(101b) Experimental and Numerical Study of Discharge Rates for Cohesive Particles in Hopper Geometries
In many industrial applications silos are used for the storage of particulate materials and products. The properties of particles, the wall material, as well as the silo geometry determine the flow characteristics inside the shaft and hopper. The flow profile can be crucial, especially for perishable goods where dead zones can lead to rotting and health hazards. Jenike (1964) established a method to determine the critical hopper angle to prevent funnel flow as well as the minimal discharge orifice size to ensure no blocking or bridging of particles. Besides the critical silo parameters to create a reliable material flow in the first place, the discharge rate of the particles is another important step to determine the corresponding process.
Overview and research questions:
During the last century, many empirical correlations of the mass flow rate for hoppers emerged. One widely used and expanded model is the Beverloo model for simple flat orifices with coarse and mostly cohesionless particles (Beverloo et al., 1961), which has been adapted for angled hoppers (Rose & Tanaka, 1956). Additionally, modifications were included for fine powders, which can be affected by air counterflow, induced by the vacuum due to the discharging particles (Crewdson et al., 1977).
Contrary to empirical correlations, Tomas (1991a, 1991b) derived an analytical approach of the outflow velocity for fine particles, which also includes the mentioned air flow due to the induced vacuum. By addressing the forces on an incremental particle bridge inside a hopper, differential equations for the stationary and instationary outflow velocities could be established. The underlying concept was also successfully expanded for vibrated geometries (Kache, 2010). One advantage of the model is that flow characteristics of the particles are used in the formulas. The cohesion and friction of particles are considered by the minimal discharge orifice diameter, which can be obtained with shear cell measurements.
Schulze (2006) proposes that the induced vacuum and therefore the counterflow during discharge is difficult to estimate; therefore leading to unreliable calculated mass flows, especially for very fine and cohesive particles. Kache (2010) used a modified Tomas model to consider those air drag forces by proposing an lower estimated pore diameter for the counterflow inside the bulk solid. While this reduces the calculated influence of the vacuum, the model is able to predict the discharge mass flow of very fine calcite MX10 for an industrial scale silo decently, which is indicative of an overestimation of the counterflow in the Tomas model.
The mentioned studies about particle flow therefore lead to the following questions for fine particle systems: Firstly, how do drag forces due to induced air flow inhibit the particle discharge. More information about this induced air counterflow could lead to models usable for Discrete Element Method (DEM) simulations in cases where a packed particle flow takes place. This could result in more realistic results without explicitly solving the fluid flow. Another question is how cohesion affects the mass flow rate and the transition between instationary and stationary flow. The effects of cohesion on potential fluctuations of the stationary flow rate, the change in porosity and flow profiles are additional issues. Also the impact of the filling height is relevant, since this can affect the particle flow rate as well (Heinrici, 2006).
In this contribution the experimental and numerical study of discharge rates for cohesive particles in hopper geometries was performed. The tested particulate systems consist of nearly monomodal spheres. The size of particles is varied to obtain different cohesive to gravitational force ratios. The flow properties of the different particle systems are measured by ring shear cells, which allows the calculation of the corresponding critical silo dimensions with the Jenike method. The test hoppers are constructed accordingly. During discharge experiments, the mass flow rate is measured by simple scales.
Besides experiments, the empirical and analytical equations for the mass flow rate are further analyzed for small particles and powders inside hoppers. Especially the model of Tomas and the appropriate equations for the counterflow are addressed thoroughly. The calculated critical discharge orifice size is used for the model to calculate the outflow velocities. Possible fluctuations during the instationary and stationary discharge are considered as well.
DEM simulations of particle discharge are also carried out to further analyze the drag forces by comparing the experiments and equations with the simulation results. The dimensions of the simulated hoppers are in line with the critical dimensions according to Jenike. The appropriate modeling of the the vacuum induced air counterflow is discussed.
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Crewdson, B. J., Ormond, A. L. und Nedderman, R. M. (1977), Air-impeded discharge of fine particles from a hopper. Powder Technology, (16), pp. 197-207.
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