Convective drying kinetics of osmotically pretreated papaya cubes

: This study assessed the fitting of mathematical models to the convective drying kinetics of osmotically pre-dehydrated papaya cubes. Papaya cubes were subjected to osmotic dehydration in sucrose solutions at 40 and 50 ºBrix, at temperatures of 50 and 60 ºC, followed by complementary convective drying in forced air circulation oven under three temperatures (50, 60 and 70 °C) and constant air velocity of 1.0 m s -1 . Ten thin-layer drying mathematical models were fitted to the experimental data. The increase in air temperature and the decrease in osmotic solution concentration resulted in increased water removal rate. Based on the statistical indices, the Two Terms model was the one that best described the drying kinetics of the samples for all evaluated conditions. The effective diffusion coefficients increased with the elevation of air temperature, ranging from 1.766 x 10 -10 to 3.910 x 10 -6 m 2 s -1 , whereas the convective mass transfer coefficients ranged from 3.910 x 10 -7 to 1.201 x 10 -6 m s -1 with Biot number from 0.001 to 12.500.


Introduction
Brazil is the second largest producer of papaya in the world, having produced a total of 1.42 million tons in 2016 (FAOSTAT, 2018). Papaya has a fast ripening, which manifests itself immediately as a structural softening that, associated with its high moisture content and water activity, makes the product highly perishable, resulting in post-harvest losses throughout its chain (Kandasamy et al., 2012).
Convective drying, for its simplicity and low cost, compared to other drying methods such as lyophilization, is one of the most used technologies for the conservation of agricultural products. However, this method causes alterations in sensory and nutritional properties and in the bioactive compounds of the dry products (Gava et al., 2008;Orikasa et al., 2014). Such alterations can be minimized using combined drying methods, as proposed in the osmo-convective drying (Prosapio & Norton, 2017;Dermesonlouoglou et al., 2018).
Mathematical modeling of the drying process is fundamental for understanding and providing information about the behavior of certain parameters that describe heat and mass transfer mechanisms (Silva et al., 2014a;Tzempelikos et al., 2015;Pacheco-Angulo et al., 2016), which can provide a solid basis for optimizing the process.
The quality of fit of the models to the experimental data can be assessed with different statistical indices; however, according to Kucuk et al. (2014), the best model to describe the drying curve of the product is the one with highest values of correlation coefficient, coefficient of determination, modeling efficiency and/or adjusted R 2 and the lowest values of chisquare, mean squared deviation, relative mean percentage error, mean polarization error, standard error of estimation, residual sum of squares, reduced sum of squared errors and/ or residuals.
In this context, the objective was to assess the mathematical modeling of convective drying kinetics, at temperatures of 50, 60 and 70 °C, of papaya cubes osmotically pre-dehydrated in sucrose solutions and to obtain the effective diffusivity coefficients and convective mass transfer coefficients.

Material and Methods
To conduct this study, the raw material used was ripe papaya fruits (Carica papaya L.) cv. Formosa, 2017 Season, purchased at the local market of the city of Campina Grande, PB, Brazil. Papaya fruits were washed with neutral detergent and subsequently sanitized with sodium hypochlorite solution (100 ppm) for 15 min. The peel was removed with a stainlesssteel knife, and the seeds were discarded. The pulp was cut into cubes with dimensions of 20 mm, measured with digital caliper (Absolute model, Mitutoyo, Brazil) with resolution of 0.01 mm. The cubes were osmotically pre-dehydrated in sucrose solution (syrup) with 40 and 50 °Brix, in a cubes:syrup proportion of 1:6 (g:g), at temperatures of 50 and 60 °C. The osmotic dehydration (OD) process was carried out in a BOD chamber and lasted 4 h, considering the maximum rate of water removal from the papaya cubes during the OD. The cubes were removed from the sucrose solution with plastic sieves and left on the bench to drain excess solution from the surface.
About 25 g of the osmotically dehydrated cubes were arranged, in a single layer, in stainless-steel rectangular baskets (15 x 12 cm) and dried, in triplicate, in a forced air circulation oven (320/5 model, Foneman, Brazil) at temperatures of 50, 60, 70 °C and air velocity of 1.0 m s -1 , determined by means of a digital anemometer (ITTHAL-300 model, Instrutemp, Brazil). Water loss was monitored by weighing on an electronic scale (AS5500C model, Marte, Brazil) with a resolution of ± 0.01 g, at regular times of 5, 10, 20, 30 and 60 min, until the samples reached constant mass. The data of drying kinetics were used to calculate the drying rates (Eq. 1) (Özdemira et al., 2017) and moisture content ratios (Eq. 2) (Galaz et al., 2017).
where: DR -drying rate, kg kg -1 h -1 ; M t0 -moisture content at previous time, kg kg -1 d.b.; M t0 + Δt -moisture content at current time, kg kg -1 d.b.; and, Δt -difference between the current time (t i ) and previous time (t 0 ) of drying, min. Different mathematical models were fitted to the experimental data of drying kinetics (Eqs. 3 to 12), using the computer program Statistica®, version 7.0, through non-linear regression, by the Quasi-Newton method (Statsoft, 2007).
-Logarithmic - Yagcioglu et al. (1999): N -number of observations; and, n -number of coefficients of the model.
The geometric shape of the samples was assumed to be that of a cube (parallelepiped with equal sides), and the analytical solution of the second Fick's law for this geometry, considering internal diffusive mass flow equal to the external convective flow in the vicinity of the samples (convective boundary condition) (Eq. 18) (Silva et al., 2014b), was fitted to the experimental data of drying kinetics for determining the effective diffusion coefficients (D ef ) and convective mass transfer coefficients (h w ), using 16 x 16 x 16 terms of the analytical solution referring to the three summations of Eq. (18)  where: MR -moisture content ratio, dimensionless; a, b, c, k, k 0 , k 1 , n -coefficients of the models; and, t -drying time, min.

Results and Discussion
In the drying curves of the osmotically dehydrated papaya cubes ( Figures 1A to D) concentration and temperature of the solution increase, the drying curves ( Figures 1C and D) become less distant from one another, denoting less relative influence of drying temperature on water removal dynamics. Although the increase in concentration and temperature of the solution causes a greater water removal during the OD process (Germer et al., 2011;Souraki et al., 2014), due to the elevation of the concentration gradient of soluble solids between the fruit and the solution, it also results in the incorporation of solute in the sample (Garcia-Noguera et al., 2010;Mendes et al., 2013), which could lead to a higher resistance to heat and mass transfers during convective drying, resulting in lower drying effectiveness. Fernandes et al. (2008) demonstrated, in experiments with pineapple, that OD in solutions with high concentrations (> 35 °Brix) results in high gain of solids by the samples, which may cause a reduction in the water removal rate during convective drying.
The drying rates (Figures 2A to D), for the same concentration of the solution, in general, were higher in samples subjected to solutions at lower temperatures, ranging from 0.85 to 1.64 kg kg -1 h -1 for the samples subjected to OD pre-treatment of 50 °Brix/60 °C and 50 °Brix/50 °C, respectively, dried at 50 °C.
The values of the drying rates changed over time, gradually increasing to the maximum value and then decreasing rapidly. This occurs because, at the beginning of the drying, liquid diffusion is the main mechanism of water transport and, as the drying progresses, vapor diffusion becomes the dominant mode, so the drying rate increases. However, with the continuity of the process, the samples become unsaturated with moisture, the vapor diffusion decreases and, consequently, the drying rate also decreases (Chen et al., 2017). In addition, drying occurred mainly in the falling rate period, and no constant rate period was observed (Figures 2A to D), indicating that the internal resistance to water movement is greater than the rate of removal from the sample surface (Pilatti et al., 2016). Similar behavior was observed by Kaushal & Sharma (2016) during convective drying at different temperatures (50-70 ºC) of osmo-dehydrated jackfruit pulp.
The indices of the models fitted to the experimental data of drying kinetics of the samples (Table 1), at different temperatures, demonstrate that the Two Terms model had the highest values of the coefficients of determination (R 2 ) (0.997-0.999) and the lowest mean squared deviations (MSD) (0.008-0.014), mean relative error (P) (1.306-6.039%), mean estimated error (SE) (0.009-0.017) and chi-square (χ 2 ) (1.0 x 10 -4 -3.0 x 10 -4 ), so it better represents the drying process of the samples under the studied conditions. However, it should be pointed out that the models of Page, Approximation of  Table 1. Values of the coefficient of determination (R 2 ), mean squared deviation (MSD), mean relative error (P), mean estimated error (SE) and chi-square (χ 2 ) of the models fitted to the experimental drying data of papaya cubes subjected to osmotic dehydration (OD) Continues on the next page Continuation of Table 1 Diffusion and Midilli, also had high R 2 values, above 0.995, and low MSD, P, SE and χ 2 , below 0.021, 8.199%, 0.023 and 5.0 x 10 -4 , respectively, indicating their adequacy to represent the drying kinetics of osmotically pre-dehydrated papaya cubes. Table 2 presents the effective diffusion coefficients (D ef ) and convective mass transfer coefficients (h w ) obtained for the drying of the samples, subjected to the temperatures of 50, 60 and 70 °C. The increase in convective drying temperature (50-70 °C) causes the increment in D ef values. In addition, it was observed that samples subjected to the solutions with the same temperature, but with higher sucrose concentration, in particular for the drying temperature of 70 ºC, offer greater resistance to external mass transfer, which in turn may be related to the reduction of h w . This behavior may be associated with increased concentration of soluble solids during the OD, on the surface of the sample (Rodríguez et al., 2015;Sangeeta Bi -Biot number; χ 2 -Chi-square Table 2. Effective diffusion coefficients (D ef ) and convective mass transfer coefficients (h w ) obtained in convective drying, at temperatures (Temp) of 50, 60 and 70 °C, of osmotically pre-dehydrated (OD) papaya cubes & Hathan, 2016;Goula et al., 2017), capable of forming, at high temperatures (≥ 70 ºC), a dense and poorly permeable layer, increasing the resistance to heat transfer to the samples and establishing an additional barrier to the water exit from its interior (Munhoz et al., 2014;Corrêa et al., 2017). Similar results have been reported in strawberry (Garcia-Noguera et al., 2010) and plum (Dehghannya et al., 2016).
It is observed that the Biot number (Bi) was within the range from 0.001 to 12.500 (Table 2), which, according to Kaya et al. (2010), is indicative of the existence of internal and external resistances to water transfer, being considered the most realistic case in practical applications. It should be pointed out that Bi tended to decrease with the elevation in the drying temperature, especially at 70 °C, indicating that there is a higher resistance to mass flow on the surface of the samples (Silva et al., 2013).
The solution of the Fick's second law equation (Eq. 18), for all OD treatments, considering convective boundary condition, showed, even in Biot number << 1 (Bi = 0.001; R 2 = 0.998; χ 2 = 4.885 x 10 -3 ), adequate fit to the experimental data of drying kinetics of the samples (R 2 > 0.996 and χ 2 < 7.161 x 10 -3 ) ( Table  2), which ensures the physical representativeness of the values of D ef and h w .

Conclusions
1. Among the fitted mathematical models, the Two Terms model was selected as the most adequate for drying kinetics of osmo-dehydrated papaya cubes.
2. The effective diffusivity in the samples increased with the increase of air temperature, whereas the convective mass transfer coefficient showed a less defined trend.