# Boltzmann-Based Empirical Model to Calculate Volume Loss during Spirit Ageing

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Data

#### 2.2. Mathematical Model and Data Fitting

^{®}, Northampton, MA, US).

_{1}, A

_{2}, x

_{0}, and dx are the Boltzmann’s model constants.

## 3. Results and Discussion

_{R}in %) plotted from the data presented in Table 1.

_{0}, A

_{1}, A

_{2}, and dT are model constants. Notice the general independent variable “x” in this case is the temperature, so x

_{0}and dx were changed to T

_{0}and dT, respectively.

_{LOSS}) versus temperature (Figure 1) can be satisfactorily linearly-correlated with the humidity of the air, with the highest correlation coefficient in comparison with the other explored models.

_{1}and A

_{2}presented a decreasing tendency with the increment of the air humidity. In contrast, the other parameter dT and k ($k={T}_{0}/dT$) exhibited values that are independent on the air humidity. Fisher’s lower significant difference (LSD) method was applied to demonstrate that there is no statistical difference between the dT and k values found at different H

_{R}. Additionally, normal distribution of the parameters dT and k was confirmed, and their means are presented in Table 2.

^{2}) were higher than 0.999 for all the cases (see the complete fitting data in Supplementary Materials). $\overline{dT}$ and $\overline{k}$ are the mean of the correspondent parameters.

_{1}and A

_{2}parameters in terms of H

_{R}

_{.}As an interesting feature of both linear correlations A

_{1}and A

_{2}, the fitting linear goodness is better in the range of 70%–95% of H

_{R}, while in the range of 40%–70%, the found dispersion is higher, as previously demonstrated in Figure 1.

_{1}and A

_{2}Boltzmann constants and H

_{R}. Both parameters present comparable correlation coefficients of around 0.99.

_{1}and A

_{2}vs. H

_{R}) and Equation (2) yield Equation (3).

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Volume loss for evaporation during ageing as function of the temperature at constant relative air humidity (H

_{R}in %). Ethanol content of the spirit: (40% v/v). White oak standard barrels of 205 L (+/− 10 L).

**Figure 2.**Linear correlation plots of the Boltzmann A

_{1}(

**a**) and A

_{2}(

**b**) constants in terms of H

_{R}.

**Figure 3.**Correlation plot between experimental data and the proposed empiric model (Equation (3)). SD = standard deviation in % of volume loss.

**Table 1.**Volume loss per year (in %) at different conditions of air temperature and relative humidity (H

_{R}).

Relative Humidity of the Air (H_{R}) (in %) | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

T °C↓ | 40 | 45 | 50 | 55 | 60 | 65 | 70 | 75 | 80 | 85 | 90 | 95 |

10 | 4.78 | 4.42 | 4.04 | 3.85 | 3.28 | 2.90 | 2.52 | 2.15 | 1.77 | 1.39 | 1.01 | 0.63 |

11 | 5.13 | 4.73 | 4.28 | 4.02 | 3.51 | 3.11 | 2.70 | 2.30 | 1.90 | 1.49 | 1.08 | 0.68 |

12 | 5.47 | 5.04 | 4.51 | 4.18 | 3.74 | 3.31 | 2.88 | 2.45 | 2.02 | 1.58 | 1.15 | 0.72 |

13 | 5.85 | 5.39 | 4.88 | 4.47 | 4.00 | 3.54 | 3.08 | 2.62 | 2.16 | 1.69 | 1.23 | 0.77 |

14 | 6.23 | 5.74 | 5.25 | 4.76 | 4.26 | 3.77 | 3.28 | 2.79 | 2.29 | 1.80 | 1.31 | 0.82 |

15 | 6.65 | 6.13 | 5.61 | 5.08 | 4.55 | 4.03 | 3.50 | 2.99 | 2.45 | 1.92 | 1.40 | 0.87 |

16 | 7.06 | 6.52 | 5.96 | 5.40 | 4.84 | 4.28 | 3.72 | 3.18 | 2.60 | 2.04 | 1.48 | 0.92 |

17 | 7.55 | 6.96 | 6.37 | 5.77 | 5.17 | 4.57 | 3.97 | 3.39 | 2.78 | 2.18 | 1.58 | 0.99 |

18 | 8.04 | 7.40 | 6.77 | 6.13 | 5.50 | 4.86 | 4.22 | 3.59 | 2.95 | 2.32 | 1.68 | 1.05 |

19 | 8.57 | 7.91 | 7.23 | 6.73 | 5.88 | 5.19 | 4.51 | 3.84 | 3.16 | 2.49 | 1.80 | 1.13 |

20 | 9.10 | 8.42 | 7.69 | 7.33 | 6.25 | 5.52 | 4.80 | 4.09 | 3.37 | 2.65 | 1.92 | 1.20 |

21 | 9.68 | 8.95 | 8.18 | 7.80 | 6.64 | 5.87 | 5.10 | 4.35 | 3.58 | 2.82 | 2.05 | 1.28 |

22 | 10.3 | 9.52 | 8.70 | 8.29 | 7.06 | 6.24 | 5.43 | 4.63 | 3.81 | 2.99 | 2.17 | 1.36 |

23 | 10.9 | 10.1 | 9.20 | 8.80 | 7.50 | 6.60 | 5.77 | 4.92 | 4.05 | 3.18 | 2.31 | 1.44 |

24 | 11.6 | 10.7 | 9.80 | 9.40 | 8.00 | 7.00 | 6.12 | 5.22 | 4.30 | 3.38 | 2.45 | 1.53 |

25 | 12.3 | 11.4 | 10.4 | 9.90 | 8.50 | 7.50 | 6.50 | 5.55 | 4.57 | 3.59 | 2.61 | 1.63 |

26 | 13.1 | 12.1 | 11.1 | 10.5 | 9.00 | 7.90 | 6.90 | 5.89 | 4.85 | 3.81 | 2.76 | 1.72 |

27 | 13.9 | 12.8 | 11.7 | 11.2 | 9.50 | 8.40 | 7.32 | 6.24 | 5.14 | 4.04 | 2.93 | 1.83 |

28 | 14.7 | 13.6 | 12.4 | 11.9 | 10.1 | 8.90 | 7.76 | 6.62 | 5.45 | 4.28 | 3.11 | 1.94 |

29 | 15.6 | 14.4 | 13.2 | 12.6 | 10.7 | 9.50 | 8.22 | 7.01 | 5.77 | 4.53 | 3.29 | 2.06 |

30 | 16.5 | 15.3 | 14.0 | 13.3 | 11.3 | 10.0 | 8.71 | 7.43 | 6.12 | 4.80 | 3.49 | 2.18 |

Relative Humidity H_{R} (%) | Fitting Parameters | ||||
---|---|---|---|---|---|

A_{1} (%) | A_{2} (%) | dT (°C) | k | R^{2} | |

40 | 1.607 | 41.71 | 10.54 | 3.37 | >0.999 |

45 | 1.536 | 40.51 | 10.57 | 3.44 | |

50 | 1.302 | 35.95 | 10.57 | 3.39 | |

55 | 1.096 | 30.06 | 10.03 | 3.31 | |

60 | 1.011 | 26.83 | 10.49 | 3.28 | |

70 | 0.846 | 21.92 | 10.51 | 3.37 | |

75 | 0.715 | 18.80 | 10.53 | 3.38 | |

80 | 0.595 | 15.63 | 10.51 | 3.40 | |

85 | 0.439 | 12.12 | 10.59 | 3.35 | |

90 | 0.341 | 8.931 | 10.51 | 3.40 | |

95 | 0.218 | 5.298 | 10.33 | 3.37 | |

$\overline{dT}$ = 10.47 | $\overline{k}$ = 3.37 |

Boltzmann Model Constants: A_{i} | Linear Fitting Parameters of H_{R} vs. A_{i} | ||||
---|---|---|---|---|---|

m | e(m) | p | e(p) | R^{2} | |

A_{1} | −0.025 | 9 × 10^{−4} | 2.57 | 0.06 | 0.989 |

A_{2} | −0.665 | 0.019 | 68.5 | 1.30 | 0.993 |

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**MDPI and ACS Style**

del Toro del Toro, N.; Fong Casas, F.; Ayan Rial, J.; Caridad Portuondo González, M.; Crespo Sariol, H.; Navarro Campa, J.; Yperman, J.; Vandamme, D.; Carleer, R. Boltzmann-Based Empirical Model to Calculate Volume Loss during Spirit Ageing. *Beverages* **2019**, *5*, 60.
https://doi.org/10.3390/beverages5040060

**AMA Style**

del Toro del Toro N, Fong Casas F, Ayan Rial J, Caridad Portuondo González M, Crespo Sariol H, Navarro Campa J, Yperman J, Vandamme D, Carleer R. Boltzmann-Based Empirical Model to Calculate Volume Loss during Spirit Ageing. *Beverages*. 2019; 5(4):60.
https://doi.org/10.3390/beverages5040060

**Chicago/Turabian Style**

del Toro del Toro, Noemí, Fredy Fong Casas, Julio Ayan Rial, Maria Caridad Portuondo González, Harold Crespo Sariol, José Navarro Campa, Jan Yperman, Dries Vandamme, and Robert Carleer. 2019. "Boltzmann-Based Empirical Model to Calculate Volume Loss during Spirit Ageing" *Beverages* 5, no. 4: 60.
https://doi.org/10.3390/beverages5040060