Introduction
Blackberry (Morus nigra L.) is a species belonging to the genus Morus, of the family Moraceae, having about 24 species and one subspecies, with at least 100 known varieties, found in temperate and subtropical regions (^{Ercisli & Orhan, 2007}). Morus nigra L. has medicinal properties in its fruits, leaves and roots (^{Grandi, 2014}), being widely used in the popular medicine.
Among the parts of Morus nigra L., leaves play a relevant role and one of the examples of use is the tea made from the leaves to relieve symptoms of climacteric, headache and irritation which occur in the premenstrual period, due to the presence of flavonoid compounds, and especially isoflavones (^{Lorenzi & Matos, 2008}). In this context, various researchers have studied the medicinal use of blackberry leaves (^{Miranda et al., 2010}; ^{Freitas et al., 2016}; ^{Rosa et al., 2016}).
Leaves of medicinal plants normally have high water content. Water is the main responsible for the increase in metabolic activities and chemical and physical changes which occur in the product during storage. Thus, it is fundamental to reduce water content through drying to maintain the quality of medicinal plants after harvest (^{Goneli et al., 2014}).
The study of thinlayer drying curves provides important data for the development of processes and dimensioning of equipment intended for the drying of agricultural products. These data allow the estimation of drying time, production planning and energy expenditure involved in the process, which will affect the final value of the product (^{Vilela & Artur, 2008}).
Water diffusivity in a product can be understood as the ease with which water is removed from this product during the drying process, and diffusivity is not an intrinsic characteristic to the product, because it varies according to the changes in the conditions of drying, temperature and air speed (^{Oliveira et al., 2006}). Thus, it becomes also fundamental to study the behavior of the diffusion coefficient, besides the drying curves as a function of the different drying air conditions.
In this context, this study aimed to fit mathematical models to predict the thinlayer drying curves of blackberry leaves at different temperatures and air speeds, as well as to determine the effective diffusion coefficient and activation energy during the drying process.
Material and Methods
The present study was carried at the Laboratory of Preprocessing and Storage of Agricultural Products, at the Faculty of Agrarian Sciences  FCA of the Federal University of Grande Dourados  UFGD, in the municipality Dourados, MS, Brazil, from October to November 2016.
Blackberry leaves were collected from a single plant, located at the Unit 2 of the UFGD, at the facilities of the FCA (22º 11’ 45” S and 54º 55’ 18” W, at altitude of 446 m), to guarantee the homogeneity of the product. Only leaves with no injuries or apparent incidence of diseases were selected. The leaves were always collected in the morning, avoiding the collection of leaves after rains or with dewdrops on the surface, to not compromise the characterization of their drying curves.
Blackberry leaves were dried in an experimental fixedbed dryer with automatic control of drying air speed and temperature (Figure 1). The thinlayer drying bed was composed of two 0.2 mdiameter trays with screened bottom, and the blackberry leaves were arranged in a thin layer on each tray of the experimental dryer.
1  Control panel; 2  Centrifugal fan; 3  Expansions; 4  Air homogenizers; 5  Electric resistances; 6  Plenum; 7  Air temperature measurement point; 8  Screened bottom; 9  Drying bed; 10  Trays for thinlayer drying. Source: Adapted from Martins (2015)
After collection and selection, the blackberry leaves had initial water content of approximately 2.03 (decimal, on dry basis  d.b.). Initial and equilibrium moisture contents were determined by the gravimetric method in the oven, at 103 ± 1 ºC for 24 h, in triplicate (^{ASABE, 2010}).
The drying tests of blackberry leaves were conducted for different drying air temperatures (40, 50, 60 and 70 °C) and speeds (0.4 and 0.8 m s^{1}) in a completely randomized design, with four replicates. Blackberry leaves were dried until they reached the equilibrium moisture content, but for mathematical modelling purposes a final moisture content of 0.11 ± 0.01 (decimal, d.b.) was considered. The moisture content of the blackberry leaves during the drying under the different conditions of the air was determined by Eq. 1.
where:
MR  moisture ratio of the product, dimensionless;
M  moisture content of the product at a certain time, decimal, d.b.;
M_{e}  equilibrium moisture content of the product, decimal, d.b.; and,
M_{i}  initial moisture content of the product, decimal, d.b.
Mathematical models traditionally used to predict the thinlayer drying of agricultural products were fitted to the moisture content data observed during the drying of blackberry leaves. The mathematical models presented in Table 1 have been used by various researchers in studies with leaves of medicinal plants (^{Prates et al., 2012}; ^{Goneli et al., 2014}; ^{Martins et al., 2015}; ^{Silva et al., 2015}; ^{Gasparin et al., 2017}; ^{Gomes et al., 2017}).
Model designation  Model  Eq. 

Approximation of diffusion 

(2) 
Two terms 

(3) 
Twoterm exponential 

(4) 
Henderson & Pabis 

(5) 
Logarithmic 

(6) 
Midilli 

(7) 
Newton 

(8) 
Page 

(9) 
q  Drying time; h; k, k_{o}, k_{1}  Drying constants; h^{1}; a, b, c, n  coefficients of the models
Effective diffusion coefficients of blackberry leaves were obtained by fitting the liquid diffusion mathematical model (Eq. 10), with eight terms, to the data observed during the drying under different air conditions.
where:
D_{i}  effective diffusion coefficient, m^{2} s^{1};
L  product thickness, m; and,
n_{t}  number of terms of the model.
Thickness (L) of blackberry leaves was measured using a digital caliper, with 0.01 mm resolution. Thickness was measured in 50 leaves, with six measurements at different points in each one. After the measurements, mean thickness was calculated and was equal to 0.427 mm. The effect of temperature on the effective diffusion coefficient was assessed using the Arrhenius equation, as described in Eq. 11.
where:
D_{o}  preexponential factor;
E_{a}  activation energy, kJ mol^{1};
R  universal gas constant, 8.314 kJ kmol^{1} K^{1}; and,
T_{a}  absolute temperature, K.
The mathematical models were fitted to the moisture ratio data observed during the thinlayer drying of blackberry leaves, under the different conditions of the air, through nonlinear regression analysis by the GaussNewton method using a statistical computer program.
The mathematical model to represent the drying of blackberry leaves was selected by assessing the degree of fit of each model, based on the magnitude of the adjusted coefficient of determination (R^{2}), mean relative error (P), standard deviation of the estimate (SE) and residual distribution behavior. P and SE values were calculated using Eqs. 12 and 13, respectively.
where:
n_{o}  number of experimental observations;
Y  value observed experimentally;
Ŷ  value estimated by the model; and,
DF  degrees of freedom of the model.
Results and Discussion
To select mathematical models to represent the thinlayer drying of agricultural products, the mean relative error (P) is considered as a parameter of exclusion of models because, according to ^{Mohapatra & Rao (2005)}, models with P higher than 10% are inadequate to represent the drying process. P values indicate the deviation of the observed data from the curve estimated by the model (^{Kashaninejad et al., 2007}). Another parameter also considered as of exclusion of mathematical models is the residual distribution because, according to ^{Goneli et al. (2009)}, if a model has biased residual distribution it is considered as inadequate to represent the phenomenon, but if it has random residuals (residual values distributed close to the horizontal strip around zero), it is considered as acceptable.
Thus, based on these two statistical parameters of the models fitted to the observed moisture ratio data of blackberry leaves (Table 2), the only acceptable model to represent the thinlayer drying, for all conditions of drying air temperature and speed evaluated in the present study, is the Midilli model (Eq. 7).
Models  0.4 m s^{1}  0.8 m s^{1}  

SE (decimal)  P (%)  R^{2} (decimal)  Residual  SE (decimal)  P (%)  R^{2} (decimal)  Residual  
40 °C  
Approximation of diffusion  0.0169  8.4960  0.9971  BS  0.0067  4.9932  0.9995  RD 
Two terms  0.0312  18.2692  0.9902  BS  0.0342  35.5409  0.9862  BS 
Twoterm exponential  0.0178  9.4502  0.9967  BS  0.0170  16.9787  0.9965  BS 
Henderson & Pabis  0.0307  18.2690  0.9902  BS  0.0311  30.5449  0.9884  BS 
Logarithmic  0.0098  5.1307  0.9990  RD  0.0079  6.6288  0.9992  RD 
Midilli  0.0082  2.9980  0.9993  RD  0.0057  3.2213  0.9996  RD 
Newton  0.0326  20.5178  0.9888  BS  0.0345  35.1279  0.9854  BS 
Page  0.0194  9.2402  0.9961  BS  0.0172  15.0631  0.9964  BS 
50 °C  
Approximation of diffusion  0.0289  25.7472  0.9906  BS  0.0081  5.5964  0.9992  RD 
Two terms  0.0262  21.8166  0.9924  BS  0.0256  20.7900  0.9928  BS 
Twoterm exponential  0.0121  7.9068  0.9983  BS  0.0137  10.0025  0.9978  BS 
Henderson & Pabis  0.0255  21.8170  0.9924  BS  0.0248  20.7895  0.9928  BS 
Logarithmic  0.0135  11.6465  0.9980  BS  0.0094  7.1292  0.9990  RD 
Midilli  0.0091  5.6663  0.9991  RD  0.0087  4.9119  0.9992  RD 
Newton  0.0282  25.7474  0.9906  BS  0.0267  23.6281  0.9914  BS 
Page  0.0131  6.8514  0.9980  BS  0.0148  9.5861  0.9974  BS 
60 °C  
Approximation of diffusion  0.1484  48.7990  0.7620  BS  0.0192  12.7678  0.9963  BS 
Two terms  0.0382  29.3516  0.9847  BS  0.0407  28.8397  0.9839  BS 
Twoterm exponential  0.0198  11.1774  0.9956  RD  0.0207  14.6769  0.9955  BS 
Henderson & Pabis  0.0370  27.6202  0.9848  BS  0.0390  28.8398  0.9839  BS 
Logarithmic  0.0210  14.6172  0.9952  RD  0.0138  9.4836  0.9981  RD 
Midilli  0.0141  4.7146  0.9979  RD  0.0111  6.1239  0.9988  RD 
Newton  0.0402  32.3802  0.9816  BS  0.0432  33.4536  0.9795  BS 
Page  0.0197  7.6632  0.9957  RD  0.0194  11.2890  0.9960  BS 
70 °C  
Approximation of diffusion  0.0280  20.4360  0.9931  BS  0.0469  30.9039  0.9808  BS 
Two terms  0.0554  41.2041  0.9747  BS  0.0525  30.9043  0.9808  BS 
Twoterm exponential  0.0296  23.2203  0.9918  BS  0.0241  16.2139  0.9949  BS 
Henderson & Pabis  0.0517  42.3852  0.9749  BS  0.0469  30.9039  0.9808  BS 
Logarithmic  0.0190  14.8630  0.9968  RD  0.0125  7.4549  0.9988  RD 
Midilli  0.0162  9.5931  0.9978  RD  0.0106  4.9279  0.9992  RD 
Newton  0.0555  48.0328  0.9692  BS  0.0494  34.5807  0.9766  BS 
Page  0.0258  15.9781  0.9938  BS  0.0213  12.2643  0.9960  BS 
BS  Biased residual distribution; RD  Random residual distribution
It can also be observed in Table 2 that, based on P values and residual distribution, for air speed of 0.8 m s^{1} two models are acceptable to represent the thinlayer drying of blackberry leaves: Logarithmic (Eq. 6) and Midilli (Eq. 7). In these cases, the model that best fits to the observed data is selected considering also the highest coefficients of determination (R^{2}) and lowest standard deviations of the estimate (SE). For all air conditions, the Midilli model (Eq. 7) has the highest R^{2} values and lowest SE values among all others, thus reinforcing its best fit to the data of thinlayer drying of blackberry leaves.
^{Gasparin et al. (2017)}, studying the drying kinetics of Mentha piperita leaves at different drying air temperatures and speeds, also found that the model which fitted best to the observed data was the Midilli model. Other researchers have also found that the Midilli model was the most adequate to represent the thinlayer drying of medicinal plants, such as ^{Gomes et al. (2017)} with Cymbopogon citratus leaves, ^{Silva et al. (2015)} with ‘jenipapo’ leaves, ^{Martins et. al (2015)} with ‘timbó’ leaves and ^{Goneli et al. (2014)} with ‘aroeira’ leaves.
The better fit of the Midilli model to the observed drying data of medicinal plants, according to ^{Goneli et al. (2014)}, is probably related to the fast water loss in the initial stages of the process in this type of product, generating a drying curve that is steeper and better characterized mathematically by this model.
The moisture ratios estimated by the Midilli model were highly correlated with the observed drying data of blackberry leaves under the different drying air conditions. This is demonstrated in Figure 2, in which it is possible to observe the proximity between the data estimated by the model and the data observed during the process, thus reinforcing the applicability of this model to estimate the drying curves of blackberry leaves.
The increase in drying air temperature considerably reduces the time required for blackberry leaves to reach the moisture content of approximately 0.11 (decimal, d.b.) (Figure 2). This phenomenon was observed by ^{Radünz et al. (2011)}, who assessed the drying kinetics of ‘carqueja’, as well as other researchers studying the drying kinetics of other medicinal plants, such as ^{Prates et al. (2012)} with ‘frutadelobo’ leaves, ^{Goneli et al. (2014)} with ‘aroeira’ leaves, ^{Martins et al. (2015)} with ‘timbó’ leaves, ^{Silva et al. (2015)} with ‘jenipapo’, ^{Gasparin et al. (2017)} with Mentha piperita leaves.
Still in Figure 2, it is also possible to note the effect of air speed on the thinlayer drying curves of blackberry leaves; the increase in air speed reduced the drying time. The effect of air speed is more accentuated at the lowest drying air temperatures; as air temperature increases, there is a reduction in the influence of the speed on the time spent to dry the product.
The more pronounced effect of drying air temperature, compared with its speed, on the reduction of blackberry leaves drying time can be attributed to the fact that the main cause of the drying process is the difference in vapor pressure between the product and the drying air. Vapor pressure difference increases with the increment in drying air temperature, and air speed does not cause alterations in the vapor pressure difference between air and product (^{Martins, 2015}).
The greater influence of drying air speed at lower temperatures, as observed in Figure 2 for the air temperature of 40 °C, can be explained by the fact that water evaporation initially occurs on the product’s surface, which causes drying air speed to have greater importance in the beginning of the process, as explained by ^{Babalis et al. (2006)}. These authors also explain that water evaporation initially on the product’s surface is replaced by an evaporation front which moves to the inside of the product, causing the effect of drying air speed to be followed by the liquid diffusion process, which becomes the most important factor for the drying process.
Since at lower temperatures the time required to remove water present on the product’s surface is longer than at higher temperatures, it causes the effect of drying air speed to be more pronounced at lower temperatures, due to the longer time during which it contributes to removing water present on the surface. Thus, the higher the drying air speed, the greater the contribution of this factor to the removal of water from the product’s surface.
The constant ‘k’ of the Midilli model increased with the increment in drying air temperature and speed (Table 3). The constant ‘k’ can be used as an approximation to characterize the effect of temperature and is related to the effective diffusivity in the decreasing period of the drying process, and liquid diffusion controls the process (^{Babalis & Belessiotis, 2004}).
The coefficients “a”, “n” and “b” of the Midilli model (Table 3) do not exhibit a defined trend in their magnitudes as a function of the increase in drying air speed, except the coefficient ‘b’ in the drying tests conducted with air speed of 0.8 m s^{1}.
Air temperature (°C)  0.4 m s^{1}  0.8 m s^{1}  

a  k  n  b  D_{i}  a  k  n  b  D_{i}  
40  0.9659  0.2055  1.1448  0.0079  0.4590  0.9719  0.3198  1.1442  0.0102  0.7038  
50  0.9696  0.7997  1.1888  0.0050  1.5008  0.9801  0.8472  1.1018  0.0150  1.5835  
60  0.9501  1.4236  1.3217  0.0071  2.4058  0.9779  1.7984  1.1927  0.0509  3.1023  
70  0.9774  2.8382  1.2413  0.1044  4.5459  0.9921  4.0350  1.1512  0.1894  6.6212 
As can be observed in Table 3, the effective diffusion coefficients (D_{i}) increase with the increment in drying air temperature and speed. According to ^{Goneli et al. (2009)}, as temperature increases there is also an increase in the level of vibration of water molecules and reduction in water viscosity, which is a measure of resistance of a fluid to flowing. Variations in this property lead to alterations of water diffusion in the capillaries of agricultural products, which contribute to a faster diffusion along with more intense vibration of water molecules.
The increase in D_{i} with the increment in drying air speed can be attributed to the fact that the increment in air speed contributes to the evaporation of water, which moves to the product’s surface (^{Martins et al., 2015}). Similar behavior was observed by ^{Kaya & Aydin (2009)} studying the drying curves of mint and nettle leaves.
The dependence of D_{i} values of blackberry leaves increased with the increment in drying air temperature and speed, as observed in Figure 3. A similar behavior was observed by ^{Kaya & Aydin (2009)}.
The slope of the Arrhenius curve for the thinlayer drying of blackberry leaves (Figure 3) is used to obtain the E_{a}/R ratio, and its intersection with the Yaxis is used to obtain the D_{o} value. Eqs. 14 and 15 present the coefficients of the Arrhenius equation fitted to the effective diffusion coefficients of blackberry leaves for the drying air speeds of 0.4 and 0.8 m s^{1}, respectively, calculated according to Eq. 11.
The activation energy for liquid diffusion in the temperature range from 40 to 70 °C during the drying of blackberry leaves was approximately of 65.94 and 66.08 kJ mol^{1} (Eqs. 14 and 15), for the drying air speeds of 0.4 and 0.8 m s^{1}, respectively. According to ^{Kashaninejad et al. (2007)}, the activation energy is a barrier that must be overcome for the diffusion process to be triggered in the product.
Activation energy values did not vary much as a function of the variation in drying air speed; these values are higher than those found by other researchers working with medicinal plants, such as Cymbopogon citratus leaves (53.76 kJ mol^{1}) (^{Gomes et al., 2017}), ‘frutadelobo’ leaves (44.60 kJ mol^{1}) (^{Prates et al., 2012}), ‘jenipapo’ leaves (33.87 kJ mol^{1}) (^{Silva et al., 2015}).
Conclusions
Among the models fitted, Midilli was the only one with satisfactory fit to the observed data for the drying air conditions studied.
The increase in drying air temperature and speed caused a reduction in drying time, but the effect of increased air speed was more pronounced, regarding the drying time, at the lowest temperatures evaluated.
Effective diffusion coefficients increased with the increment in drying air temperature and speed, whereas the activation energy increased slightly with the increment in drying air speed.