A discrete element model for simulation of a spinning disc fertilizer spreader P. Van Liedekerke; E. Tijskens; E. Dintwa; H. Ramon Departement of Agro-engineering and Economics, K.U.Leuven, Kasteelpark Arenberg 30, B-3001 Leuven; e-mail of corresponding author: paul.vanliedekerke@biw.kuleuven.be 1. introduction In this paper, a DEM (Discrete Element Method) model is presented and a series of computer experiments is analyzed and compared to experimental validations. Also, the simulations are compared to experimental results. The model uses a 3 parameter contact force to calculate impact between particles and other object such as vanes. The components of the contact forces are typically modeled in terms of a scalar quantity measuring the material deformation at the contact point. In this paper we use the Hertz-Kono-Kuwabara model [Kuwabara,1987] for the normal force component [Schäfer et al., 1996] experienced by the particle : 1 N = min 0, δ 2 N knδn + c & NδN n. (1) ( ( )) Here, k N and c N are the non-linear contact stiffness and damping parameter, respectively. The quantity δ = d r is termed the virtual overlap of the contacting bodies. N 2. Experiments s First, an estimation of the model parameters was done. For measuring methods of particle stiffness, damping and friction, we refer to (Van liedekerke, 2006). The particles used for the experiment are from a domestic fertiliser (quite round shape), with an average radius of 1.2 mm. The model was then validated using a domestic centrifugal spreader disc of 0.15m radius with 4 vanes which is driven by an electrical motor at 400 rpm. The feeding of the fertiliser particles was controlled by a funnel shaped bin with a circular orifice of 0.011 m radius. Figure 1 (left) : experimental set-up of the mini spread hall with collector tray, (right) :measured and experimental amount of fertiliser particles in each basket with the collector tray located at 1m from the disc centre.
DEM for centrifugal spreaders Paul Van Liedekerke Katholieke Universiteit Leuven (K. U. Leuven) Kasteelpark Arenberg 30, B-3001 Leuven, Belgium http://www.agr.kuleuven.ac.be/aee/amc/amc.htm
overview Situation to model : from bin to field >10 6 particles! field?
overview DEM Simulations : 1. Single particle simulations 2. Multi particle simulations (reality)
1.Single particle system Why single particle simulations? 1.Comparision of DEM contact forces withthe components of the Patterson and Reece (1962) equations -Centrifugal force, Coriolis force, Friction,.. 2.Experimental verification of the trajectory
1.Single particle system DEM Contact-force description of two objects 0.7 N 0.6 0.5 0.4 0.3 T F 0.2 0.1 T= min(kx, F) (slip-stick) 0 0 0.2 0.4 0.6 0.8 1 1.2 x 10-4 N=C x 1/2 dx/dt + Kx 3/2, N>0 (Hertz) F=T+N function of,c,k
1. Single particle system : measuring model parameters Measuring stifness : Compression tests of fertiliser particles force N -Breakage typically around 55 N break Hertz: F=Kx 3/2 Intrusion mm
1. Single particle system : measuring model parameters Measuring damping : rebounce experiments of fertiliser particles v1 v2 Use of High speed camera C=f(v1/v2) (theoretical) or C via model parameter optimization
1. Single particle system : measuring model parameters : measuring angle ( ) of slope at constant speed tan ( )= Use of High speed camera
1. Single particle system : measuring model parameters Patersson equations versus DEM Ma=-(5/7µ)Mg + (5/7)Mω²R (rolling against vanes) or Ma= -µmg +Mω²R- 2M ω µv (sliding against vanes) F=C x 1/2 dx/dt + Kx 3/2, F>0 T= min(kx, F)
Single particle system forces involved -rolling force R=? F total force =F-R Line is analytical solution
Single particle system forces involved centrifugal force F=mω²R R Line is analytical solution F R
Single particle system Forces involved -Coriolis force F=2m ωv ω V=radial speed Bouncing against vane (dotted is lower damped particle)
Single particle system Experimental verification : Measuring the radial trajectory of a particle on a flat disc
Single particle system particle feeder
Single particle system Radial trajectory plots o : experiment - : DEM ω=470tr
Particle rotation? (spin) Single particle system vane disc Rotational speed against vane : a) high friction, b) low friction ->a) rolling, b) sliding
Single particle system Particle rotation (spin) 140rad/s Sliding on the disc : theoretical should be 220 rad/s ->DEM simulation shows lifting of particle,probably due to conservation of spin perpendicular to vane
Conclusions : Single particle system 1.DEM shows a general very good agreement for 1- particle situations 2.Tangential contact properties should be further verified (->lifting of particle??)
2. Multi particle system -Difference with 1-particle : - Conical disc - 2 or more vanes - Vanes with border are needed - Multiple collisions involved - Feeding of the spreader (bin)
2. Multi particle system - Conical disc - 2 or more vanes with border Cylinders on the vane edges to incorporate random reflection
- Multiple collisions 2. Multi particle system # Particles Computation time Use of efficient contact detection algorithms is required
Feeding : fill-up of a beam 2.Multi particle system In stead of a bin, the particles start from a beam which has the same shape on its base as the orifice of the bin
2.Multi particle system Feeding : adjustable input flow (kg/s) No Gravity Constant speed V Bin orifice Random velocity components added Gravity Flow(kg/s) = V x (particle density) x (orifice surface)
Model validation : 1. Spread pattern measurements 2.Multi particle system: measurements
2.Multi particle system: measurements Summary of the experiment Domestic fertilizer Domestic centrifugal spreader Disc radius : 0.15m Disc speed : 400rpm Basket resolution : 0.25m X 0.25m Number of baskets : 14 Amount of fertiliser per basket-line : 1.5kg Particle flow : 0.1kg/s
2.Multi particle system: measurements 1.5m Collector tray 1m 0.5m spreader
2.Multi particle system simulation results Longitudinal measurements(-) and simulations(--) at 0.5m,1m and 1.5m from disc
Multi particle system 2. Mini tester bin Collector tray
Multi particle system Particle feeding area 90 collector 0 vanes 270
Cilindrical plots Multi particle system
Multi particle system
simulation
simulation
Conclusions Multi particle system -There is a good agreement of DEM simulations with experiments
steel plate
When the particles are released on the disc, they are thrown away by the vanes and collected by a tray of baskets, which is 3.5m long and consists of 14 baskets of 0.25m x 0.25m. This collecting tray can be replaced to obtain information about the spread pattern at different locations from the disc (see figure1, left). Each experiment consists of releasing 1500g of fertiliser on the spinning disc. The flow rate of the particles through the bin is constantly kept at 0.1 kg/s. This experiment is repeated for 3 times for 3 different distances (1.5m, 1m and 0.5m) from the disc in order to have an idea of the static spread pattern. 3. Simulations In the simulations, 10000 spherical particles are used with the same particle distribution, representing a total mass of 150g and using the same flow rate. Although the total particle mass is 10 times less than in the experiment, it was investigated that introducing more particles in the simulation has little effect on the result. The trajectory through the air was calculated by simple ballistics, using an air resistance coefficient of 0.5. Using an efficient contact detection algorithm, one simulation typically takes 30.000 seconds and represents 1 second of real time. 4. Conclusions Figure 1 (right) shows a reasonable agreement between simulation and experiment, especially in a qualitative way. Anyway, a discrete element model might provide interesting information about how a spread pattern behaves when different geometry is introduced for the spreader without having to do any experiments. It might also be used as an optimization tool to obtain better spread patterns. 5. References Kuwabara G; Kono K (1987) Restitution coefficient in a collision between 2 spheres. Japanese Journal of Applied Physics 26(8):1230-1233 Schäfer J; Dippel S; Wolf D E (1996). Force Schemes in simulations of granular materials. Journal de Physique (France), 6, 5-20 Van Liedekerke P; Tijskens E; Ramon H (2006). A discrete element model for centrifugal spreaders. I: single particle simulations. Powder Technologie (in press)