Modelagem matemática da cinética de secagem de folhas de chicória

hb Horticultura Brasileira Hortic. Bras. 0102-0536 1806-9991 Associação Brasileira de Horticultura RESUMO Chicória é um vegetal tradicionalmente consumido como salada crua ou cozida. É uma fonte de importantes compostos nutricionais e um dos procedimentos para sua industrialização é a secagem, que aumenta a vida útil, preserva seus nutrientes e reduz as perdas por microrganismos. Esta pesquisa avaliou a cinética de secagem de folhas de chicória em diferentes temperaturas e ajustou os dados experimentais de acordo com modelos matemáticos. O delineamento experimental foi inteiramente casualizado, em triplicata, onde cada unidade amostral era uma bandeja de alumínio perfurada, contendo cerca de 100 g de folhas frescas. As folhas de chicória foram secas em estufa, a 50, 60, 70 e 80°C. Os modelos matemáticos foram ajustados de acordo com os dados experimentais e, a partir deles, a análise de regressão não-linear foi realizada pelos métodos de Gauss-Newton e Quasi-Newton. Em todas as condições, os modelos matemáticos que melhor se ajustaram à cinética de secagem das folhas de chicória foram o Midilli, o Logarítmico e o Valcam. O modelo logarítmico, nessas condições de secagem, pode ser descrito com precisão como adequado para prever e simular a cinética de secagem das folhas de chicória, pois apresentou melhores resultados nos parâmetros estatísticos avaliados. Endive is a leafy vegetable, rarely explored industrially. It contains important nutrients, mainly inulin, and is an important source of phytochemicals. Its leaves are traditionally eaten as raw and cooked salad in Mediterranean cuisine (Bayazid et al., 2020). Therefore, it becomes interesting to reduce its moisture content by drying method, which promotes an increase in shelf life, preserves its nutrients and prevents microbial growth. This processing can also enable the industrial use of the endive, especially in products that use its soluble form (Schneider et al., 2016; Santos et al., 2017). The term equilibrium is sought in all areas of knowledge; however, there is a relative portion of variables that hinder the knowledge and standardization of this equilibrium condition for different types of products. To assess drying conditions, hygroscopic equilibrium is essential. Therefore, mathematical modeling has been used to identify and optimize industrial operations (Gasparin et al., 2017). Mathematical modeling can be used as an aid tool to identify the equilibrium moisture content and the mathematical model that best describes the drying kinetics. The mathematical models contribute to the sizing of drying procedures, optimizing time, standardizing mechanized activities and reducing financial costs (Brooker et al., 1992; Berbert et al., 1995). Therefore, the aim of this research was to (i) investigate the drying kinetics of endive leaves, (ii) obtain drying curves by fits of mathematical models, (iii) determine the best fit using statistical analysis. MATERIAL AND METHODS The endive leaves were purchased at a store from Rio Verde, Goias, Brazil and then, selected according to similar visual aspects and absence of physical injuries. The leaves were sanitized in chlorinated water for 15 min and dried on paper towels. Approximately 100 g of fresh leaves were uniformly distributed in a thin layer on the perforated rectangular aluminum trays (30x15x2 cm), and dried in an oven, using forced air circulation at 50, 60, 70 and 80°C, to determine the drying kinetics. The temperature and relative humidity of the ambient air were monitored using a data logger, and the relative humidity inside the oven was obtained through the basic principles of psychometrics with the GRAPSI computer program. To determine the drying curves and fit the models, the leaves were dried to a constant mass. The moisture content was determined in an oven at 105 ± 3°C, in three replicates, until constant mass was reached (Zenebon et al., 2008). At the end of each drying process, the final equilibrium moisture content was determined, being 9.1; 7.4; 5.61 and 4.28% (d.b.), respectively, for the drying temperatures of 50, 60, 70 and 80°C. The product’s moisture content ratios were determined by Equation 1. RX= X * - X e * X i * - X e * (1) where RX is the moisture ratio (non-dimensional); is the moisture content (% d.b.); is rhe initial moisture content (% d.b.); and, is the equilibrium moisture content (% d.b.). The experimental data obtained from the drying of the endive leaves were used to adjust the mathematical models, with twelve empirical and non-empirical equations (Equations 2 to 13) commonly used to describe the drying processes of vegetable products (Box 1). Box 1 Mathematical models (Equations. 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 and 13) applied to drying kinetics data. Rio Verde, IF Goiano, 2020. t = Time (h); k, k0, k1 = Equation constants (h-1); a, b, c, n = Equation parameters; RX = Moisture ratio The mathematical models were adjusted according to the experimental data, and, from these, non-linear regression analysis was carried out by the Gauss-Newton and Quasi-Newton methods. The models were selected based on the magnitude of the coefficient of determination (R2), the chi-squared test (χ2), the mean relative error (P) and the estimated standard deviation (SE) (Equations. 14, 15and 16), at the 5% significance level and the 95% confidence interval (p<0.05), according to adaptations presented by Martins et al. (2018). P = 100 n ∑ |Y-Ŷ| Y (14) SE = ∑(Y-Ŷ 2 ) DF (15) χ 2 = Σ (Y-Ŷ) 2 DF (16) Where Y is the experimental value; Ŷ is the estimated value; n is the number of data sets; and, DF is the degree freedom. In addition to the above parameters, Akaike’s information criterion (AIC) and Schwarz’s Bayesian information criterion (BIC) were used (Equations 17 and 18). The AIC is used to compare non-nested models or to compare three or more models. Lower AIC values reflect a better fit (Akaike, 1974). AIC = -2ln(L) + 2k (17) Where k is the number of estimated parameters; and, L is the value of the likelihood. BIC also considers the degree of parameterization of the model, and therefore, the smaller the BIC value, the better the model adjustment (Schwarz, 1978). BIC = -2ln(L) + 2ln(N)k (18) where: k is the number of estimated parameters; L is the value of the likelihood; and, N is the number of recorded measurements. RESULTS AND DISCUSSION Figure 1 shows the moisture ratio of endive leaves (Cichorium intybus) during drying time at 50, 60, 70 and 80°C. Figure 1 Moisture ratio of endive leaves during drying at 50, 60, 70 and 80°C. Rio Verde, IF Goiano, 2020. As shown, the drying time of leaves decreases with increases of drying air temperature. In accordance to Gomes et al. (2017), this is due to the pressure difference of the air vapor saturated with the water inside the vegetable product, allowing the water to move from inside the leaves to the drying air in a shorter period. This fact directly influences the quality of the leaves, and a long drying can cause color deterioration, losses in nutritional composition and lead to the growth of molds (Babu et al., 2018). In addition, it is necessary to evaluate the bioactive compounds of the formed product at different drying temperatures, as since most of these compounds are known to be susceptible to degradation at elevated temperatures. There was a faster loss of moisture content at the beginning of the drying process at all temperatures, due to the greater ease of water removal on the product surface. According to Babalis et al. (2006), after the evaporation of surface water, the velocity of drying air is minimized due to the water gradually moving to the outermost layers of the product, prevailing the process of liquid diffusion that will be influenced by the temperature of drying air. At higher temperatures, the moisture ratio of the final content was lower. According to Sagrin & Chong (2013), this occurred because more heat was supplied to the samples at higher temperatures, thus inducing more evaporation of leaf moisture. The average times to complete the drying process were 6.5; 4; 2.33 and 2 hours, at temperatures of 50, 60, 70 and 80ºC, respectively. Similar drying times were observed in drying of salvia leaves (Radünz et al., 2010), Azadirachta indica (Vidal et al., 2016), Bauhinia forficata (Silva et al., 2017) and thin layer drying of scent leaves (Ocimum gratissimum) and lemon basil leaves (Ocimum africanum) (Mbegbu et al., 2021). The statistical parameters for each drying condition obtained using empirical and non-empirical equations are shown in Table 1. According to the Chi-square test, all the adjusted models showed values within the 95% confidence interval. As this is an analysis that evaluates the difference in the model’s estimate, some authors evaluate the values of this parameter and recommend adjustments with lower values (Günhan et al., 2005; Oliveira et al., 2018). The Midilli and Valcam generally showed lower values for this parameter under all drying conditions. All the adjustments of the models presented coefficients of determination above 98% (Table 1). Madamba et al. (1996) indicated magnitude values of coefficient of determination (R²) greater than 95% for fitting mathematical models to experimental data. However, these criteria cannot be used as decisive. The recommended fit of mathematical drying models indicates that the average relative error (P) should be less than 10%, and this criterion is often used in the mathematical modeling of drying processes (Mohapatra & Rao, 2005; Vidal et al., 2016; Martins et al., 2018; Oliveira et al., 2018). Thus, for drying of endive leaves at 50ºC, the Thompson, Wang & Singh, and Two-terms exponential models aren’t recommended, while at temperatures of 60 and 80ºC only the models of Midilli, Logarithmic and Valcam are recommended to estimate the drying kinetic of this product. At temperature of 70ºC, only Midilli, Logarithmic, Two-term and Valcam models are recommended for drying kinetics adjustments. Table 1 Statistical parameter values obtained in the drying of the endive leaves. Rio Verde, IF Goiano, 2020. Temperature (ºC) Model SE (%) χ² P (%) R² (%) 50 Page modified 0.00042 0.00010 9.65239 99.88680 Page 0.00042 0.00010 9.65242 99.88680 Midilli 0.00007 0.00002 2.12671 99.98098 Newton 0.00168 0.00041 6.42475 99.52973 Thompson 0.00088 0.00022 11.71069 99.76177 Henderson and Pabis 0.00089 0.00022 5.68852 99.75868 Two-term 0.00019 0.00005 6.44606 99.95226 Verna 0.00179 0.00046 6.42469 99.52973 Logarithmic 0.00084 0.00021 8.20583 99.77097 Wang and Singh 0.01726 0.00432 43.68686 99.31953 Two-term Exponential 0.00086 0.00022 12.49230 99.76681 Valcam 0.00019 0.00005 4.37606 99.94725 60 Page modified 0.00157 0.00047 17.50407 99.52739 Page 0.00157 0.00047 17.50362 99.52739 Midilli 0.00041 0.00014 8.16207 99.88822 Newton 0.00150 0.00043 17.56482 99.52738 Thompson 0.00157 0.00047 17.57557 99.52722 Henderson and Pabis 0.00149 0.00045 18.45762 99.55138 Two-term 0.00164 0.00055 18.45779 99.55138 Verna 0.00146 0.00046 19.37658 99.52738 Logarithmic 0.00080 0.00025 4.53503 99.77023 Wang and Singh 0.00568 0.00171 22.80828 98.28564 Two-term Exponential 0.00139 0.00042 19.36830 99.52738 Valcam 0.00043 0.00014 5.46965 99.88239 70 Page modified 0.00116 0.00032 10.04876 99.65315 Page 0.00116 0.00032 10.04881 99.65315 Midilli 0.00025 0.00008 2.98136 99.93051 Newton 0.00123 0.00033 11.89703 99.61603 Thompson 0.00128 0.00036 11.90049 99.61586 Henderson and Pabis 0.00128 0.00036 11.88703 99.61604 Two-term 0.00098 0.00030 8.28228 99.61604 Verna 0.00976 0.00282 32.98195 99.61603 Logarithmic 0.00050 0.00014 2.63217 99.82722 Wang and Singh 0.00341 0.00095 12.54784 98.61603 Two-term Exponential 0.00128 0.00036 11.89713 99.61603 Valcam 0.00025 0.00008 2.28777 99.92985 80 Page modified 0.00245 0.00074 26.87871 99.29912 Page 0.00245 0.00074 26.87878 99.29912 Midilli 0.00048 0.00016 9.07189 99.87539 Newton 0.00356 0.00103 39.87322 98.93866 Thompson 0.00371 0.00112 39.87827 98.93837 Henderson and Pabis 0.00359 0.00108 38.54671 98.97394 Two-term 0.00397 0.00132 38.54814 98.97394 Verna 0.02403 0.00760 93.70589 93.45119 Logarithmic 0.00061 0.00019 5.41558 99.83280 Wang and Singh 0.00239 0.00072 15.53913 99.31747 Two-term Exponential 0.00371 0.00112 39.87430 98.93866 Valcam 0.00042 0.00014 5.76498 99.89207 R² = Coefficient of determination, χ² = Chi-square, P = Mean relative error, SE = Estimated standard deviation After analyzing all the mathematical parameters, the Midilli, Logarithmic and Valcam models showed the best fit for drying endive leaves under all conditions. The AIC and BIC parameters were used to define the best model among them (Table 2). Table 2 Akaike information criterion (AIC) and Bayesian Schwarz information criterion (BIC) of the models with best adjustments of endive leaves drying kinetics at different temperatures. Rio Verde, IF Goiano, 2020. T (ºC) Midilli Logarithmic Valcam AIC BIC AIC BIC AIC BIC 50 138.20530 133.75340 95.35620 91.79471 119.80960 115.35770 60 73.56327 70.73852 66.18397 63.92417 72.88560 70.06085 70 94.35863 90.81838 85.53293 82.70073 94.20406 90.66380 80 71.47202 68.64727 69.64292 67.38312 73.33870 70.51396 The AIC and BIC parameters are currently used to indicate the best model among those with a good fit to the experimental data (Quequeto et al., 2019; Souza et al., 2019). Gomes et al. (2018) used these parameters to adjust mathematical models for the drying of crushed jambu dough, defining that the Logarithmic model was more suitable for most drying conditions. As stated by Akaike (1974) and Schwarz (1978), lower values for these criteria indicate a better fit, therefore, the logarithmic model is the one that best represents the drying kinetics of endive leaves under all the conditions studied, and is therefore the model selected to estimate the drying curves of the product under the different conditions (Figure 2). Figure 2 Estimated values from logarithmic model of drying kinetics of endive leaves. Rio Verde, IF Goiano, 2020. Figure 2 shows a good fit between the data estimated by the model and the experimental data. The Logarithmic model was also recommended to estimate the drying of coriander both under the action of direct and diffuse radiation, as well as drying in an oven (Sousa et al., 2018). The Logarithmic model was the best fit for the drying curves of the palm fruit (Santos et al., 2016), foamed mango pulp (Silva Filho et al., 2016), crushed jambu mass (Gomes et al., 2018), and slices of acuri (Santos et al., 2019). Table 3 presents the values of the coefficients of the Logarithmic model used in the adjustment of the equations. Table 3 Estimated values of logarithmic model coefficients to predict the drying curve of endive leaves. Rio Verde, IF Goiano, 2020. Coefficient 50ºC 60ºC 70ºC 80ºC a 0.95499 1.03069 1.05149 1.2733 k 0.52995 0.63899 0.93750 0.98224 c 0.00944 -0.05740 -0.06973 -0.14409 It can be seen that the coefficients (a) and (k) increase with increasing temperature, while the coefficient (c) shows the opposite behavior. According to Corrêa et al. (2010) this behavior of the k coefficient, which is a drying constant, is expected, as higher temperatures lead to a higher drying rate. The trend of coefficients (a) and (c) has not been described in the literature, as these coefficients are empirical and have no theoretical relationship to the fit of the equation to experimental data. The conditions influence the drying behavior, because with the increase in temperature, there was a decrease in the drying time necessary to reduce the moisture content of the endive leaves. The mathematical models of Midilli, Logarithmic and Valcam were the ones that best fitted the drying curves for all conditions. From the parameters AIC and BIC, it was determined that the best among them was the Logarithmic model, which was used to estimate the drying kinetics of the endive leaves. ACKNOWLEDGMENTS To Instituto Federal Goiano, Post-Harvest of vegetable products and Phytochemistry laboratories. REFERENCES AKAIKE, H. 1974. A new look at the statistical model identification. IEEE Transactions on Automatic Control19: 716-723. AKAIKE H 1974 A new look at the statistical model identification IEEE Transactions on Automatic Control 19 716 723 BABALIS, SJ; PAPANICOLAOU, E; KYRIAKIS, N; BELESSIOTIS, VG. 2006. 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