0.4 | 54 | |

0.5 | 29 | |

0.6 | 12 | |

0.7 | 0.8 | |

0.8 | 0 | |

0.9 | 0 | |

1 | 0 | |

Model Validation | 0.25 | 96 |

Result | Fractional Order Model | Equation | Integer Order Model | Equation |
---|---|---|---|---|

NE (Number of Experiments) | 11 | - | 11 | - |

NP (Number of Parameters) | 2 (k, α) | - | 1 (k) | - |

F_{OBJ} | 312.4 | (4) | 1987.2 | (5) |

$\sqrt{{\sigma}_{{B}^{\mathrm{EXP}}}^{2}}=\sqrt{\frac{{F}_{\mathrm{OBJ}}}{\mathrm{NE}-\mathrm{NP}}}$ | 5.89 | (9) | 14.09 | (9) |

k | 3.42 | - | 3.11 | |

k (standard deviation) | 0.12 | (6) | 0.28 | (7) |

k (confidence interval) | [3.14; 3.69] | (19) | [2.48; 3.75] | (20) |

α | 1.196 | - | - | - |

α (standard deviation) | 0.027 | (6) | - | - |

α (confidence interval) | [1.14; 1.26] | (19) | - | - |

$\left[\underset{\_}{\underset{\_}{{\mathbf{V}}_{\mathrm{param}}}}\right]$ (parametric covariance matrix) | $\left[\begin{array}{cc}1.459\times {10}^{-2}& 9.778\times {10}^{-4}\\ 9.778\times {10}^{-4}& 7.325\times {10}^{-4}\end{array}\right]$ | (6) | $\left[8.170\times {10}^{-4}\right]$ | (7) |

$\left[\underset{\_}{\underset{\_}{{\mathbf{r}}_{\mathrm{param}}}}\right]$ (parametric correlation matrix) | $\left[\begin{array}{cc}1& 0.299\\ 0.299& 1\end{array}\right]$ | (8) | - | - |

r | 0.997 | (25) | 0.987 | (25) |

r^{2} | 0.993 | - | 0.972 | - |

Experimental | Fractional Order Model Prediction | Integer Order Model Prediction | |
---|---|---|---|

Olive oil mass fraction | Color Component B | Color Component B | Color Component B |

0.25 | 96 | 93.1 ± 6.6 | 80.2 ± 15.2 |

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**MDPI and ACS Style**

Lenzi, E.K.; Ryba, A.; Lenzi, M.K.
Monitoring Liquid-Liquid Mixtures Using Fractional Calculus and Image Analysis. *Fractal Fract.* **2018**, *2*, 11.
https://doi.org/10.3390/fractalfract2010011

**AMA Style**

Lenzi EK, Ryba A, Lenzi MK.
Monitoring Liquid-Liquid Mixtures Using Fractional Calculus and Image Analysis. *Fractal and Fractional*. 2018; 2(1):11.
https://doi.org/10.3390/fractalfract2010011

**Chicago/Turabian Style**

Lenzi, Ervin K., Andrea Ryba, and Marcelo K. Lenzi.
2018. "Monitoring Liquid-Liquid Mixtures Using Fractional Calculus and Image Analysis" *Fractal and Fractional* 2, no. 1: 11.
https://doi.org/10.3390/fractalfract2010011