Comparison of mathematical models of the kinetic of drying pennyroyal leaves

Mentha pulegium L. , popularly known as pennyroyal, has simple leaves that give off a pleasant aroma when crushed. The main objective of this work was to carry out the drying of pennyroyal leaves, to estímate the effective diffusion coefficient through drying kinetics in forced convection, and to, determine the best mathematical model at four different temperatures (40, 50, and 60 ºC) inflow. 1.5 m/s air. Analyzing the drying curves, it was observed that the drying kinetics were strongly influenced by temperature. The thin layer models that best fit the experimental data were Approximate Diffusion, Two Terms, and Logarithmic for the temperatures of 40, 50, and 60 °C, respectively. The evaluation method used the R² (coefficient of determination), RMSE (root-mean-square), and X² (chi-square), and the coefficient of determination parameter remained >0.90. The effective diffusion coefficient decreased 74% with increasing temperature from 40 ºC to 60 ºC and enthalpy and entropy decreased with increasing temperature, while Gibb's free energy increased 5% for each increment of 10 ºC in temperature.


Introduction
The drying of food products can be deliberated as a simultaneous procedure of heat and mass transfer between the product and the drying air, which consists in the removal of excessive moisture contained within the material through dissipation, caused by forced air convection heated, to allow for greater conservation of quality during storage for long seasons (Chua et al., 2000;Da Silva Morais et al., 2013;Tavone et al., 2021).
The use of mathematical models in the drying process helps researchers to better optimize, integrate and control energy during the drying process. When the water activity (Aw) is reduced to the minimum level, where a balance and stability point of free water within the food is found, chemical and biological deterioration tend to be minimized, a process that helps in food preservation and storage for long periods (Mghazli et al., 2017).
Mentha pulegium L. known in some places in Brazil as pennyroyal is a plant used as a drug in several countries due to its medicinal properties. Its composition is rich in bioactive compounds, such as antioxidants and phenolic compounds, antiinflammatory, analgesic, digestive action (Ahmed et al., 2018;Mollaei et al., 2020;Yakoubi et al., 2021). The use of dried plant extracts turns out to be more viable, as it increases shelf life and facilitates the storage of this plant material. Therefore, the aim of this study was(1) to evaluate the effect of drying by convection at different temperatures, (2) to verify which is the best mathematical model that fits in drying, (3) to calculate the effective diffusion coefficient and thermodynamic properties for drying pennyroyal leaves.

Plant Material
Mentha pulegium L.(penny royal) was acquired through the living laboratory of Alternative Agriculture practices at Faculdade Intercultura Indígena -FAIND, located at the Federal University of Grande Dourados (Mato Grosso do Sul -Brazil).
The penny royal was selected according to the color of the leaves (preferably green), which eliminated those with physical Research, Society andDevelopment, v. 11, n. 6, e33011621924, 2022 (CC BY 4.0) | ISSN 2525-3409 | DOI: http://dx.doi.org/10.33448/rsd-v11i6.21924 3 damage, then they were washed and sanitized in 1% sodium hypochlorite for 15 min. After this time, the penny royal leaves were stored under refrigeration at a temperature of 5 ºC in polyethylene bags.

Drying
The drying process was carried out with a tray dryer, at temperatures of 40, 50, and 60 ºC and a speed of 1.5 m/s, until obtaining a constant temperature, which varied for each temperature studied. Before drying, the initial moisture of the penny royal leaves was determined through the method of drying in an oven until reaching constant weight (AOAC, 1990). The dryer used is at the laboratory level, consisting of a drying chamber where the trays (NG Scientific brand) are placed, and the samples are deposited in these. The air pre-heating system is provided by a set of electrical resistances and an air circulation system consisting of a fan. An anemometer is used to control the speed of hot air that was inserted inside the dryer.
Before drying the samples, the dryer was turned on half an hour beforehand to stabilize the temperature. Once the temperature was stabilized, the trays containing the penny royal leaves were placed inside the dryer compartment, to start the drying process. The samples were removed from the dryer during the first hour at 15-minute intervals, then at 1-hour intervals until constant dynamic equilibrium in the Ubu samples (moisture on a wet basis) was lower than 10%, drying was carried out in triplicate for every temperature.

Moisture content and mathematical models
The different moisture contents according to the time interval and their weighing was calculated from the difference between the initial weight and the weighing, considering the weight loss. The moisture contents at different temperatures were converted about moisture (MR), according to equation 1. It is noteworthy that the MR is dimensionless.
(eq. 1) where MR is the water content ratio (dimensionless value), Mx is the water content of the product represented on a dry basis (bs); Mx 0 is the equilibrium water content of the product (bs), and Mxi is the initial water content of the product (bs).

Statistical Parameters
The mathematical models used in the analysis of the drying kinetics were selected according to the literature, and the approximate diffusion, Two Terms, Logarithmic, Henderson & Pabis, Newton, and Page models were chosen, as shown in Table   1. Table 1. Mathematical Models used in the drying kinetics of pennyroyal leaves.

Diffusion coefficient (Def) and influence of temperature on Def
The diffusion coefficient was calculated using Equation 7, based on the theory of liquid diffusion, and the Arrhenius equation was used to evaluate the influence of temperature on the effective diffusion coefficient.

Thermodynamic properties (ΔH, ΔS, ΔG)
The thermodynamic properties associated with the drying process were determined according to the method proposed by Jideani and Mpotokwana (2009). Arranged in Equations 10, 11, and 12 respectively, specific enthalpy, specific entropy, and Gibbs Free energy.

Statistical analysis
For the experimental adjustment of the drying kinetics, the computer program Statistic version 8.0 was used, using the non-linear regression analysis, by the Quasi-Newton method. The goodness of fit of the mathematical models to the observed statistical data was evaluated by the coefficient of determination (R²), chi-square (x²) and the root-mean-square error (RMSE, Root Mean Square Error).  According to Fiorentin et al. (2010), this difference between drying times from 40 to 60 °C occurs because at higher temperatures the sample starts to reduce its moisture more quickly at the beginning of drying, and therefore the drying time required will be shorter. The aforementioned effect is observed by several authors in their research as in the drying of plantains Milk et al. (2015), fermented grape pomace (Deamici et al., 2016), and strawberry drying (Oliveira, 2015).

Results and Discussion
With the drying process, the moisture contents found for each time were used to calculate the experimental moisture ratio (MR) values, which, in turn, were used to adjust the five chosen mathematical models (Table 1)     Source: Authors.
The coefficient of determination (R2) was high in all mathematical models tested in this work, above 98%, indicating quality in the adjustments. According to Madamba et al. (1996), these results indicate a satisfactory representation of the phenomenon under study, as the minimum value to have an acceptable reproduction of the models is R2 greater than 95%. Research, Society andDevelopment, v. 11, n. 6, e33011621924, 2022 (CC BY 4.0) | ISSN 2525-3409 | DOI: http://dx.doi.org/10.33448/rsd-v11i6.21924 8 The coefficients of the mathematical models are related to the drying temperature and the moisture content of the sample.
According to Azevêdo et al. (2014), This phenomenon indicates that as drying temperatures increase, the speed at which water is removed from the sample is accelerated, an action attributed to the increase in the drying rate. That is, the values of k gradually increase as the temperature increases, as it is associated with the ease of removing moisture from the sample. This phenomenon can be easily seen in Table 3, where for all the models described, there was an increase in the values of gradual k for each drying temperature. The Logarithmic mathematical model presented the lowest drying constant (k min−1) for the temperature of 40 ºC in drying with airflow with an air velocity of 1.5 m/s (k = 0.004366 ), followed by the Two Terms model for the temperature of 50 ºC (k1 = 0.00494), and Page for the temperature of 60 ºC (k = 0.00754), thus, these models correspond to the lowest drying rates.
For the Approximate Diffusion model, at the three drying temperatures, the highest values of k were found in comparison to the other models, and at 60 ºC the maximum of k was reached (0.50828).
The effect indicates that with the increase in the temperature of the drying air, there was a decrease in the time needed for the penny royal leaves to reach the equilibrium water content.
Fick's second law describes very well the dynamic behavior of the drying process during the period of decreasing moisture transfer rate over time, since effective diffusion (Def) is the main mass transfer mechanism (Henríquez et al., 2014).
The increase in temperature directly affected the effective diffusion of the sample, as shown in Table 4, there was a decrease in Def when compared to temperatures of 40 and 60 ºC. Regarding thermodynamic properties, Table 4, it can be seen that the specific enthalpy (ΔH) decreased as the temperature used in the drying kinetics increased (40, 50 and 60 ºC), confirming that the higher the temperature used, the less energy will be worn out during the drying process. On the other hand, specific entropy ( GiS) and Gibbs free energy had a reverse behavior to that of enthalpy, with an increase in values with increasing temperatures. The low entropy variation, between 0.3476 and 0.3481, is related to the low variation in the temperatures used (10ºC), and negative values are usually related to changes in the material's structure.

Final Considerations
Among the drying models studied, the Logarithmic model and the Page model satisfactorily adjusted to the drying curves obtained experimentally for penny royal leaves. The drying temperature was influenced by the kinetics, and there was a decrease in Def when compared to temperatures of 40 and 60 ºC. If we consider the most efficient time/temperature binomial for future use as conservation of penny royal leaves, the ideal for this type of plant material, sliced at 0.4 cm, would be at a temperature of 60 ºC. For future work, the research group will use the data from the work to intensify the applicability of penny royal in food matrices.