Drying Kinetics of Noni Seeds

Noni seeds have been used for years as an important medicinal source, with wide use in the pharmaceutical and food industry. Drying is a fundamental process in the post-harvest stages, where it enables the safe storage of the product. Therefore, the present study aimed to fit different mathematical models to experimental data of drying kinetics of noni seeds, determine the effective diffusion coefficient and obtain the activation energy for the process during drying under different conditions of air temperature. The experiment used noni seeds with initial moisture content of 0.46 (decimal, d.b.) and dehydrated up to equilibrium moisture content. Drying was conducted under different controlled conditions of temperature, 40; 50; 60; 70 and 80 oC and relative humidity, 24.4; 16.0; 9.9; 5.7 and 3.3%, respectively. Eleven mathematical models were fitted to the experimental data. The parameters to evaluate the fitting of the mathematical models were mean relative error (P), mean estimated error (SE), coefficient of determination (R), Chi-square test (), Akaike Information Criterion (AIC) and Schwarz’s Bayesian Information Criterion (BIC). Considering the fitting criteria, the model Two Terms was selected to describe the drying kinetics of noni seeds. Effective diffusion coefficient ranged from 8.70 to 23.71 × 10 m s and its relationship with drying temperature can be described by the Arrhenius equation. The activation energy for noni seeds drying was 24.20 kJ mol for the studied temperature range.


Introduction
Noni (Morinda citrifolia L.) is native to southeast Asia and Australia. Commercial plantations can be found in Tahiti, Hawaii and other countries of Polynesia, where most juices commercialized worldwide are produced (Silva et al., 2012).
Noni has been traditionally used for more than 2,000 years by Polynesia and such use is attributed to the effects related to antibacterial, antioxidant, antiviral, antifungal, antitumor, analgesic, anti-inflammatory, hypotensive and immunostimulant activity (Wang et al., 2002;Costa, Oliveira, Silva, Mancini-Filho, & Lima, 2013;Lemos, Queiroz, & Figueirêdo, 2015). Since there are no selected cultivars, commercial exploitation of noni is carried out using plants grown from seeds.
Drying consists in the removal of excess water contained in the seed by evaporation, which is usually obtained by hot-air forced convection. It can also be defined, according to Goneli, Nasu, Gancedo, Araújo and Sarath (2014), as a process which involves the simultaneous transfer of energy in the form of heat and mass between the product and the drying air, being one of the main steps of post-harvest.
Mathematical modeling is used to represent the drying kinetics of various products and involves conditions such as air temperature, relative humidity, air speed and characteristics of the product. Resende, Rodrigues, Siqueira, and Arcanjo (2010) report that these studies can be applied to drying processes and systems, dimensioning, optimization and evaluation of viability of the execution on commercial scale.
Several mathematical models have been successfully used by various researchers in agricultural products, such as annatto flour (Santos, Queiroz, Figueirêdo, & Oliveira, 2013); rice grains (Corrêa, Oliveira, Oliveira, Botelho, & Goneli, 2016); peanut fruits (Araujo, Goneli, Corrêa, Hartmann Filho & Martins, 2017); sunflower grains (Smaniotto, Resende, Sousa, Oliveira, & Campos 2017); common bean grains (Quequeto, Siqueira, Ferranti, Schoeninger, & Leite, 2017); soybean grains (Botelho, Hoscher, Hauth, & Botelho, 2018); potato (Lisboa et al., To assess the fitting of the mathematical models to the drying data of plant products, several criteria can be used, such as the magnitudes of the mean relative error and mean estimated error, coefficient of determination, residual distribution and Chi-square test. However, some of these parameters have limitations, thus requiring the adoption of complementary criteria in the selection of the model to emphasize and endorse the decision-making. Thus, the Akaike Information Criterion (AIC) and Schwarz's Bayesian Information Criterion (BIC) consist in evaluating the models based on the parsimony principle, since the number of parameters in the models is variable (Gomes, Resende, Sousa, Oliveira, & Araújo Neto, 2018;Ferreira Junior, Resende, Oliveira, & Costa, 2018).
Given the above, the present study aimed to fit different mathematical models to the experimental data of drying kinetics of noni seeds, determine the effective diffusion coefficient and obtain the activation energy for the process during drying under different air temperature conditions.

Conduction of the Research
The study was carried out at the Laboratory of Post-Harvest of Plant Products of the Federal Institute of Education, Science and Technology of Goiás, Campus of Rio Verde, located in the municipality of Rio Verde, GO, Brazil.

Drying Kinetics
Noni (Morinda citrifolia L.) seeds with initial moisture content of 0.46 (decimal, d.b.) were used. The moisture contents of the product were determined by the gravimetric method in an oven at 105±1 °C, for 24 hours, in two repetitions (Brasil, 2009). Drying was conducted under different controlled conditions of temperature, 40; 50; 60; 70 and 80 °C and relative humidity, 24.4; 16.0; 9.9; 5.7 and 3.3%, respectively. Temperature and relative humidity of the ambient air were monitored by means of a data logger.
The seeds were dried on unperforated trays containing 0.15 kg of product in a completely randomized design, in four repetitions. During the drying process, the trays with samples were periodically weighed on a scale with resolution of 0.01 g until the product reached its equilibrium moisture content, i.e., constant mass.
Moisture content ratios of noni seeds during the drying, under the different air conditions, were determined using Equation 1 (Smaniotto et al., 2017): where, RX: moisture content ratio, dimensionless; X: moisture content of the product (decimal, d.b.); X e : equilibrium moisture content (decimal, d.b.); X i : initial moisture content (decimal, d.b.).

Mathematical Modeling
Mathematical models traditionally used to describe the thin-layer drying kinetics of agricultural products were fitted to the experimental data of drying, as described in Table 1.
Where, Y: value observed experimentally; Ŷ: value estimated by the model; n: number of experimental observations; and DF: degrees of freedom of the model.
In order to select a single model to describe with satisfaction the drying process of noni seeds under different air condition, the models which obtained the best fits were subjected to Akaike Information Criterion (AIC) and Schwarz's Information Criterion (BIC). According Wolfinger (1993), lower values of AIC and BIC indicate better fit of the model, and BIC is the strictest criterion. Gomes et al. (2018) state these criteria can be additionally included in the selection of drying models. These information criteria were determined by Eqs. 16 and 17.
Where, L: maximum likelihood; p: number of parameters of the model; N: total number of observations; and r: rank of the matrix X (incidence matrix for fixed effects).

Effective Diffusion Coefficient
The effective diffusion coefficient for the drying conditions was calculated by fitting the model, based on the liquid diffusion theory, to the observed data. This equation is the analytical solution for the second Fick's law, considering a cylindrical shape with six-term approximation (value established when diffusion coefficient variation is lower than 0.1 × 10 -13 m 2 s -1 ), disregarding the volumetric shrinking of the seeds, according to Brooker, Bakker-Arkema, and Hall, (1992), using the following expression: Where, t: drying time; D: liquid diffusion coefficient, m 2 s -1 ; r: equivalent radius (0.0051 m); n: number of terms; and λ n : roots of the zero-order Bessel's equation.
The volume (V s , mm 3 ) of each seed was obtained by measuring the three orthogonal axes (length, width and thickness) in thirty seeds before drying, using a digital caliper, according to the expression proposed by Mohsenin (1986): Where, a: seed longest axis; b: seed middle axis; c: seed shortest axis.
Equivalent sphere radius (r, mm) was determined using Equation 20: The influence of temperature on the effective diffusion coefficient was evaluated using the Arrhenius equation, described as follows: Where: D 0 : pre-exponential factor; R: universal gas constant, 8.314 kJ kmol -1 K -1 ; Ta: temperature, K; and E a : activation energy, kJ mol -1 .
The coefficients of the Arrhenius equation were linearized by applying the following logarithm:  Table 2 shows the magnitudes of the mean relative error (P, %), mean estimated error (SE, decimal), coefficient of determination (R 2 , %) and Chi-square test ( 2 , decimal) for the eleven models fitted, during the drying of noni seeds under the different air conditions. In relation to the mean relative error (P), it can be observed that only the models Midilli (3), Two Terms (7) and Valcam (12) had values lower than 10% for all drying temperatures studied. According to Mohapatra and Rao (2005), this parameter can be used to recommend or not a model. Mean relative error values reflect the deviation of the observed values relative to the curve estimated by the model (Kashaninejad, Mortazavi, Safekordi & Tabil, 2007). Thus, in this case, the deviation can be considered as acceptable for the models evaluated.

Results and Discussion
According to Draper and Smith (1998), the mean estimated error (SE) indicates the capacity of a model to accurately describe a certain physical process, and the lower its magnitude, the better the fitting quality of the model relative to the experimental data. Thus, the model Two terms stood out among the others, for showing the lowest values under all different conditions of the drying air (Table 2), hence demonstrating a good fit.
Based on the coefficient of determination (R 2 ), only the models Newton, Henderson and Pabis, Logarithmic and Wang and Singh were below 99%. According to Kashaninejad et al. (2007), models with coefficients of determination above 98% can satisfactorily represent the drying phenomenon. Nevertheless, Mohapatra and Rao (2005) report that the coefficient of determination as single criterion of evaluation to select drying models is not a good parameter to represent the drying phenomenon. Schwarz's used as c previously

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, and 015), the biological and physical characteristics of the different agricultural products can influence the variations of the values activation energy.

Conclusions
The model Two terms showed the best fit to the data and was selected to describe the drying kinetics of noni seeds.
The effective diffusion coefficient tends to increase with elevating temperature, showing values from 8.6968 to 23.7089 × 10 -10 m 2 s -1 .
The activation energy was equal to 24.20 kJ mol -1 , obtained through the Arrhenius equation, which establishes the dependence of the diffusivity in relation to the temperature.