The Canadian Society for Bioengineering The Canadian society for engineering in agricultural, food, environmental, and biological systems. La Société Canadienne de Génie Agroalimentaire et de Bioingénierie La société canadienne de génie agroalimentaire, de la bioingénierie et de l environnement Paper No. CSBE13-31 Modeling Airflow Path through Grain Bulks Using the Discrete Element Method Rong Yue, Graduate Student Department of Biosystems Engineering, University of Manitoba, Winnipeg, MB, Canada, R3T 5V6. Email: yuer@cc.umanitoba.ca Qiang Zhang, Professor Department of Biosystems Engineering, University of Manitoba, Winnipeg, MB, Canada, R3T 5V6. Email: zhang@cc.umanitoba.ca Written for presentation at the CSBE/SCGAB 013 Annual Conference Saskatoon, Saskatchewan 7-10 July 013 ABSTRACT Knowledge of airflow through agricultural bulk solids, such as bulk grain, is important for designing storage and processing equipment. The complex and variable connectivity of pore structures of agricultural bulk solids makes airflow difficult to predict. This study aims to simulate the pore-structure of a grain bed and calculate the airflow paths at a microscopic scale. A discrete element model developed in PFC 3D (Particle Flow Code in 3 Dimensions) was used to simulate the pore-structure of the grain bed. Then a mathematical algorithm was developed to calculate the tortuosity of the widest airflow path through the pore structure of the grain bed. The calculated tortuosity was in good agreement with the range of the tortuosity values reported in the literature for porous beds consisting of spherical particles. Keywords: airflow, grain bed, pore structure, tortuosity, discrete element method Papers presented before CSBE/SCGAB meetings are considered the property of the Society. In general, the Society reserves the right of first publication of such papers, in complete form; however, CSBE/SCGAB has no objections to publication, in condensed form, with credit to the Society and the author, in other publications prior to use in Society publications. Permission to publish a paper in full may be requested from the CSBE/SCGAB Secretary, 08 Calico Crescent, Orleans, ON, K4A 4L7 or contact secretary@bioeng.ca. The Society is not responsible for statements or opinions advanced in papers or discussions at its meetings.
INTRODUCTION Much research has been done to study the airflow through grain beds. Those studies are generally focused on developing empirical equations to represent the airflow resistance characteristics of various agricultural products (Shedd 1953; Hukill and Ives 1995; Ergun 195). The ASABE Standards D7. (007) provides a collection of the data for estimating the resistance to airflow through bulk grains, seeds and other agricultural products. Some researchers also developed simple mathematic models to predict airflow patterns and distribution in grain bins (Brooker 1961, 1969; Jindal and Thompson 197; Pierce and Thompson 1974; Segerlind 198; Smith 198, 1995; Chapman et al. 1989; De Ville and Smith 1996; Khatchatourian and Binelo 008). However, most of those models were developed based on the assumption that airflow paths in grain beds are straight tubes, and the interconnectivity and tortuosity of pore structures of grain beds are not considered. Airflow resistance through grain bulks is affected by many factors, including airflow rate, bed depth, shape and size of grain kernels, bulk density, moisture content, method of filling, foreign materials, and direction of airflow. The complexity and irregularity of porous media makes it difficult to simulate the pore structure of porous media. Neethirajan and Jayas (008) used X-ray technology studied the 3D-distribution of air paths in grain bulks and the results showed the connectivity of pore network brought the difference of airflow resistance between horizontal and vertical directions. One of the most important variables affecting airflow through porous media is tortuosity. Tortuosity is usually defined as the ratio of the actual length of flow path and the straight length or thickness of the porous bed (Jacob, 1988). Some geometry models were proposed for calculating the tortuosity of streamlines in porous media with spherical, cubic and cylindrical, square particles (Yu and Li 004; Yun et al. 005, 010; Matyka et al. 008). A represent unit cell (RUC) model was proposed by Du Plessis and Woudberg (008) to simplify the pore structure of porous media in studying the resistance to airflow through RUC. Wu et al. (008) developed a model for predicting resistance of flow through granular media as a function of tortuosity, porosity, diameters of pores and particles, and fluid properties. In most existing models, the attention is focused on the flow behavior (velocity, pressure, etc.), while assuming idealized pore structures for porous media. There is a lack of research effort in linking the characteristics of pore structures (sizes, location and orientations, connectivity, etc) to airflow resistance. The objective of this study was to develop a DEM model to predict the pore structures of grain beds, from which the tortuosity of airflow paths in grain beds is quantified. METHDOLOGY DEM Simulation of pore-structure of grain beds The PFC 3D (version 4.0 EV) software package (Itasca Consulting Group Inc., Minneapolis, MN) was used to construct models for simulating the pore structure consisting of randomly packed spheres. PFC 3D is a powerful tool to model complicated problems in solid mechanics and granular flow. It can generate particles randomly and model the movement and interaction of three-dimensional assembles of particles by contact and frictional forces. The PFC 3D model was developed to simulate a test grain bin of 0.5 0.0 0.75 m filled with soybeans (Fig. 1). The model generated 5146 spherical particles with sizes randomly distributed from 5.5 to 7.5 mm in diameter to fill to a depth of 0.5 m. Soybeans were selected for simulations because of their smooth surface and relatively round shape.
Fig.1 PFC 3D model of the grain bed. Parameters of soybeans used in the simulations were obtained from a previous study of PFC simulations reported by Liu et al. (008a) (Table 1). When all particles in the porous bed reached the steady state (equilibrium) during simulations, the coordinates of all the particles were obtained. Table 1. PFC simulation parameters reported by by Liu et al. (008a). Parameters value Wall normal stiffness (k n ), N/m 1.8 10 7 Wall shear stiffness (k s ), N/m 0.9 10 7 Particle normal stiffness (k pn ), N/m Particle shear stiffness (k ps ),, N/m Friction coefficient between particles Friction coefficient between particle and wall (Plexiglass) Particle density (ρ p ), kg/m 3 Bulk density (ρ b ), kg/m 3 Porosity (ψ), % Particle diameter (d p ), mm 4.0 10 6.0 10 6 0.47 0.3 185 761 40.8 5.5-7.5 Algorithm for tortuosity calculation The tortuosity was calculated as the ratio of the actual length of flow path to the physical depth of a porous bed as follows: Le τ = (1) L where τ = tortuosity 0 3
L e = actual length of flow path (m) L 0 = the depth of porous bed (m) Air flows through the pore spaces between particles in grain beds. To determine the airflow paths, the grain bed is treated as a collection of tetrahedron units, and each tetrahedron consists of four particles (Fig. ) (Sobieski et al. 01). Air enters the tetrahedron unit through the space between particles A, B and C ( ABC is termed the base triangle), and exits in three possible paths: between particles A, B, and D (represented by ABD); between particles A, C, and D ( ACD); or between particles B, C, and D ( BCD). To study the lowest resistance of airflow though porous beds, the widest airflow path was investigated and the corresponding tortuosity was calculated in this study. Fig. A tetrahedron unit representing grain mass in the bin. Because air flows from the bottom to the top of the grain bin, a random point at the bottom of the grain bin is first selected as an air entrance point (AEP). An AEP is physically the pore space between particles at the bin bottom. In this study, three particles nearest to the AEP are selected to form the pore space for AEP, and these three particles also form the initial base triangle. The next step is to find a particle as the fourth vertex to construct a tetrahedron. All the particles in the bin whose coordinates have been determined by the DEM model are sorted by their distances to the centroid of the base triangle and the particle that is closest to the base triangle is selected as a candidate for the fourth particle to construct the tetrahedron. The particle is confirmed to be the fourth particle if the following two conditions are met: (1) the center of the fourth particle should be higher than the centroid of the base triangle because the air flows upwards, and () the space among the four particles that form the tetrahedron should be sufficiently small (not be able to accommodate another particle inside the tetrahedron), i.e., the inscribed sphere within the tetrahedron must be smaller than a particle. If these conditions are not met, the next closest particle is considered, and the process is repeated until a suitable particle is found. Once a tetrahedron is constructed, the air is assumed to enter the tetrahedron through the base triangle and exit through other three triangular faces of the tetrahedron. The next step is to determine the widest path of the three possible paths by comparing the areas of the three exit triangles. The area of each triangle is calculated by the Huron Equation as follows: S = p ( p a)( p b)( p c) () 4
where p = a half of perimeter of triangle 1 p = ( a + b + c) (3) a, b, c = lengths of three sides of triangle The triangle with the largest area is selected as the exit triangle. In other words, the widest airflow path through a tetrahedron unit follows the line that connects the centroid of the base triangle (inlet) to the centroid of the exit triangle. The cross-sectional area of the pore (path width) is calculated as the area of exit triangle subtracting the area occupied by the particles (Fig. 3). Fig.3 Pore size for airflow through a tetrahedron unit. Fig.4 Illustration of the cross-sectional area for airflow through a triangle. Take ACD (Fig. 4) as an example, the total area of triangle is calculated as follows: 5
1 p = ( AC + AD + CD) (4) S( ACD) = p ( p AC) ( p AD) ( p CD) (5) The areas occupied by the three particles A, C, D are calculated respectively as follows: where 1 S( A1 AA ) = CAD r 1 S( C1CC ) = ACD r 1 S( D1DD ) = ADC r S ( A AA 1 ), S ( C ) 1 CC, S ( D 1 DD ) = the areas occupied by particles A, C, D, respectively, within triangle ACD r A, r C, r D = radius of particles A, B, C, respectively Therefore, the area available for air flow through ACD is: A C D S = S( ACD) S( A1 AA ) S( C1CC ) S( D1DD ) (7) Once the widest path within a tetrahedron is determined, the exit triangle is used as the base triangle to construct the next tetrahedron. The process repeats from the bottom to the top surface of the grain bin. The total airflow path length is then calculated as the sum of distances connecting the centroids of base triangles in all tetrahedron units. There are two cases that require reselecting the exit triangle or tetrahedron unit. In the first case, when the exit triangle from the current tetrahedron is selected as the base triangle for the next tetrahedron, the exit triangle area S may overlap with other triangles of the previous tetrahedron unit. As shown in Fig.5, air enters the current tetrahedron unit from ABC. If the area of ABD is the biggest area, it is selected as the base triangle to construct the next tetrahedron unit with particle E. But when air flows through ABD it also goes through ACD. In this case, another triangle from the current tetrahedron would be selected to as the base triangle for the next tetrahedron unit. (6) Fig.5. One example which requires reselecting the exit triangle. 6
In the second case, a particle penetrates into the exit triangle, causing the flow area (calculated by equation 7) to be negative ( ACD in Fig. 6). The tetrahedron unit is not stable and the pore space in the unit is narrow. In this case, another particle would be selected to construct a new tetrahedron unit. Fig. 6. One example which requires reselecting the tetrahedron unit. RESULTS AND DISCUSSIONS Because different initial start point (ISP) for AEP generates different airflow paths, three ISP located at the bottom of the test grain bin were selected to calculate the widest airflow paths. The results are shown in Table. of the corresponding airflow tetrahedron units and the airflow paths are shown in Fig.7 and Fig. 8. Table. ISP The calculation results of tortuosity for 3 ISP. Location (m) Path length (m) Le Bed depth L0 (m) Tortuosityτ No. of tetrahedron units n p ISP1 (0.0,0,0) 0.367 0.516 1.46 157 ISP (-0.07,0,0) 0.368 0.516 1.46 160 ISP3 (0.09,0,0) 0.385 0.516 1.53 16 Fig. 7 A series of tetrahedron units for three widest airflow paths. 7
Fig. 8 Widest airflow paths through the grain bed. The calculated tortuosity values are higher than that has been reported in the literature (e.g., Yun et al., 005). One of the reasons is that the calculated airflow paths include sharp turns. The airflow path is constructed as a series of straight lines connecting the centroids of base triangles in tetrahedrons, and therefore, sharp angles may exist between two adjacent line segments. However, flowing air cannot make sharp turns. In other words, the airflow path should be a smooth curve. Therefore, the sharp angles are replaced by an arc to smooth the airflow path (Fig. 9), as recommended by Sobieski et al. (01). Removing the sharp angles causes the reduction in the airflow path length and Sobieski et al. (01) proposed a set of equations for calculating the reduction in path length. Fig. 9. Schematic representation of smoothing the airflow path by replacing the sharp turning angle with an arc (Sobieski et al. 01). After smoothing, the calculated tortuosity values ranged from 1.35 to 1.45 (Table 3), which compare favorably with the values reported by Yun et al. (005) 1.0 and 1.4 with the decrease of porosity from 1.0 to 0.4. Sobieski (01) obtained an average tortuosity of 1.14 for the shortest airflow path 8
in porous beds. It is expected that tortuosity for the widest path calculated in this study should be greater than that for the shortest path. Table 3. The comparison of tortuosity before and after smoothing. ISP Tortuosity before smoothing τ Tortuosity after smoothing ISP1 1.46 1.40 ISP 1.46 1.35 ISP3 1.53 1.45 Average 1.48 1.40 cor τ CONCLUSIONS This paper proposes a method to study the airflow path in grain beds at a microscopic scale. The discrete element model is shown to be capable of simulating the pore structure of grain beds, which provides a base for geometrically constructing airflow paths through the pore space between particles (grain kernels). The tortuosity predicted by the model is in good agreement with the experimental data reported in the literature. However, this research only studies the airflow paths geometrically in grain bulks without considering the properties of airflow. Further research is needed to study the relationship of flow properties and the geometrical flow path. REFERENCES ASAE Standards. 007. D7.3: Resistance to airflow of grains, seeds, other agricultural products, and perforated metal sheets. St. Joseph, Mich.: ASAE. Brooker, D. B. 1961. Pressure patterns in grain-drying systems established by numerical methods. Trans. ASAE 4(1): 7-74. Brooker, D. B. 1969. Computing air pressure and velocity distribution when air flows through a porous medium and non-linear velocity-pressure relationships exist. Trans. ASAE 1(1): 118-10. Chapman, J. E., R. Vance Morey, H. A. Cloud, and J. L. Nieber. 1989. Airflow patterns in flat storage aeration systems. Trans. ASAE 3(4): 1368-1373. Du Plessis, J. P., and S. Woudberg. 008. Pore-scale derivation of the Ergun equation to enhance its adaptability and generalization. Chemical Engineering Science, 63(9): 576-586. Ergun, S. 195. Fluid Flow through Packed Columns. Chemical Engineering Progress 48(): 89-94. Hukill, W. V., and N. C. Ives. 1955. Radial airflow of grain. Agricultural Engineering 36(5): 33-335. Jacob, B. 1988. Danymics of Fluids in Porous Media, New York, USA: Elsevier. Jindal, V. K., and T. L. Thompson. 197. Air pressure patterns and flow paths in two-dimensional triangular-shaped piles of sorghum using forced convection. Trans. ASAE 15(4): 737-744. Khatchatourian, O. A., and M. O. Binelo. 008. Simulation of three-dimensional airflow in grain storage bins. Biosystems Engineering, 101(): 5-38. 9
Liu, C. Y., Q. Zhang, and Y Chen. 008a. PFC3D simulation of vibration characteristics of bulk solids in storage bins. ASABE Paper No. 083339. St. Joseph, MI.: ASABE. Matyka, M., A. Khalili, and Z. Koza. 008. Tortuosity-porosity relation in the porous media flow. Physical Review E, 78(): 1-8. Neethirajan, S., and D. S. Jayas. 008. Analysis of pore network in three dimensional (3D) grain bulks using X-ray CT images. Transport in Porous Media 73(3): 319-33. Pierce, R. O., and T. L. Thompson. 1974. Airflow patterns in conical shaped piles of grain. ASAE paper No. 74-3015. St. Joseph, Mich.: ASABE. Segerlind, L. J. 198. Solving the non-linear air flow equation. ASAE paper No. 8-3017. St. Joseph, Mich.: ASAE. Shedd, C. K. 1953. Resistance of grains and seeds to airflow. Agricultural Engineering 33(9): 616-619. Smith, E. A. 198. 3-dimensional analysis of air velocity and pressure in beds of grain and hay. Journal of Agricultural Engineering Research 7(): 101-117. Smith, E. A. 1995. Forced air distribution for in-bin drying and aeration. In Stored-Grain Ecosystems. Edited by D.S. Jayas, N.D.G. White and W.E. Muir. Marcel Dekker, Inc., New York: 569-607. Sobieski, W., Q. Zhang, and C. Liu. 01. Predicting tortuosity foe airflow path through porous beds consisting of randomly packed spherical particles. Transport in Porous Media 93(3): 431-451. De Ville, A., and E. A. Smith. 1996. Airflow through beds of cereal grains. Applied Mathematical Modelling 0(4): 83-89. Wu, J. S., B. M. Yu, and M. J. Yun. 008. A resistance model for flow through porous media. Transport in Porous Media 71: 331-343. Yu, B. M., and J.H. Li. 004. A Geometry Model for Tortuosity of Streamlines in Porous Media. Chinese Physical Latters 1(8): 1569-1571. Yun, M. J., B. M. Yu, B. Zhang, and M. T. Huang. 005. A Geometry Model for Tortuosity of Streamlines in Porous Media with Sphere Particles. Chinese Physical Latters (6): 1464-1467. Yun, M. J., Y. Yue, B. M. Yu, J. D. Lu, and W. Zhang. 010. A Geometry Model for Tortuosity of Streamlines in Porous Media with Cylindrical Particles. Chinese Physical Latters 7(10): 104704-1-104704-4. 10