Introduction
Buckwheat (Fagopyrum esculentum Moench) is a dicotyledonous plant belonging to the Polygonaceae family, with no kinship with common wheat. Its flour does not contain gluten and is mainly recommended for people with celiac disease; in addition, it positively affects human organism (^{Wronkowska et al., 2010}).
Buckwheat protein is of excellent quality and has high content of lysine, an essential amino acid which is deficient in most common cereals (^{Rodrigues & Oliveira, 2010}). It contains dietary fiber, resistant starch, rutin and mineral salts, besides a high level of essential polyunsaturated fatty acids and several vitamins (^{Sedej et al., 2011}).
Given the economic potential, up to present time some studies have been conducted with the crop, related to yield (^{Görgen et al., 2016}), adaptation (^{Alves et al., 2016}), cultivation (^{Vazhov et al., 2013}) and nutritional properties (^{Zhu, 2016}). However, information on drying and physical properties of this species is very vague. Variations in true density, bulk density, porosity and volumetric shrinkage are observed in several agricultural products along the drying process (^{Payman et al., 2011}; ^{Mir et al., 2013}; ^{Botelho et al., 2016}; ^{Smaniotto et al., 2016}; ^{Sousa et al., 2016}). These properties are of extreme importance to optimize the postharvest process because they are necessary in the calculations of the static capacity of the silo, dimensioning of dryers, transporters and fans for drying and aeration systems.
Given the lack of information on the physical properties of buckwheat during drying, the present work aimed to determine the volumetric shrinkage, porosity, bulk density and true density of buckwheat grains along the drying process, as well as to fit mathematical models to the experimental values of true and bulk volumetric shrinkage.
Material and Methods
The study was conducted at the Laboratory of Physical Properties of Agricultural Products of the Faculty of Agrarian Sciences, belonging to the Federal University of Grande Dourados, located in the municipality of Dourados, MS, Brazil.
Grains of the buckwheat cultivar IPR 91 – Baili, mechanically harvested, with initial moisture content of 0.250 (decimal, d.b.) were used in the experiment.
The initial moisture content of the samples was determined using the oven method described by the Rules for Seed Analysis (^{Brasil, 2009}), modified, using a forced ventilation oven at 105 ± 1 ºC for 24 h, in three replicates.
The buckwheat samples used to determine physical properties were subjected to drying in forced air circulation oven stabilized at temperature of 40 ± 1 °C, providing relative humidity of 22 ± 2.3%, and this value was obtained by basic principles of psychrometry.
Drying was monitored based on mass difference, by knowing the initial moisture content of the product. As drying began, the mass of the samples was weighed periodically on a semianalytical scale with 0.01 g resolution. When the product reached preestablished moisture contents (0.250, 0.234, 0.219, 0.204, 0.190, 0.176, 0.162, 0.149 and 0.136, decimal d.b.), the samples were removed and their physical properties were determined.
The true density of the grain mass (ρ_{t}) was determined by the direct method, with addition of hexane, in a known mass of grains. This solvent was used because it has low density and is not absorbed by the grains (^{Donadon et al., 2012}). All tests were conducted using a 30mL flask and a 50mL burette. By knowing the mass of the grains, mass and volume of the flask and solvent density, it was possible to obtain the total volume of the sample, which was divided by the number of grains to determine the individual volume.
Bulk density (ρ_{b}), in kg m^{3}, for the product was determined using a scale of hectoliter weight, with 1 L volume, with three replicates for each moisture content, as described by ^{Siqueira et al. (2012a)}.
Porosity (ε, in %) of buckwheat seeds was indirectly calculated by Eq. 1 (^{Mohsenin, 1986}), traditionally used for agricultural products.
The experimental data relative to the bulk density, true density and porosity of buckwheat grains were subjected to linear regression analysis. Regression models were fitted using the computer program SigmaPlot 11.0 (^{Systat Software Inc., 2008}). The criteria to select the models were their significance level by F test, along with the coefficients of determination (R^{2}).
To determine the coefficients of true and bulk volumetric shrinkages of buckwheat grains, the ratio between the product volume at a certain instant and its initial volume was calculated using Eq. 2 (^{Siqueira et al., 2012b}).
where:
Ѱ  coefficient of volumetric shrinkage, dimensionless;
V  volume (of the mass or grain) at a certain instant, cm^{3}; and,
V_{0}  initial volume (of the mass or grain), cm^{3}.
Bulk volumetric shrinkage was monitored based on an initial volume of 150 cm^{3} (150 mL) of buckwheat grains. The reduction of this volume was verified along the drying process by transferring the grain mass to a 150cm^{3} cylinder and recording the volume for the predetermined moisture contents during the drying. For each moisture content sampled, three measurements of grain mass volume were taken in the 150cm^{3} cylinder, and the grain mass volume was represented by the mean of these three measurements.
Mathematical models (Table 1) were fitted to the experimental data of coefficients of true and bulk volumetric shrinkage of buckwheat grains.
Model designation  Model  

Rahman (1995) 

(3) 
Corrêa et al. (2004) 

(4) 
Exponential 

(5) 
Linear 

(6) 
2^{nd} order polynomial 

(7) 
Modified Bala & Woods (1984) 

(8) 
Where: U  Moisture content of the product, decimal d.b.; U_{0}  Initial moisture content of the product, decimal d.b.; a, b, c  Parameters depending on the product, dimensionless; β  Volumetric shrinkage coefficient, dimensionless
The experimental data of true volumetric shrinkage and bulk volumetric shrinkage were subjected to nonlinear regression analysis by the GaussNewton method. The models were fitted to the experimental data using the computer program Statistica 7.0 (^{Statistica, 2005}).
The models to represent the volumetric shrinkage coefficient were selected considering the values of the relative mean error (P), standard error of the estimate (SE), coefficient of determination (R^{2}) and residual distribution. Relative mean error and standard error of the estimate were calculated according to Eqs. 9 and 10 (^{Siqueira et al., 2012b}):
where:
n  number of experimental observations;
Y  experimental value;
Ŷ  value estimated by the model; and,
DF  degrees of freedom of the model.
Results and Discussion
Bulk density and true density of buckwheat grains increased with the reduction in moisture content (Figure 1). The estimated values of bulk density and true density of buckwheat grains varied from 633.24 to 663.91 kg m^{3} and from 1089.28 to 1252.17 kg m^{3}, respectively, for a variation in product moisture content from 0.250 to 0.136 (decimal, d.b.). Such behavior is similar to those of most agricultural products studied (^{Firouzi et al., 2009}; ^{Payman et al., 2011}; ^{Oliveira Neto et al., 2012}; ^{Jesus et al., 2013}; ^{Theertha et al., 2014}).
With the reduction in moisture content and consequently in the volume, the grains were more easily arranged within the container, increasing the bulk density, as described by ^{Siqueira et al. (2012a)}. True density tends to increase because the densities of the main compounds of the grain (carbohydrates and proteins) are higher than the density of water (^{USDA, 1975}), which was removed by the drying process.
It was also observed in Figure 1 (A and B) that the variation in bulk density and true density as a function of the reduction in moisture content can be satisfactorily represented by linear models, which indicates that, with the drying, the densities increased proportionally, as observed by other researchers (^{Solomon & Zewdu, 2009}; ^{Sousa et al, 2016}).
Porosity values increased as moisture content decreased (Figure 2). ^{Kingsly et al. (2006)} and ^{Kibar & Öztürk (2008)} reported a similar increment in the voids of the grain masses of pomegranate and soybean, respectively. This behavior is probably associated with the pyramid shape of the product (^{Oliveira et al., 2015}). The linear model satisfactorily represents the experimental values, showing high coefficient of determination (R^{2}). Porosity in the mass of buckwheat grains had estimated values between 41.98 and 47.06% within the studied range of moisture content. Thus, it was found that, in air movement systems, the lower the moisture content, the lower the resistance offered by the grain mass to the passing of the air, since there is formation of more voids.
Table 2 shows the regression equations for bulk volumetric shrinkage and true volumetric shrinkage of buckwheat grains as a function of moisture content, with their respective coefficients of determination, relative mean error, standard error of the estimate, and residual distribution. All models fitted had low values of relative mean error, all below 10. According to ^{Mohapatra & Rao (2005)}, the relative mean error is a parameter that can be used to recommend or not a model. Relative mean error values reflect the deviation of the observed values with respect to the curve estimated by the model (^{Kashaninejad et al., 2007}). Thus, in this case, the deviation can be considered as acceptable.
Models  P  SE  R^{2}  Residual 

Bulk  
3  0.4999  0.0062  0.9863  R 
4  0.4550  0.0057  0.9940  R 
5  0.5011  0.0062  0.9931  R 
6  0.4660  0.0057  0.9941  R 
7  0.4860  0.0058  0.9939  R 
8  0.4418  0.0059  0.9875  R 
True  
3  0.8725  0.0116  0.9546  R 
4  0.9282  0.0119  0.9758  R 
5  0.8414  0.0106  0.9810  R 
6  0.8183  0.0100  0.9832  R 
7  0.7217  0.0095  0.9870  R 
8  0.8725  0.0116  0.9546  R 
R – Random
The standard error of the estimate (SE) demonstrates the capacity of a model to accurately describe a certain physical process, and the lower its value, the better the quality of fit of the model in comparison to the experimental data (^{Siqueira et al., 2012b}). Thus, all models used showed a good fit, for bulk volumetric shrinkage and true volumetric shrinkage, since their values are very low.
The good behavior of the models in comparison to the experimental data can also be observed in the residual distribution, which was random in all cases, indicating good fit (^{Goneli et al., 2011}), and in the coefficients of determination, which were above 0.95 for all conditions. ^{Madamba et al. (1996)} claim that it indicates satisfactory representation by the models used.
Based on a joint analysis of the statistical parameters, it is possible to state that all models used to represent bulk volumetric shrinkage and true volumetric shrinkage of buckwheat grains can be recommended with high reliability, since all models evaluated showed low SE values, P values below 10% and R^{2} values above 95% (^{Draper & Smith, 1998}; ^{Goneli et al., 2011}; ^{Siqueira et al., 2012b}).
Among all models fitted to the experimental data, the linear model is the simplest one and was therefore used to describe the phenomenon of shrinkage. In addition, it has been recommended to represent the volumetric shrinkage of other agricultural products, such as soybean (^{Smaniotto et al., 2016}), common beans (^{Oliveira Neto et al., 2012}) and peanut (^{Payman et al., 2011}). The excellent fit of the model, reliably representing the behavior of bulk and true volumetric shrinkage of buckwheat grains, can be seen in Figure 3.
According to the results, the estimated values of the mass of buckwheat grains showed a reduction in the initial volume of about 14.47% and in the true volumetric shrinkage, following the same behavior, was 14.70% for the moisture content range from 0.250 to 0.136 (decimal, d.b.). If the linear model is adopted to represent the shrinkage, for every 0.01 variation (decimal, d.b.) in the moisture content, there was a 0.0128% variation in the true volumetric shrinkage. For bulk volumetric shrinkage, there was a variation of 0.0129%. These data are of extreme importance to optimize the use of dryers, because they can be used to calculate the volume necessary to refill the dryer’s drying chamber, so that it works always full, ensuring maximum productivity.
Conclusions
Reduction in moisture content during drying causes increase in bulk density, true density and porosity.
Reduction in moisture content influences bulk volumetric shrinkage and true volumetric shrinkage of buckwheat grains, causing reductions in their values of approximately 14.47 and 14.70%, respectively, and both variables can be satisfactorily represented by the linear model of shrinkage.