Bean moisture diffusivity and drying kinetics : a comparison of the liquid diffusion model when taking into account and neglecting grain shrinkage

The aim of this work was to compare the results of the liquid diffusion model with respect to bean (Phaseolus vulgaris L.) drying when taking into account or neglecting grain shrinkage. Bean grains were harvested with a moisture content of 0.92 kg water/kg dry matter and dried at air temperatures of 25-55°C and relative humidities of 20-75%. The volume of each grain, understood as a sphere, was determined several times over the drying process, taking the diameter to be the mean length of the three orthogonal axes. Grain shrinkage was determined by examining the relationships between the volume associated with each moisture content and the initial volume. The results show that the liquid diffusion model describes the drying kinetics of beans satisfactorily, and that grain shrinkage can be ignored. The diffusion coefficient increases with air temperature, with values ranging between 10.8 × 10 and 67.0 × 10 m s. This is described by the Arrhenius equation, with an activation energy of 40.08 kJ mol. Additional key words: diffusion coefficient, grains, mathematical models, moisture, Phaseolus vulgaris.


Introduction
Water is the main component of most agricultural products and it has enormous influence on their physical properties.Agricultural products are usually dried to ensure their quality and stability.Diminishing moisture levels lead to a reduction in biological activity and induce chemical and physical changes in grains.
of agricultural products were developed without taking into account the importance of volumetric contraction during dehydration (Brooker et al., 1992).According to Ramos et al. (2005), this should be included in models for a complete description and analysis of the phenomenon to be made.Drying models based on the liquid diffusion theory have attracted special attention from researchers.The liquid diffusion mechanism is extremely complex due to the diversity of the chemical composition and physical structure of different products.In addition, the use of different experimental methodologies renders comparisons difficult.
According to Brooker et al. (1992), Fick's second law is used in liquid diffusion theory to establish moisture diffusion as a function of the concentration gradient.The following equation describes a one dimensional drying situation (in Cartesian coordinates): [1] where U * is the product moisture content (kg water/kg dry matter), t is the time (in s), x is the distance between two points of reference in the product (in m), and D is the liquid diffusion coefficient (in m 2 s -1 ).
The variation in moisture content as a function of the drying time in homogeneous matter with a constant diffusion coefficient is represented by the following equation (Brooker et al., 1992): [2] where r is the radial distance (in m), and c = «0» for flat bodies, «1» for cylindrical bodies and «2» for spherical bodies.
Many solutions to Equation [2] for different geometric shapes have been used to describe the process of drying in agricultural products in boundary conditions (Brooker et al., 1992): where U* i is the initial moisture content (kg water/kg dry matter), and U* e is the moisture content equilibrium (kg water/kg dry matter).Brooker et al. (1992) provided the analytical solution to Equation [2] for spherical shapes as follows: [5] where MR is the product moisture ratio (dimensionless), and n the number of terms.
The analytical solution of these equations is based on an inf inite series, with the number of terms determining the precision of the results.Afonso-Júnior and Corrêa (1999) evaluated the results of drying the bean (Phaseolus vulgaris L.) cultivar Ouro Negro using the diffusion model adjusted for a spherical geometry and by approximating with a series of 8 terms.This was found to be adequate and provided satisfactory estimates of the product drying rate.
The diffusion coefficient generally increases as the air temperature increases (Ramesh, 2003), and can be described by the Arrhenius equation as a function of temperature: [6] where D o is the pre-exponential factor (in m 2 s -1 ), E a is the activation energy (in kJ mol -1 ), R the universal gas constant (8,314 J mol -1 K -1 ), and T a the absolute temperature (in ºK).
Arrhenius expression coefficients are easily obtained by making the equation linear, applying the following logarithm: [7] Due to the importance of the study of the drying of tropical agricultural products, and the lack of theoretical information regarding the phenomena that occur during bean drying, the aim of this work was to compare the results of the liquid diffusion model under different conditions, taking into account or neglecting grain shrinkage.

Material and Methods
The beans used in this work were of the red group, cv.Vermelhinho.These were manually harvested at a moisture content of around 0.92 kg water/kg dry matter.All experiments were performed at the Laboratory of Physical Properties and Quality of Agricultural Products-CENTREINAR, Universidade Federal de Viçosa, Viçosa-MG, Brazil.Product moisture was determined using the greenhouse method (MARA, 1992), leaving the beans at 105 ± 1°C for 24 h until a constant mass was achieved.
Drying experiments were performed in a controlled environment chamber (Aminco) under different conditions of temperature (25, 35, 45 or 55°C) and relative humidity (20 or 75%).The total number of combinations of conditions was 14. Drying proceeded until the product reached its equilibrium moisture content under the conditions set.
Two removable trays -each containing 50 g of beans-with netted bottoms to allow air to pass through the product mass, were placed inside the environment chamber.Airflow was monitored with an anemometer and kept constant at around 4 m 3 s -1 m -2 .Air temperature and relative humidity were monitored using a psychrometer installed near the sample trays.
During drying, these trays were periodically weighed.Hygroscopic equilibrium was deemed to have been reached when the mass variation of the containers remained approximately constant over three consecutive weighings (two replicates).
To determine the moisture ratio (MR) of the beans drying under the different conditions, the following expression was used: [8] As a control for non-shrinkage, eight beans, wrapped in a fabric permeable to air were individually placed inside each tray.
The volume of each grain (V g ), considered to be a sphere, was obtained using the average value of the three orthogonal axes (a, b and c, in mm, Figure 1) as the diameter, as proposed by Mohsenin (1986).This measurement was taken for eight beans over the drying process, using a digital calliper.The volume was calculated using the following expression: [9] Grain shrinkage (ψ g ) during drying was determined as the relationship between grain volume (V g ) at each time point and the initial volume (V o ) using the following expression: [10] The liquid diffusion model (employing an approximation of eight terms [see Eq. 5]) was used with the experimental drying data, either taking into account grain shrinkage or neglecting it.The number of terms in the model was established when the variation of the diffusion coefficient was less than 0.1 × 10 -13 m 2 s -1 .
The experimental data were interpreted by nonlinear regression analysis using the Quasi-Newton method and employing STATISTICA 6.0 ® software (http://www.statsoftinc.com).The validity of the models with and without grain shrinkage was tested based on the significance of the regression coefficients (Student t test, P = 0.01), the mean relative error (MRE), the standard error of estimation (SEE), and the determination coefficient (R 2 ).Residual distribution plots were constructed to evaluate the fitting quality.The mean relative error and the standard error of estimation for the models with and without grain shrinkage were calculated according to the following expressions: where n is the number of experimental observations, Y the experimental value, Y ^the predicted value, and GLR the degrees of freedom of the model (number of observations minus the number of variables in the model).
To compare the diffusion models (as adjusted by [Eq.5]) with and without grain shrinkage under each drying condition, the values estimated by the models were plotted.Linear regression analysis was then performed, with the straight line passing through the origin.The consistency of each model was assessed by determining the coefficient of determination (R 2 ) and the agreement expressed by the «d» indices proposed by Willmott et al. (1985).The «d» index indicates the degree of accuracy between the values estimated by the models, i.e., the closer to 1, the smaller the difference between the models.The «d» index is represented by the following expression: [13] where

Results
Figure 2 shows the experimental grain shrinkage values obtained during drying under the different conditions.Volume reduction varied between 18.9 and 35.2%, with the moisture content changing from 0.92 (kg water/kg dry matter) until the equilibrium moisture content specific for each set of conditions was reached.
Table 1 shows the mean diffusion coefficients, the relative and estimated mean errors, and the coefficients of determination for the liquid diffusion model with and without grain shrinkage under the drying different conditions.The SEE were small, confirming the validity of both models.However, the relative mean errors for both were > 10%, a less satisfactory result.Therefore, neither model appeared to be better than the other: these variables were insufficient to distinguish between the liquid diffusion models with and without grain shrinkage under the different drying conditions.Thus, «d» index values were calculated (Table 2).
Figure 3 shows the experimental moisture ratio values compared to values estimated by the liquid diffusion model for drying beans, taking into account the grain shrinkage under the different conditions of tem-  perature and relative humidity.The correspondence between the experimental and estimated values show that the liquid diffusion model without grain shrinkage appropriately describes the phenomenon studied.
The dependence of the diffusion coefficient on the drying temperature can be described by an Arrheniustype relationship (Fig. 4).The coefficient of diffusion increased with the drying temperature.The slope [Eq. 6] provides the relationship E a /R, while its intersection with the «y» axis indicates the value of D o .Thus, the diffusion coefficient can be calculated as follows: where E a in [Eq.6] was 40.08 kJ mol -1 for temperatures ranging from 35ºC to 55°C.
The Arrhenius factor D o is a constant equivalent to the diffusivity at infinitely high temperatures, and has the value 3.0 × 10 -3 in drying beans.At this magnitude, this value can be neglected.

Discussion
When analysing grain shrinkage, Afonso Júnior et al. ( 2004) reported a reduction of 39% in the volume   of coffee berries, with a reduction in their moisture content of 2.27 to 0.11 (kg water/kg dry matter).Ramos et al. (2005) observed a reduction of 35% of the radius of grapes during the drying process and indicated that the significant variation in the shrinkage of different products should be included in mass transfer models.
Table 1 shows that the liquid diffusion model (with or without grain shrinkage) satisfactorily describes the drying of the beans for all the air conditions studied.The regression coefficient was significant (P < 0.05) according to the Student t test, and the coefficient of determination was > 97.8%.
Table 1 also shows that the lowest drying air temperatures offer more internal resistance to water transport.Thus, a higher air temperature leads to increased diffusion coefficients and a greater outward movement of moisture (Babalis and Belessiotis, 2004;Sharma and Prasad, 2004).
The liquid diffusion theory assumes that there is no influence of capillarity, neglects the effects of mass and energy transfer from one body to another (given the difficulty of quantifying these multiple effects on the product mass), and considers that thermal equilibrium with the air is reached instantaneously; this could lead to the discrepancies seen between the experimental and model-predicted results.
Table 2 shows a high «d» index (> 0.999) for all the temperature and relative humidity combinations, and excellent agreement between the values estimated by both models.The simple linear regression for the humidity ratio estimated by the models obtained high R 2 values (> 99.96%).This confirms the marked similarity and correlation between the liquid diffusion models that either take into account the volumetric contraction of the grains or neglect it.Carmo and Lima (2004) studied the drying of lentils using a liquid diffusion model that took shrinkage into account, and found it to predict experimental data very well.
Thermodynamically, E a is the ease with which water molecules overcome the energy barrier when migrating within the product.In the drying processes, the smaller the E a , the greater the water diffusivity within the product.The E a value found in this work agrees with those reported for a number of biological materials, e.g., (in kJ mol -1 ) soybean 28.8-30, wheat kernels 54-70.2, starch gel 18.8-50, scallion 29.05-42.05, canned mushroom 23.89-31.45, Zea mays indentata corn, 29.56, and rice 36.4 (Becker and Sallans, 1955;Fish, 1958;Park et al., 1996;Bróvia et al., 1997;Doymaz and Pala, 2003;Ramesh, 2003).Zogzas et al. (1996) noted that the E a for these agricultural products ranged between 12.7 and 110 kJ mol -1 .
Based on the present results it is concluded that: i) the liquid diffusion model satisfactorily represents the kinetics of drying beans under a number of air conditions; ii) based on the analysed statistical variables, grain shrinkage can be neglected: the correlation between the models taking this into account and neglecting it was strong; iii) the diffusion coefficient increases with temperature, with values of 67.00 × 10 -10 to 10.84 × 10 -10 m 2 s -1 seen for air temperatures ranging from 25ºC to 55°C and relative humidities of between 20 and 75%; iv) the relationship between the diffusion coeff icient and temperature is described by the Arrhenius equation, with an activation energy for liquid diffusion of 40.08 kJ mol -1 .

Figure 1 .
Figure 1.Schematic diagram of a bean grain showing the characteristic dimensions.

Figure 4 .
Figure 4. Arrhenius-type relationship between effective moisture diffusivity and temperature.
d is the agreement index, Y e is the i th value estimated by model 1, Y o is the i th value estimated by model 2, and Y o the average of the values estimated by model 2.

Table 1 .
Coefficients of diffusion, mean relative error (MRE), standard error of estimation (SEE), and determination coefficient (R 2 ) for the liquid diffusion model with and without grain shrinkage (ψ g ), during the drying of beans under different conditions of temperature (T) and relative humidity (RH).Without ψ ψ g With ψ ψ g Without ψ ψ g With ψ ψ g Without ψ ψ g With ψ ψ g Without ψ ψ g * Significant at P < 0.05 (Student t test).

Table 2 .
«d» 1 indices used to compare the liquid diffusion models with and without shrinkage, the coefficients of determination (R 2 ), and the linear regressions (z), during drying under different temperatures (T) and relative humidities (RH)