Introduction
Yacon (Smallanthus sonchifolia), native to the Andes, began to be cultivated in Brazil through the Japanese colonization, in the region of Capão Bonito, SP, where the root became popularly known as ‘batata yacon’ or ‘batata diet’, being cultivated in other regions (^{Moscatto et al., 2005}; ^{Santana & Cardoso, 2008}).
Yacon roots are particularly abundant source of fructooligosaccharides (FOS) and inulin, with beneficial effect on human health for having low energetic value, reducing cholesterol and glucose levels in the blood (^{Genta et al., 2009}; ^{Ojansivu et al., 2011}).
Despite its nutritive potential, this tuber is still produced on small scale, having fast postharvest deterioration due to the high moisture content. For this reason, developing new products is highly promising, and one of the main products from this tuber, for instance, is yacon flour, has good possibility of use as functional ingredient in the formulation of various industrialized foods (^{Rodrigues et al., 2011}).
The use of mathematical modeling to predict the drying phenomenon and the evaluation of effective diffusivity in various products has been object of numerous studies (^{Kingsly et al., 2007}; ^{TorregrozaEspinosa et al., 2014}; ^{Fernandes et al., 2015}; ^{Lopes et al., 2015}; ^{Araújo et al., 2017}). It is known that effective diffusivity represents the speed with which water moves from the interior to the surface of the product; consequently, the higher the temperature the faster the water movement from the food to the environment (^{Menezes et al., 2013}).
Thus, the drying of this product emerges as a viable alternative to reverse this situation, increasing the storage period because the process reduces water activity, which is higher than 80% in plants (^{Silva et al., 2013}).
This study aimed to evaluate the effective diffusivity in yacon cylinders during the drying process.
Material and Methods
The raw material was yacon (Smallanthus sonchifolia), purchased at the Center of Supply and Logistics of Pernambuco  CEASA, in the city of Recife, PE. Samples cut in cylinders approximately 0.5 cm thick and 1.0 cm wide were subjected to a treatment of enzymatic inactivation with 3% citric acid, according to the methodology adopted by ^{Silva et al. (2013)}. Then, the samples were placed on screen trays and kept at room temperature on the workbenches of the Laboratory of Storage and Processing of Products, at the Academic Unit of Agricultural Engineering, Federal University of Campina Grande (7º 13’ S, 35º 52’ W). After draining the yacon cylinders, the initial moisture content was determined and the samples were immediately taken to the oven at 105 ºC for 24 h. Then, the cylinders were subjected to drying in a convective tray dryer (fixedbed) at different temperatures (50, 60 and 70 ºC) and air speeds (1.0; 1.5; and 2.0 m s^{1}). Drying was conducted in mean times of about 13.3, 10.3 and 7.2 h for the temperatures of 50, 60 and 70 ºC, respectively.
The material was continually weighed, counting from time 0 until reaching constant weight. Moisture content (X) was calculated based on the initial mass (m) and dry mass (ms) of the yacon cylinders, according to Eq. 1.
The experimental data were expressed as moisture ratio (RX), as shown in Eq. 2.
where:
X  moisture content of the product, d.b.;
X_{i}  initial moisture content of the product, d.b.; and
X_{e}  equilibrium moisture content of the product, d.b.
The models Approximation of Diffusion, Henderson & Pabis, Page and Two Terms, evaluated for the fit to the drying kinetics experimental data (Table 1), have also been used by ^{Radünz et al. (2010)} and ^{Araújo et al. (2017)}.
Models  Equation  

Approximation of diffusion 

(3) 
Henderson & Pabis 

(4) 
Page 

(5) 
Two Terms 

(6) 
RX  Moisture ratio (dimensionless); a, b, k, n, k_{0}, k_{1}  Constants of the equation; t  Time (min)
The parameters of the models were obtained by nonlinear regression analysis through the QuasiNewton numerical method for all models and using the program Statistica 7.0. (^{Statistica, 2005}). The model with best fit was evaluated using coefficient of determination (R2) and mean square deviation (MSD), as presented in Eq. 7.
where:
MSD  mean square deviation;
RX_{pred}  moisture ratio predicted by the model;
RX_{exp}  experimental moisture content; and
n  number of observations.
The diffusion model based on the Fick’s Law (Eq. 8) was applied to estimate effective diffusivity (Def) of yacon cylinders considering the geometry of the product as an infinite cylinder for the four first terms of the series, using the program Statistica 7.0 (^{Statistica, 2005}).
where:
RX  moisture ratio, dimensionless;
Def  effective diffusivity, m^{2} s^{1};
n  number of terms;
t  drying time, s;
R  cylinder radius, m; and,
μ_{n}  roots of the Bessel’s equation.
Results and Discussion
Table 2 presents the parameters, coefficients of determination and means square deviations of the models Approximation of Diffusion, Two Terms, Henderson & Pabis and Page fitted to the experimental drying data of yacon cylinders, at air speed of 1.0 m s^{1} and drying air temperatures of 50, 60 and 70 ºC.
Model  Temperature (ºC)  Parameter  R^{2}  MSD  

a  k  b  
Approximation of Diffusion  50  0.0133  0.1124  0.0625  0.9999  0.0049  
60  1.0031  0.0133  0.8353  0.9998  0.0060  
70  0.0276  0.0858  0.1454  0.9996  0.0093  
a  k_{0}  b  k_{1}  
Two Terms  50  0.4930  0.0068  0.4930  0.0068  0.9998  0.0052  
60  0.5246  0.0094  0.4846  0.0094  0.9996  0.0069  
70  0.4962  0.0128  0.4962  0.0128  0.9995  0.0103  
a  k  
Henderson & Pabis  50  0.9918  0.0070  0.9998  0.0052  
60  1.0092  0.0094  0.9998  0.0069  
70  0.9925  0.0128  0.9995  0.0103  
k  n  
Page  50  0.0077  0.9839  0.9998  0.0056  
60  0.0080  1.0342  0.9998  0.0060  
70  0.0143  0.9758  0.9995  0.0738 
Among the models fitted (Table 2), Approximation of Diffusion showed the highest coefficients of determination and lowest mean square deviations. However, all other models tested can be used to represent the yacon drying process. TorregorzaEspinosa et al. (2014) fitted the Page model to the drying curve of cassava cv. ‘Corpoica Verônica’, at temperature of 70 ºC with air speed of 1.0 m s^{1} and observed good fits of the model to the experimental curve. As to the parameter k, it was observed that between the temperatures of 60 and 70 ºC there was an increase in k and, according to ^{Martins et al. (2015)}, the higher its value the higher the effective diffusivity in the drying process.
For the Two Terms model, k_{0} and k_{1} increased as the temperature increased; on the other hand, the parameters a and b showed undefined behavior. A study conducted by ^{Ruhanian & Movagharnejad (2016)}, on the drying of potato slices, also found good fit of this same model to the drying data. The parameters k of the models Henderson & Pabis and Page increased with the increment in temperature, corroborating the results of ^{Lopes et al. (2015)}. The constant n of the Page model increased between 50 and 60 ºC. ^{Reis et al. (2015)} observed no clear trend of variation in the parameter k of the Page model with the increase in drying air temperature.
Table 3 presents the parameters, coefficients of determination and mean square deviations of the models Approximation of Diffusion, Two Terms, Henderson & Pabis and Page fitted to the experimental drying data of yacon cylinders, at air speed of 1.5 m s^{1} and drying air temperatures of 50, 60 and 70 ºC.
Model  Temperature (ºC)  Parameter  R^{2}  MSD  

a  k  b  
Approximation of Diffusion  50  0.0397  0.0829  0.9750  0.9999  0.0027  
60  0.1967  0.0310  0.2756  0.9999  0.0034  
70  0.0109  0.0604  0.2038  0.9999  0.0037  
a  k_{0}  b  k_{1}  
Two terms  50  0.0403  0.0849  0.9604  0.0080  0.9999  0.0027  
60  0.8004  0.0085  0.1983  0.0304  0.9999  0.0034  
70  0.4987  0.0124  0.4987  0.0124  0.9999  0.0039  
a  k  
Henderson & Pabis  50  0.9808  0.0083  0.9997  0.0061  
60  0.9678  0.0101  0.9992  0.0134  
70  0.9974  0.0124  0.9999  0.0039  
k  n  
Page  50  0.0109  0.9479  0.9999  0.0047  
60  0.0169  1.8952  0.9999  0.0045  
70  0.0129  0.9917  0.9999  0.0038 
All models satisfactorily represented the drying process of yacon cylinders (Table 3). These results corroborate those of ^{Reis et al. (2015)}, obtained in the drying of pepper. For the model Approximation of Diffusion, the constant k decreased with the increment in temperature between 50 and 60 ºC, and the parameter b decreased as temperature increased. For the Henderson & Pabis model, the constant k increased with the increment in temperature. Similar behavior was observed by ^{Santos et al. (2010)} for the drying of potato slices at temperatures of 50 and 60 ºC and air speed of 1.5 m s^{1}. In addition, the constants k and n of the Page model showed an increase between 50 and 60 ºC, followed by slight reduction when the drying temperature increased from 60 to 70 ºC. ^{Santos et al. (2010)} observed, for the Page model, that the value of n increased with the increment in temperature.
Table 4 presents the parameters, coefficients of determination (R^{2}) and mean square deviations (MSD) of the models Approximation of Diffusion, Two Terms, Henderson & Pabis and Page fitted to the experimental drying data of yacon cylinders, at air speed of 2.0 m s^{1} and drying air temperatures of 50, 60 and 70 ºC.
Models  Temperature (ºC)  Parameter  R^{2}  MSD  

a  k  b  
Approximation of Diffusion  50  0.0153  0.0942  0.0689  0.9975  0.0175  
60  0.0074  0.2132  0.0419  0.9999  0.0038  
70  0.1317  0.0438  0.3279  0.9999  0.0036  
a  k_{0}  b  k_{1}  
Two Terms  50  0.4959  0.0065  0.4963  0.0065  0.9975  0.0178  
60  0.4882  0.0089  0.5067  0.0089  0.9999  0.0040  
70  0.1303  0.0454  0.8714  0.0143  0.9999  0.0035  
a  k  
Henderson & Pabis  50  0.9922  0.0065  0.9987  0.0178  
60  0.9950  0.0080  0.9999  0.0040  
70  0.9827  0.0159  0.9996  0.0085  
k  n  
Page  50  0.0067  0.9961  0.9986  0.0181  
60  0.0093  0.9938  0.9999  0.0044  
70  0.0207  0.9405  0.9999  0.0043 
Based on the obtained results (Table 4), all models fitted well to the experimental data. Once again, the model Approximation of Diffusion behaved as the best model at the temperatures of 50 and 60 ºC, and the Two Terms was the best model at 70 ºC.
For the model Approximation of Diffusion, the constant k increased between the temperatures of 50 and 60 ºC and decreased between 60 and 70 ºC. For the Two Terms model, the constants k_{0}, k_{1} and b increased with the increment in temperature. For the model Henderson & Pabis, there was an increment in the constant k with the increase in temperature.
Comparing all models at the three drying air speeds, there was a trend of increase in k values, the drying rate constant, which represents the effect of the external drying conditions, with the increase of temperature, a result confirmed in each increment of temperature in the model Henderson & Pabis. For the parameter n, which represents the effects of the internal conditions of the material on the drying process (^{Perez et al., 2013}), the condition is clearly observed at the air speeds of 1.0 and 1.5 m s^{1}, possibly denoting a decreasing influence of the internal medium with the increment in air speed.
Based on the results presented in Tables 2, 3 and 4, both drying air speed and temperature influenced the drying time of the samples, which has also been confirmed by ^{Kingsly et al. (2007)}, ^{Lopes et al. (2015)}, ^{Araújo et al. (2017)}, and other authors. There was a gradual reduction in the drying time with the increase in temperature and air speed, proving that these variables significantly influenced yacon drying. According to ^{Shi et al. (2013)}, the higher the heat transfer rate, the more easily water molecules move, and it facilitates water movement from inside to the surface of the product.
Table 5 presents the values of water effective diffusivity and their respective coefficients of determination (R^{2}) as a function of temperature and speed of the drying air.
Air speed (m s^{1})  Def (m^{2} s^{1})  

50 ºC  R^{2}  60 ºC  R^{2}  70 ºC  R^{2}  
1.0  1.18 x 10^{9}  0.986  1.28 x 10^{9}  0.984  1.94 x 10^{9}  0.980 
1.5  0.96 x 10^{9}  0.985  1.57 x 10^{9}  0.980  2.15 x 10^{9}  0.976 
2.0  1.08 x 10^{9}  0.983  1.61 x 10^{9}  0.983  1.90 x 10^{9}  0.973 
There was a progressive increase in the effective diffusivity with the increase of temperature, at all air speeds (Table 5). The effect of increasing temperature on the increment of effective diffusivity, a consequence of the progressive increase in the molecular agitation in the liquid phase, perhaps followed by increase in cell wall permeability, has been found by various authors, such as ^{Mercali et al. (2008)}, in the osmotic dehydration of banana cylinders, in which water effective diffusivity increased from 5.37 x 10^{10} to 6.30 x 10^{10} m^{2} s^{1} as temperature increased from 25 to 55 °C; and ^{Portela et al. (2014)}, who dried 2.5mmthick slices of macambira (Bromelia laciniosa) at temperatures of 43, 49 and 56 ºC, at air speed of 1.0 m s^{1}, and observed that the increase of temperature caused increment in effective diffusivity, showing values of 1.81 x 10^{10}, 2.37 x 10^{10} and 4.09 x 10^{10} m^{2} s^{1}, respectively.
At the temperature of 60 ºC, the effective diffusivity increased as air speed increased; however, this increment was only observed at 50 ºC between the speeds 1.5 and 2.0 m s^{1}; whereas at the temperature of 70 ºC, the diffusivity increased with the increment in air speed between 1.0 and 1.5 m s^{1}. With this, it was observed that air speed irregularly influenced the values of effective diffusivity among the samples. According to ^{Kingsly et al. (2007)}, there is a complex of phenomena involved in the simultaneous transfer of heat and mass, such as the structural nature of the material and the magnitude of the moisture content in the product. ^{Fernandes et al. (2015)} evaluated the effect of drying air speed on the effective diffusivity of apple cubes dried in convective dryer at the temperatures of 45 and 60 ºC with air speeds of 1, 2, 3 and 5 m s^{1}, and observed an increase in diffusivity with the increment in air speed at both drying temperatures, and also that the diffusivity increased with the increment in temperature, ranging from 0.58 x 10^{9} to 1.89 x 10^{9} m^{2} s^{1}.
The reduction in effective diffusivity which occurred as air speed increased at the temperatures of 50 and 70 ºC can be explained by the amount of sugars present in the yacon. The study conducted by ^{Shi et al. (2013)}, with heat pump drying of 4mmdiameter yacon cylinders at temperature of 45 ºC and air speed of 1.5 m s^{1}, found water diffusivity of 7.388 x 10^{7} m s^{1}, which was much higher than that found in the present study at temperature of 50 ºC for the same air speed. These authors point out that the difference between the values of effective diffusivity is due to a series of factors such as variety, moisture content, sample geometry, chemical pretreatment, type of dryer and temperature used. ^{Alves (2010)} also found in the drying kinetics of avocado that the effective diffusivity increased with the increment of temperature, with values of 8.3 x 10^{11}, 11.3 x 10^{11} and 12.7 x 10^{11} m^{2} s^{1} in the temperatures of 50, 60 and 70 ºC, respectively.
Figure 1 presents the curves fitted with the Fick’s diffusion model for the different drying temperatures and drying air speeds.
According to Figure 1, the experimental data at temperatures of 50 and 60 ºC remained very similar along the entire drying process and, at all temperatures, the fitted curves were slightly away from the experimental data from 100 min on, denoting lower predictive accuracy at higher times.
Conclusions
The models Approximation of Diffusion, Two Terms, Henderson & Pabis and Page fitted well to the experimental data, with coefficient of determination (R^{2}) higher than 0.990, and the best fit was represented by the model Approximation of Diffusion.
The influence of air speed on the increase in effective diffusivity proved to be more efficient in the increments from 1.0 to 1.5 m s^{1} and only at higher drying temperatures.
Effective diffusivity increased with the increment of temperature, and the temperature of 70 ºC is recommended for drying processes aiming at shorter residence times.