# Application of Mathematical Models and Thermodynamic Properties in the Drying of Jambu Leaves

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Material and Methods

#### 2.1. Obtaining of Raw Material and Drying

^{®}) combined with 2% of an emulsifier (Emustab

^{®}). The mass of leaves and the foam were subjected to thin-layer convective drying.

^{−1}(measured in a digital anemometer Homis Mod 489). The materials (mass of leaves and foam) were spread evenly in rectangular stainless steel trays (25.5 × 13.5 cm), forming a thin layer of 1.0 cm thickness measured with a digital caliper (King Tools).

#### Physicochemical Characterization

#### 2.2. Mathematical Modeling

_{i}: initial moisture content of the product (d.b.); X

_{e}: equilibrium moisture content of the product (d.b.).

^{2}), relative mean error (P), estimated mean error (SE) and the mean chi-square ($\chi $

^{2}).

^{2}, P and SE) were subjected to the selection criteria of Akaike Information (AIC) and Schwarz’s Bayesian Information (BIC).

^{2}s

^{−1}; S: equivalent plate area, m

^{2}; V: equivalent plate volume, m

^{3}; L

_{0}: mass thickness, m; n: number of terms of the Equation; t: time, s.

^{−1}; R: universal constant of gases, 8.314 kJ kmol

^{−1}. K

^{−1}; Ta: absolute temperature, K.

#### 2.3. Thermodynamic Properties

^{−1}; ΔS-specific entropy, J mol

^{−1}K

^{−1}; ΔG-Gibbs free energy, J mol

^{−1}; KB-Boltzmann constant, 1.38 × 10

^{−23}J K

^{−1}; hp-Planck constant, 6.626 × 10

^{–34}J s

^{−1}; T-temperature, °C.

## 3. Results and Discussion

#### 3.1. Physicochemical Characterization

#### 3.2. Mathematical Modeling

^{2}) and chi-square test (χ

^{2}) for the mathematical models fitted to the experimental data of the drying kinetics of the mass of jambu leaves and foam at temperatures of 50, 60 and 70 °C and thickness of 1.0 cm Table 3.

^{2}higher than 99%, lower estimated mean error (SE) and chi-square test (χ

^{2}), as well as relative mean error (P) lower than 10%, which is considered as an adequate representation of the model [21].

^{−1}(samples without foam mat) to 43–48 kJ mol

^{−1}(samples with foam mat). These differences in activation energy may result from the variation in effective diffusivity, depending on the variability and physical structure of the sample, chemical composition, geometry and air drying temperature [28].

#### 3.3. Thermodynamic Properties

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Moisture content ratio in the drying of crushed mass of jambu leaves, obtained experimentally and estimated by the Wang & Singh and Midilli models for the different drying conditions.

**Figure 2.**Mean value of the effective diffusion coefficient (D) obtained in the drying of the crushed mass of jambu leaves and foam at temperatures of 50, 60 and 70 °C.

Model Designation | Model | Equation | |
---|---|---|---|

1 | Page | $\mathrm{RX}=expexp\left(-k\ast {t}^{n}\right)$ | (2) |

2 | Midilli | $\mathrm{RX}=a\ast expexp\left(-k\ast {t}^{n}\right)+b\ast t$ | (3) |

3 | Henderson & Pabis | $\mathrm{RX}=a\ast expexp\left(-k\ast t\right)$ | (4) |

4 | Approximation of Diffusion | $\mathrm{RX}=a\ast expexp\left(-k\ast t\right)+\left(1-a\right)\ast expexp\left(-k\ast b\ast t\right)$ | (5) |

5 | Two Terms | $\mathrm{RX}=a\ast expexp\left(-{k}_{0}\ast t\right)+b\ast expexp\left(-{k}_{1}\ast t\right)$ | (6) |

6 | Two-Term Exponential | $\mathrm{RX}=a\ast expexp\left(-k\ast t\right)+\left(1-a\right)\ast expexp\left(-k\ast a\ast t\right)$ | (7) |

7 | Logarithmic | $\mathrm{RX}=a\ast expexp\left(-k\ast {t}^{n}\right)+c$ | (8) |

8 | Thompson | $\mathrm{RX}=\frac{(-a-{\left({a}^{2}+4\ast b\ast t\right)}^{0,5})}{2}\ast b$ | (9) |

9 | Newton | $\mathrm{RX}=expexp\left(-k\ast t\right)$ | (10) |

10 | Verma | $\mathrm{RX}=a\ast expexp\left(-k\ast t\right)+\left(1-a\right)\ast expexp\left(-{k}_{1}\ast t\right)$ | (11) |

11 | Wang & Singh | $\mathrm{RX}=1+a\ast t+b\ast {t}^{2}$ | (12) |

12 | Valcam | $\mathrm{RX}=a+b\ast t+c\ast {t}^{1,5}+d\ast {t}^{2}$ | (13) |

_{0}, k

_{1}-Drying constants; h

^{−1}; a, b, c, n-Coefficients of the models; t-Drying time, h.

**Table 2.**Mean values of the physicochemical composition of the mass of jambu leaves, foam and powder obtained under different drying conditions.

Material | Temperature °C | Analyses | ||||
---|---|---|---|---|---|---|

Moisture Content (% w.b.) | Protein % | Lipids % | Ash % | Total Titratable Acidity * % | ||

Fresh mass of jambu leaves | --- | 92.71 ± 0.29 | 3.39 ± 0.23 | 0.24 ± 0.08 | 1.34 ± 0.04 | 0.03 ± 0.0 |

Foam | --- | 90.31 ± 0.05 | 3.30 ± 0.22 | 0.26 ± 0.07 | 1.31 ± 0.01 | 0.04 ± 0.0 |

Dried mass of jambu leaves | 50 | 5.70 aA | 28.33 aA | 0.78 aB | 17.18 aA | 0.26 aA |

60 | 3.79 aB | 30.44 aA | 0.68 aB | 16.28 aA | 0.29 aA | |

70 | 2.21 aB | 28.48 aA | 0.69 aB | 16.32 aA | 0.27 aA | |

Dried foam | 50 | 6.29 aA | 24.75 aA | 4.72 aA | 14.20 aB | 0.24 aA |

60 | 7.67 aA | 22.98 aB | 4.71 aA | 13.74 aA | 0.24 aA | |

70 | 6.58 aA | 23.37 aA | 4.09 aA | 13.00 aB | 0.20 aB |

**Table 3.**Estimated mean error (SE), relative mean error (P), coefficient of determination (R

^{2}) and chi-square test (χ

^{2}) for the twelve models analyzed in the drying of crushed mass of jambu leaves.

Model | Mass of Leaves | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

50 °C | 60 °C | 70 °C | ||||||||||

SE (Decimal) | P (%) | χ^{2} (Decimal) × 10^{−}³ | R^{2} (%) | SE (Decimal) | P (%) | χ^{2} (Decimal) × 10^{−}³ | R^{2} (%) | SE (Decimal) | P (%) | χ^{2} (Decimal) × 10^{−}³ | R^{2} (%) | |

Wang & Singh | 0.0074 | 6.90 | 0.055 | 99.94 | 0.009 | 7.075 | 0.09 | 99.91 | 0.011 | 4.2 | 0.13 | 99.87 |

Verma | 0.0881 | 75.95 | 7.7532 | 91.79 | 0.180 | 167.9 | 32.51 | 68.31 | 0.012 | 4.9 | 0.15 | 99.85 |

Valcam | 0.0363 | 35.56 | 1.3184 | 98.60 | 0.047 | 54.3 | 2.19 | 97.86 | 0.042 | 10.7 | 1.77 | 97.19 |

Thompson | 0.0528 | 46.59 | 2.7902 | 96.96 | 0.051 | 56.8 | 2.65 | 97.33 | 0.060 | 27.5 | 3.54 | 96.36 |

Page | 0.0234 | 20.44 | 0.5494 | 99.40 | 0.024 | 26.2 | 0.56 | 99.44 | 0.021 | 10.1 | 0.46 | 99.53 |

Newton | 0.0521 | 46.58 | 2.7099 | 96.96 | 0.051 | 56.8 | 2.57 | 97.33 | 0.058 | 27.5 | 3.42 | 96.36 |

Midilli | 0.0069 | 6.13 | 0.0480 | 99.95 | 0.009 | 8.2 | 0.07 | 99.93 | 0.006 | 2.4 | 0.03 | 99.97 |

Logarithmic | 0.0083 | 8.22 | 0.0686 | 99.93 | 0.009 | 9.3 | 0.08 | 99.92 | 0.008 | 3.6 | 0.07 | 99.93 |

Henderson & Pabis | 0.0460 | 41.13 | 2.1155 | 97.69 | 0.043 | 49.5 | 1.88 | 98.11 | 0.049 | 22.9 | 2.39 | 97.54 |

Two-term exponential | 0.0528 | 46.58 | 2.7896 | 96.96 | 0.051 | 56.8 | 2.65 | 97.33 | 0.060 | 27.5 | 3.54 | 96.36 |

Two terms | 0.0236 | 21.84 | 0.5576 | 99.43 | 0.045 | 49.5 | 2.01 | 98.11 | 0.024 | 11.5 | 0.57 | 99.46 |

Approximation of diffusion | 0.0094 | 8.98 | 0.0882 | 99.91 | 0.011 | 10.6 | 0.13 | 99.87 | 0.012 | 4.9 | 0.15 | 99.85 |

Model | Foam | |||||||||||

50 °C | 60 °C | 70 °C | ||||||||||

SE (decimal) | P (%) | χ^{2} (decimal) × 10^{−}³ | R^{2} (%) | SE (decimal) | P (%) | χ^{2} (decimal) × 10^{−}³ | R^{2} (%) | SE (decimal) | P (%) | χ^{2} (decimal) × 10^{−}³ | R^{2} (%) | |

Wang & Singh | 0.006 | 4.1 | 0.03 | 99.97 | 0.010 | 6.2 | 0.10 | 99.90 | 0.016 | 7.3 | 0.27 | 99.77 |

Verma | 0.242 | 154.6 | 58.38 | 44.48 | 0.346 | 217.0 | 119.56 | 0.00 | 0.439 | 313.1 | 192.32 | 0.00 |

Valcan | 0.050 | 37.2 | 2.50 | 97.62 | 0.043 | 29.4 | 1.83 | 98.40 | 0.052 | 38.2 | 2.72 | 97.80 |

Thompson | 0.049 | 35.8 | 2.43 | 97.61 | 0.060 | 39.8 | 3.59 | 96.73 | 0.064 | 46.4 | 4.08 | 96.52 |

Page | 0.022 | 15.6 | 0.50 | 99.50 | 0.023 | 15.0 | 0.52 | 99.52 | 0.020 | 15.2 | 0.39 | 99.67 |

Newton | 0.048 | 35.8 | 2.35 | 97.61 | 0.059 | 39.8 | 3.44 | 96.73 | 0.062 | 46.4 | 3.87 | 96.52 |

Midilli | 0.008 | 5.0 | 0.06 | 99.95 | 0.007 | 4.8 | 0.05 | 99.95 | 0.008 | 4.3 | 0.06 | 99.95 |

Logarithmic | 0.010 | 7.2 | 0.09 | 99.91 | 0.010 | 7.0 | 0.10 | 99.91 | 0.013 | 7.5 | 0.16 | 99.87 |

Henderson & Pabis | 0.043 | 31.5 | 1.86 | 98.17 | 0.050 | 33.6 | 2.48 | 97.74 | 0.050 | 37.5 | 2.47 | 97.89 |

Two-term exponential | 0.049 | 35.8 | 2.43 | 97.61 | 0.060 | 39.8 | 3.59 | 96.73 | 0.064 | 46.4 | 4.07 | 96.52 |

Two terms | 0.021 | 15.8 | 0.46 | 99.58 | 0.025 | 17.0 | 0.62 | 99.48 | 0.053 | 37.5 | 2.76 | 97.89 |

Approximation of diffusion | 0.010 | 7.4 | 0.10 | 99.91 | 0.013 | 8.3 | 0.17 | 99.86 | 0.019 | 10.4 | 0.37 | 99.70 |

**Table 4.**Akaike Information criterion (AIC) and Schwarz’s Bayesian Information criterion (BIC) for the models that best fitted to the drying data of the crushed mass of jambu leaves.

Model | Wang & Singh | Midilli | Logarithmic | ||||
---|---|---|---|---|---|---|---|

Drying | Temperature °C | BIC | AIC | BIC | AIC | BIC | AIC |

Mass of Leaves | 50 | −242.58 | −247.33 | −242.15 | −250.07 | −231.76 | −238.10 |

60 | −205.62 | −210.11 | −207.35 | −214.83 | −207.87 | −213.86 | |

70 | −150.17 | −154.48 | −173.45 | −180.62 | −147.70 | −153.43 | |

Foam | 50 | −224.32 | −228.62 | −189.15 | −194.88 | −189.15 | −194.88 |

60 | −156.83 | −160.61 | −169.75 | −176.04 | −154.81 | −159.84 | |

70 | −106.23 | −109.36 | −133.10 | −138.32 | −114.71 | −118.88 |

**Table 5.**Coefficients of the models that best fitted to the drying data of crushed mass of jambu leaves and foam.

Model | Temperature (°C) | Mass of Leaves | Foam | ||||||
---|---|---|---|---|---|---|---|---|---|

a | b | k | n | a | b | k | n | ||

Midilli | 50 | 0.997247 | −0.014930 | 0.073879 | 1.142397 | 0.990898 | −0.017696 | 0.157579 | 1.160437 |

60 | 1.006883 | −0.019378 | 0.127761 | 1.088324 | 0.998051 | −0.032604 | 0.242910 | 1.185451 | |

70 | 1.003061 | −0.031494 | 0.143453 | 1.178146 | 1.008538 | −0.038754 | 0.444715 | 1.204884 | |

Wang & Singh | 50 | −0.099148 | 0.002023 | −−−− | −−−− | −0.186675 | 0.008153 | −−−− | −−−− |

60 | −0.146170 | 0.004782 | −−−− | −−−− | −0.275789 | 0.016182 | −−−− | −−−− | |

70 | −0.181939 | 0.005765 | −−−− | −−−− | −0.435303 | 0.041703 | −−−− | −−−− |

**Table 6.**Mean values of enthalpy (ΔH), entropy (ΔS) and Gibbs free energy (ΔG) obtained in the drying of the crushed mass of jambu leaves with and without foam mat at temperatures of 50, 60 and 70 °C.

Mass of Jambu Leaves | |||
---|---|---|---|

Temperature (°C) | ΔH (KJ mol^{−1}) | ΔS (KJ mol^{−1} K^{−1}) | ΔG (KJ mol^{−1}) |

50 | 40.79223 | −0.27725 | 130.3847 |

60 | 40.70909 | −0.2775 | 133.1584 |

70 | 40.62595 | −0.27775 | 135.9347 |

Foam | |||

Temperature (°C) | ΔH (KJ mol^{−1}) | ΔS (KJ mol^{−1} K^{−1}) | ΔG (KJ mol^{−1}) |

50 | 28.62219 | −0.32029 | 132.1241 |

60 | 28.53905 | −0.32054 | 135.3282 |

70 | 28.45591 | −0.32079 | 138.5349 |

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**MDPI and ACS Style**

Gomes, F.P.; Resende, O.; Sousa, E.P.d.; Célia, J.A.; de Oliveira, K.B. Application of Mathematical Models and Thermodynamic Properties in the Drying of Jambu Leaves. *Agriculture* **2022**, *12*, 1252.
https://doi.org/10.3390/agriculture12081252

**AMA Style**

Gomes FP, Resende O, Sousa EPd, Célia JA, de Oliveira KB. Application of Mathematical Models and Thermodynamic Properties in the Drying of Jambu Leaves. *Agriculture*. 2022; 12(8):1252.
https://doi.org/10.3390/agriculture12081252

**Chicago/Turabian Style**

Gomes, Francileni Pompeu, Osvaldo Resende, Elisabete Piancó de Sousa, Juliana Aparecida Célia, and Kênia Borges de Oliveira. 2022. "Application of Mathematical Models and Thermodynamic Properties in the Drying of Jambu Leaves" *Agriculture* 12, no. 8: 1252.
https://doi.org/10.3390/agriculture12081252