1 Introduction
Grains and cereals are important staple foods that are cropped seasonally and have to be properly processed and stored to meet the market demand throughout the year. According to ^{Food and Agriculture Organization of the United Nations (2018)} statistics, cereal use in the world is forecast to be 2,641 million tonnes in 2018/19, and the estimated production will not be sufficient to meet this expected use, therefore the deficit must be supplied by accumulated stocks. The importance of storage has grown significantly as a result of the increase in volumes traded and the rise in worldwide food demand. Aeration in silos is commonly used to improve the storability of grains, because forced ventilation helps to keep uniform temperatures throughout the silo, and contributes to avoiding moisture migration and removing the odours of the stored grain masses, and can also be used to apply fumigation (^{Khatchatourian & Savicki, 2004}; ^{Khatchatourian et al., 2017}).
The static pressure requirements of the fan used for aeration, drying or cooling of a packed bed of grains, is determined by the resistance of the packed grains to the airflow. The pressure drop across a homogeneous packed bed of granular material can be predicted by Forchheimer’s equation, given by (^{Coulson & Richardson, 2002}):
where
Although Equation 1 establishes a simple relationship between the pressure drop and the aeration velocity, the challenges involved in using it are that the permeability and inertial resistance factor are parameters dependent on the size and shape of the particles as well as on structural properties of the packed bed, such as bed porosity (^{German, 1989}). Darcy’s permeability can be interpreted as a measure of the resistance to flow offered by the porous medium. It is associated with the energy loss due to viscous flow and fluidparticle friction during flow through the porous medium, and is a characteristic of the porous structure. The inertial factor is associated with the energy loss under highvelocity flow and is enhanced by turbulence and bed tortuosity (^{German, 1989}). Since these parameters are not easy to determine, many empirical or semitheoretical relationships have been proposed to estimate the pressure drops through packed beds composed of either grains/seeds or particles with irregular size distribution (^{Park et al., 2018}; ^{Cruz et al., 2014}; ^{Li & Sokhansanj, 1994}; ^{Matthies & Petersen, 1974}; ^{Shedd, 1953}; ^{Yang & Williams, 1990}). However in most of these studies, the effect of the moisture content of the solids on the pressure drop was neglected, but this moisture content can vary significantly during processing and storage. In food grains, variation in moisture content can lead to changes in particle size and shape that could affect the structural properties of the packed bed and modify the pressure drops (^{Felizardo & Freire, 2018}; ^{Górnicki & Kaleta, 2015}; ^{Kobus et al., 2011}). In addition, assessing how the grain dimensions change with moisture content is important because the variation in their volume and weight due to shrinkage has a relevant impact on transportation and storage costs, particularly in a country with a recognized deficit of silos, like Brazil (^{Rabello, 2018}; ^{Bala, 2016}).
The purpose of this study was to investigate the influence of the moisture content of the solids on the pressure drops through packed beds of four widely consumed staple grains, namely soy, barley, lentils and oats, with moisture contents ranging from the equilibrium moisture up to 0.24 g.g^{1}. The linear dimensions, surface area, volume, sphericity and density of the particles were determined as a function of the moisture content, as well as the porosity and bulk density of the packed beds. The parameters of Equation 1,
2 Material and methods
2.1 Particulate materials
The materials were selected to cover different particle shapes and sizes, namely: soy (Glycine max L.), green lentils (Lens culinaris), barley (Hordeum vulgare L.) and oats (Avena sativa). The dehydrated grains were purchased from a local market in São Carlos, SP, Brazil. Spherical alumina particles (type F200) were provided by Alcoa^{©}. Alumina particles have a porous structure, diameter of 3.2 mm, surface area of 32. mm^{2}, volume of 17.3 mm^{3} and sphericity close to 1.0. As the water is absorbed mainly by the internal pores, the particle dimensions and shape do not change with moisture content. For these reasons, they were included in this study to serve as a standard particulate for comparison.
The initial moisture contents of the biological products ranged from 0.091 to 0.138 g.g^{1}. The grains were either moistened or further dried to reach the specified moisture content levels (
2.2 Particle characterization
The dimensions of the biological grains were determined assuming a triaxial spheroid shape (Figure 1). The lengths (
Images of grains with moisture contents of 0.24 g g^{1} and
2.3 Packed bed characterization
The bulk densities (
2.4 Pressure drop measurements
Prior to the experiments, the materials were allowed to rest at ambient temperature for 1 h and the assays then carried out in triplicate. The experimental set up is illustrated in Figure 2. Compressed air (1) from a blower (2) was filtered (air filter F2000, Arprex^{®}) (3) and transported through a 5 cm diameter pipe towards the packed bed column. The air flowrate could be adjusted by a valve and was measured using a rotameter (E 9869, Gilmont^{®}) (4). The packed bed column was a cylindrical vessel (5) 5 cm in diameter (
The pressure drops (
In Equation 9,
where
2.5 Statistical analysis
The influence of the moisture levels of the solids on the properties and fitted parameters was evaluated using a oneway analysis of variance (ANOVA) and Tukey’s test (
In order to evaluate the differences between the experimental data (
3 Results and discussion
3.1 Influence of the moisture content of the solids on the physical properties of the particles
Figures 3ah show the linear dimensions, volume, surface area, sphericity and density of the particles at different moisture levels. The dependence of the biological properties of the particles on the moisture content can be observed from these figures. Since the dimensions and shape of alumina particles do not change significantly with the moisture content, only the variation in particle density was shown for this particle (Figure 3h).
A statistically significant decrease in the linear dimensions
Based on ANOVA, in most cases it was observed that shrinkage caused a statistically significant decrease in the volume (
Figure 3h shows that the
3.2 Influence of the moisture content of the solids on the characteristics of the packed beds
Figures 55b show, respectively, the values for
Figure 5b shows that the dependence of
3.3 Influence of the moisture content of the solids on the fluid flow parameters of the packed beds
Figure 6 shows the values of the parameters
Figures 66b show that the values for
The values obtained for
The variation in
Under the conditions tested, the
3.4 Influence of the moisture content of the solids on the pressure drop
Figure 7 shows the values determined for
With a view to incorporating the effect of the moisture content of the solids in Equation 1,
where
Material 







Soy  5.489x10^{7}  2.831x10^{8}  0.654  1.324  0.99  62.50 
Lentils  4.630x10^{7}  1.909x10^{9}  5.273  2.525  0.99  236.75 
Barley  9.834x10^{8}  4.405x10^{9}  1.178  1.064  1.00  82.11 
Oats  3.757x10^{8}  6.697x10^{9}  0.883  1.372  1.00  84.29 
Alumina  1.236x10^{7}  4.292x10^{8}  1.648  1.235  0.98  125.47 
A, B, C and D are parameters in Equation 15 and R^{2} is the coefficient of determination and RMSE is the root mean square error.
The fitted equations were verified using experimental data reported in the literature for products and operational conditions similar to those used in the present study. Figure 8 shows a comparison of the experimental and predicted values for
4 Conclusions
An investigation concerning the influence of the moisture content of the solids on the pressure drop through packed beds of biological particles (soy, oats, lentils and barley) was carried out for moisture levels varying from the equilibrium moisture up to 0.24 g g^{1}. The results demonstrated that the airflow resistance in aeration can change significantly as the moisture content of the solids decreases. As water was removed, the pressure drops increased significantly, mainly because the particle size decreased, resulting in static beds with lower bulk voidage and lower permeability. By regressing the permeability and inertial factor parameters with the moisture content, a set of simple, accurate and reliable equations could be obtained to predict the pressure drops through packed beds of biological products as a function of the moisture content of the solids.
Nomenclature

Empirical constant of Equation 15  [m^{2}.g^{1}.g] 

Equivalent surface area of the particle  [mm^{2}] 

Surface area of the spherical particle  [mm^{2}] 

Empirical constant of Equation 15  [m^{2}] 

Empirical constant of Equation 15  [s m^{3} g^{1} g] 

Inertial resistance factor  [s m^{3}] 

Empirical constant of Equation 15  [s m^{3}] 

Cylindrical vessel diameter  [cm] 

Mean diameter of the particle  [mm] 

Diameter of the spherical particle  [mm] 

Sauter mean diameter  [m] 

Distance between the pressure taps  [cm] 

Length of the particle  [mm] 

Mean hydraulic radius  [mm] 

Reynolds number for the porous media  [] 

Thickness of the particle  [mm] 

Air temperature  [°C] 

Surface air velocity  [m s^{1}] 

Equivalent volume of the particle  [mm^{3}] 

Volume of the spherical particle  [mm^{3}] 

Width of the particle  [mm] 

Moisture content of the solids  [g g^{−1}] 

Moisture content of the solids at equilibrium  [g g^{−1}] 

Shape factor  [] 

Shape factor related to the crosssection of the tube  [] 

Mean bulk voidage  [] 

Permeability  [m^{2}] 

Air viscosity  [g m^{1} s^{1}] 

Air density  [g m^{3}] 

Apparent density  [g m^{3}] 

Bulk density  [g m^{3}] 

Sphericity  [] 

Pressure drop  [g m^{1} s^{2}] 