# Mathematical Modelling of Muña Leaf Drying (Minthostachys mollis) for Determination of the Diffusion Coefficient, Enthalpy, and Gibbs Free Energy

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Raw Material

#### 2.2. Raw Material Conditioning

#### 2.3. Drying Process

#### 2.4. Mathematical Models

#### 2.5. Activation Energy and Thermodynamic Properties

_{eff}is effective diffusivity coefficient (m/s

^{2}), and L

_{0}is the semi-thickness of the sheet to be dried (m).

_{eff}was determined through the graph of ln(MR) versus time of the experimental data by means of the slope $\left(\frac{{\pi}^{2}{D}_{eff}}{4{L}_{0}^{2}}\right)$ of Equation (10).

_{a}) and initial diffusion constant (D

_{0}) were estimated from the slope and intercept of the graph ln(D

_{eff}) versus 1/T [16].

_{eff}is a rate constant to evaluate empirical constants, R is the universal gas constant (8.314 J/mol$\xb7$K), E

_{a}is the activation energy (kJ/mol), D

_{0}is the Arrhenius factor (m

^{2}/s), and Ta is the absolute temperature (K).

_{B}is Boltzmann constant (1.38 × 10

^{−23}JK

^{−1}, the Planck constant is h

_{p}(6.626 × 10

^{−34}J$\xb7$s), and T is the absolute temperature (°K).

#### 2.6. Statistical Analysis

^{2}(Equation (17)), and the mean square error (MSE, Equation (18)). This was accomplished by using the XLSTAT 2022 software. The lowest values of SSE and RMSE (≈0.0) and the highest values of the coefficient of determination r

^{2}(≈1.0) were considered as criteria to select the best fit between the models.

_{e,i}is the experimental moisture content, MR

_{c,i}is the calculated moisture content, I is the number of terms, z is a constant number, c is the value given by the model, and N is the number of data.

## 3. Results and Discussion

#### 3.1. Drying Kinetic Curves

#### 3.2. Mathematical Modeling of Drying Kinetics

^{2}, SSE, MSE, and RMSE of the tests are shown in Table 2, Table 3 and Table 4. The highest values of R

^{2}and the lowest values of RMSE were selected as criteria for the accuracy of the fit.

^{2}values > 0.990, with the exception of the No Treatment (SB) sample dried at 40 °C, where the best fit was the model Page. Because the values of SSE, MSE, and RMSE are much closer to zero, with respect to the logarithmic model, despite the fact that its R

^{2}is 0.924 and the logarithmic is 0.994, this demonstrated a good data fit. Quequeto et al. [23], Martins et al. [24], Martins et al. [18], and Gasparin et al. [20] found similar values of R

^{2}> 0.990 in drying Piper aduncuma leaves at 40–60 °C, blackberry leaves at 40–70 °C, Serjania marginata leaves at 40–70 °C, and mint leaves at 30 −70 °C, respectively. However, the model with the best fit to describe the drying kinetics in these investigations was the Midilli model. Similarly, Silva et al. [19] and Da Silva et al. [22] dried Genipa americana leaves at 35–65 °C and boldo leaves at 20–60 °C, respectively, reporting that the model with the best fit was the modified Henderson and Pabis model. A similar behavior was reported by Eneighe et al. [25] on Xymalos monospora leaves at 50–70 °C, indicating that their data fit the Page and modified Page model. The fit to the best model of the drying data observed in medicinal plants is related to the rapid loss of water in the initial stages of the process in this type of leaves, which generates a more pronounced drying curve and is better characterized by the mathematical model logarithmic.

#### 3.3. Diffusion Coefficient, Thermodynamic Properties and Activation Energy

#### 3.3.1. Water Diffusion Coefficient

_{eff}) can be determined. The D

_{eff}values obtained for each sample at different drying temperatures are presented in Table 5. The diffusivity values of water increased as the drying temperature increased, so values between 3.098 and 7.744 × 10

^{−10}were obtained. m

^{2}/s in the range of 40–60 °C. These values were similar to those reported by Doymaz et al. [26] and Kaya and Aydin [27] in mint leaves drying 0.307–1.941 × 10

^{−8}m

^{2}/s between 35–60 °C and 1.975–6.172 × 10

^{−9}m

^{2}/s between 35–55 °C, respectively. Kaya and Aydin [27] recorded values in nettle leaves between 1.744–4.992 × 10

^{−9}m

^{2}/s between 35–55 °C, while Doymaz et al. [28] recorded values in parsley leaves, which were reported as 0.900–2.337 × 10

^{−9}m

^{2}/s, involving drying with hot air between 50–70 °C. Therefore, the values found agree with the water diffusivity data during the drying of the different types of leaves. At low diffusion coefficient, higher temperatures can be used to speed up the drying process, as long as it is ensured that the temperature does not damage the material. On the other hand, if the diffusion coefficient is high, lower temperatures can be used to dry the material more gently and preserve its quality. The diffusion coefficient is important in the drying process because it helps to improve the efficiency of the drying process and to preserve the quality of the material.

#### 3.3.2. Thermodynamic Properties

#### 3.3.3. Activation Energy

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Variations of the moisture ratio (MR) as a function of time at different drying temperatures: (

**a**) SB: without pretreatment, (

**b**) BAA: immersion in a 1% ascorbic acid solution for 30 s at 40 °C, and (

**c**) B60: bleaching process at 60 °C for 30 s ((

**c**), adjusted to the logarithmic model).

Model Name | Model Equation | Equation |
---|---|---|

Page | MR = exp (−k·t^{n}) | (1) |

Modified Page | MR = exp (−k·t)^{n} | (2) |

Midilli | MR = a·exp (−k·t^{n}) + b·t | (3) |

Lewis | MR = exp (−k·t) | (4) |

Wang and Singh | MR = 1 + (a·t) + (b·t^{2}) | (5) |

Logarithmic | MR = a·exp (-k·t) + c | (6) |

Peleg | MR = (1 − t)/(k_{1} + (k_{2}·t^{2})) | (7) |

Henderson and Pabis | MR = a·exp (−k·t) | (8) |

Moisture ratio | MR = (w_{t} − w_{e})/(w_{o} − w_{e}) | (9) |

_{t}is the real-time moisture content (g water/g sample), w

_{o}is the initial moisture content (g water/g sample), and w

_{e}is the equilibrium moisture content (g water/g sample).

**Table 2.**Empirical models, constants, and regressive statistical parameters for muña (Minthostachys mollis) drying at 40 °C.

Sample | Model | Statistics | Parameters | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

r^{2} | SSE | MSE | RMSE | ||||||||||

SB | Midilli | 0.425 | 5.269 | 0.258 | 0.499 | a | 1.085 | B | 7.473 | k | −57.716 | n | −5.971 |

Logarithmic | 0.994 | 4.117 | 0.274 | 0.524 | a | 0.575 | C | 1.563 | k | 0.012 | |||

Page | 0.924 | 0.073 | 0.004 | 0.062 | k | 51.380 | N | 51.380 | |||||

Modified Page | 0.521 | 0.003 | 0.000 | 0.014 | k | 2.618 | N | −1.036 | |||||

Henderson and Pabis | 0.700 | 0.131 | 0.009 | 0.093 | a | 1.908 | K | 0.001 | |||||

Wang and Singh | 0.597 | 4.117 | 0.257 | 0.507 | a | 0.006 | B | 0.000 | |||||

Peleg | 0.659 | 2.616 | 0.174 | 0.418 | k_{1} | 0.473 | k_{2} | −0.606 | |||||

Lewis | 0.465 | 0.233 | 0.016 | 0.125 | k | −0.001 | |||||||

Fick’s second law | 0.923 | 0.0523 | 0.073 | 0.042 | D_{eff} | 3.182 × 10^{−10} | |||||||

BAA | Midilli | 0.963 | 0.114 | 0.006 | 0.078 | a | −0.585 | B | 20.399 | k | −24.721 | n | −3.494 |

Logarithmic | 0.998 | 0.001 | 0.000 | 0.007 | a | 0.578 | C | 1.555 | k | 0.009 | |||

Page | 0.405 | 46.497 | 3.100 | 1.761 | k | 51.380 | N | 51.380 | |||||

Modified Page | 0.627 | 4.539 | 0.303 | 0.550 | k | 2.618 | n | −1.028 | |||||

Henderson and Pabis | 0.806 | 0.089 | 0.006 | 0.077 | a | 0.001 | k | 1.959 | |||||

Wang and Singh | 0.556 | 2.768 | 0.185 | 0.430 | a | 0.006 | b | 0.000 | |||||

Peleg | 0.381 | 0.284 | 0.019 | 0.138 | k_{1} | 0.474 | k_{2} | −0.595 | |||||

Lewis | 0.704 | 4.539 | 0.284 | 0.533 | k | −0.001 | |||||||

Fick’s second law | 0.921 | 0.146 | 0.0352 | 0.077 | D_{eff} | 3.098 × 10^{−10} | |||||||

B60 | Midilli | 0.969 | 0.114 | 0.006 | 0.078 | a | −0.449 | b | 27.008 | k | −50.981 | n | −5.544 |

Logarithmic | 0.999 | 0.000 | 0.000 | 0.005 | a | 0.578 | c | 1.553 | k | 0.009 | |||

Page | 0.433 | 45.937 | 3.062 | 1.750 | k | 5.682 | n | 5.682 | |||||

Modified Page | 0.604 | 4.419 | 0.295 | 0.543 | k | 2.618 | n | −1.031 | |||||

Henderson and Pabis | 0.790 | 0.096 | 0.006 | 0.080 | a | 1.946 | k | 0.001 | |||||

Wang and Singh | 0.572 | 2.719 | 0.181 | 0.426 | a | 0.006 | b | 0.000 | |||||

Peleg | 0.409 | 0.270 | 0.018 | 0.134 | k_{1} | 0.473 | k_{2} | −0.599 | |||||

Lewis | 0.688 | 4.419 | 0.276 | 0.526 | k | −0.001 | |||||||

Fick’s second law | 0.944 | 0.029 | 0.230 | 0.012 | D_{eff} | 3.075 × 10^{−10} |

**Table 3.**Empirical models, constants and regressive statistical parameters for muña (Minthostachys mollis) drying at 50 °C.

Sample | Model | Statistics | Parameters | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

r^{2} | SSE | MSE | RMSE | ||||||||||

SB | Midilli | 0.792 | 0.058 | 0.010 | 0.088 | a | 16.218 | b | −1.564 | k | −13.986 | n | −3.666 |

Logarithmic | 1.000 | 0.000 | 0.000 | 0.003 | a | 0.561 | c | 1.572 | k | 0.024 | |||

Page | 0.772 | 22.841 | 3.263 | 1.806 | k | 43.649 | n | 43.649 | |||||

Modified Page | 0.545 | 4.616 | 0.474 | 0.674 | k | 2.618 | n | −0.885 | |||||

Henderson and Pabis | 0.698 | 0.085 | 0.012 | 0.110 | a | 0.001 | k | 1.937 | |||||

Wang and Singh | 0.772 | 1.738 | 0.248 | 0.498 | a | 0.011 | b | 0.000 | |||||

Peleg | 0.958 | 5.656 | 0.808 | 0.899 | k_{1} | −8.129 | k_{2} | −0.538 | |||||

Lewis | 0.589 | 2.474 | 0.309 | 0.556 | k | −0.003 | |||||||

Fick’s second law | 0.872 | 1.496 | 0.262 | 0.991 | D_{eff} | 6.195 × 10^{−10} | |||||||

BAA | Midilli | 0.849 | 0.051 | 0.007 | 0.081 | a | 0.330 | b | −17.883 | k | −12.486 | n | −1.413 |

Logarithmic | 0.990 | 0.003 | 0.001 | 0.024 | a | 0.012 | c | 1.528 | k | 0.012 | |||

Page | 0.506 | 24.891 | 3.556 | 1.886 | k | 42.657 | n | 42.657 | |||||

Modified Page | 0.753 | 5.520 | 0.566 | 0.737 | k | 2.618 | n | −0.858 | |||||

Henderson and Pabis | 0.885 | 0.039 | 0.006 | 0.074 | a | 0.001 | k | 2.059 | |||||

Wang and Singh | 0.486 | 2.008 | 0.287 | 0.536 | a | 0.013 | b | 0.000 | |||||

Peleg | 0.484 | 0.174 | 0.025 | 0.158 | k_{1} | 0.467 | k_{2} | −0.583 | |||||

Lewis | 0.789 | 3.101 | 0.388 | 0.623 | k | −0.003 | |||||||

Fick’s second law | 0.898 | 0.957 | 0.168 | 0.205 | D_{eff} | 4.646 × 10^{−10} | |||||||

B60 | Midilli | 0.809 | 0.061 | 0.008 | 0.088 | a | 1.273 | b | −0.459 | k | −15.893 | n | −2.460 |

Logarithmic | 0.995 | 0.002 | 0.000 | 0.017 | a | 0.613 | c | 1.545 | k | 0.013 | |||

Page | 0.561 | 24.426 | 3.489 | 1.868 | k | 6.561 | n | 6.561 | |||||

Modified Page | 0.714 | 5.312 | 0.546 | 0.724 | k | 2.618 | n | −0.865 | |||||

Henderson and Pabis | 0.856 | 0.046 | 0.007 | 0.081 | a | 0.001 | k | 2.033 | |||||

Wang and Singh | 0.567 | 1.955 | 0.279 | 0.528 | a | 0.012 | b | 0.000 | |||||

Peleg | 0.539 | 0.148 | 0.021 | 0.145 | k_{1} | 0.468 | k_{2} | −0.588 | |||||

Lewis | 0.752 | 2.955 | 0.369 | 0.608 | k | −0.003 | |||||||

Fick’s second law | 0.866 | 0.649 | 0.109 | 0.352 | D_{eff} | 4.459 × 10^{−10} |

**Table 4.**Empirical models, constants, and regressive statistical parameters for muña (Minthostachys mollis) drying at 60 °C.

Sample | Model | Statistics | Parameters | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

r^{2} | SSE | MSE | RMSE | ||||||||||

SB | Midilli | 0.628 | 0.098 | 0.016 | 0.123 | a | 0.020 | b | −16.572 | k | −19.213 | n | −1.645 |

Logarithmic | 0.995 | 0.001 | 0.000 | 0.016 | a | 0.558 | c | 1.570 | k | 0.031 | |||

Page | 0.868 | 19.703 | 3.284 | 1.812 | k | 21.007 | n | 21.007 | |||||

Modified Page | 0.521 | 4.015 | 0.478 | 0.677 | k | 2.618 | n | −0.833 | |||||

Henderson and Pabis | 0.672 | 0.087 | 0.014 | 0.120 | a | 0.002 | k | 2.064 | |||||

Wang and Singh | 0.786 | 1.587 | 0.265 | 0.514 | a | 0.012 | b | 0.000 | |||||

Peleg | 0.854 | 0.039 | 0.006 | 0.080 | k_{1} | 0.470 | k_{2} | −0.615 | |||||

Lewis | 0.567 | 2.193 | 0.313 | 0.560 | k | −0.003 | |||||||

Fick’s second law | 0.931 | 0.037 | 0.0246 | 0.084 | D_{eff} | 7.744 × 10^{−10} | |||||||

BAA | Midilli | 0.976 | 0.007 | 0.002 | 0.041 | a | 0.002 | b | −15.807 | k | −31.810 | n | −2.262 |

Logarithmic | 0.997 | 0.001 | 0.000 | 0.013 | a | 0.569 | c | 1.565 | k | 0.026 | |||

Page | 0.801 | 20.030 | 3.338 | 1.827 | k | 88.354 | n | 88.354 | |||||

Modified Page | 0.565 | 4.153 | 0.495 | 0.688 | k | 2.618 | n | −0.852 | |||||

Henderson and Pabis | 0.713 | 0.079 | 0.013 | 0.114 | a | 0.001 | k | 1.950 | |||||

Wang and Singh | 0.798 | 1.653 | 0.275 | 0.525 | a | 0.012 | b | 0.000 | |||||

Peleg | 0.968 | 5.599 | 0.933 | 0.966 | k_{1} | −8.479 | k_{2} | −0.530 | |||||

Lewis | 0.602 | 2.291 | 0.327 | 0.572 | k | −0.003 | |||||||

Fick’s second law | 0.946 | 0.358 | 0.029 | 0.071 | D_{eff} | 7.357 × 10^{−10} | |||||||

B60 | Midilli | 0.859 | 0.042 | 0.007 | 0.081 | a | 0.628 | b | 0.521 | k | −14.952 | n | −2.068 |

Logarithmic | 0.988 | 0.004 | 0.001 | 0.027 | a | 0.636 | c | 1.524 | k | 0.012 | |||

Page | 0.840 | 22.004 | 3.667 | 1.915 | k | 21.007 | n | 21.007 | |||||

Modified Page | 0.776 | 5.014 | 0.596 | 0.756 | k | 2.618 | n | −0.833 | |||||

Henderson and Pabis | 0.893 | 0.032 | 0.005 | 0.073 | a | 0.002 | k | 2.064 | |||||

Wang and Singh | 0.505 | 1.869 | 0.312 | 0.558 | a | 0.015 | b | 0.000 | |||||

Peleg | 0.789 | 5.922 | 0.987 | 0.994 | k_{1} | −7.390 | k_{2} | −0.514 | |||||

Lewis | 0.798 | 2.838 | 0.405 | 0.637 | k | −0.003 | |||||||

Fick’s second law | 0.969 | 0.682 | 0.032 | 0.458 | D_{eff} | 7.583 × 10^{−10} |

Sample | Temperature (°C) | Effective Diffusivity (D_{eff} × 10^{−10} m^{2}/s) | Δh (kJ/mol) | Δs (kJ/mol × K) | ΔG (kJ/mol) | Activation Energy (kJ/mol) | R^{2} |
---|---|---|---|---|---|---|---|

SB | 40 | 3.182 ± 0.049 | 37.332 | −0.229 | 109.288 | 39.935 | 0.929 |

50 | 6.195 ± 0.040 | 37.249 | −0.230 | 111.587 | |||

60 | 7.744 ± 0.012 | 37.166 | −0.230 | 113.889 | |||

BAA | 40 | 3.098 ± 0.162 | 37.329 | −0.228 | 108.955 | 39.315 | 0.992 |

50 | 4.646 ± 0.854 | 36.992 | −0.229 | 111.252 | |||

60 | 7.357 ± 0.014 | 36.909 | −0.230 | 113.487 | |||

B60 | 40 | 3.075 ± 0.035 | 37.075 | −0.229 | 108.957 | 39.678 | 0.922 |

50 | 4.459 ± 0.475 | 36.984 | −0.229 | 111.305 | |||

60 | 7.583 ± 0.014 | 36.709 | −0.230 | 113.551 |

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## Share and Cite

**MDPI and ACS Style**

Silva-Paz, R.J.; Mateo-Mendoza, D.K.; Eccoña-Sota, A.
Mathematical Modelling of Muña Leaf Drying *(Minthostachys mollis)* for Determination of the Diffusion Coefficient, Enthalpy, and Gibbs Free Energy. *ChemEngineering* **2023**, *7*, 49.
https://doi.org/10.3390/chemengineering7030049

**AMA Style**

Silva-Paz RJ, Mateo-Mendoza DK, Eccoña-Sota A.
Mathematical Modelling of Muña Leaf Drying *(Minthostachys mollis)* for Determination of the Diffusion Coefficient, Enthalpy, and Gibbs Free Energy. *ChemEngineering*. 2023; 7(3):49.
https://doi.org/10.3390/chemengineering7030049

**Chicago/Turabian Style**

Silva-Paz, Reynaldo J., Dante K. Mateo-Mendoza, and Amparo Eccoña-Sota.
2023. "Mathematical Modelling of Muña Leaf Drying *(Minthostachys mollis)* for Determination of the Diffusion Coefficient, Enthalpy, and Gibbs Free Energy" *ChemEngineering* 7, no. 3: 49.
https://doi.org/10.3390/chemengineering7030049