(112b) Finite Element Analysis for Incipient Flow of Bulk Solid in a Diamondback Hopper

Storing of bulk materials is essential in a large number of industries. Powders are often stored in containers such as a bin, hopper or silo and discharged through an opening at the bottom of the container under the influence of gravity. This research tackles the frequent problems encountered in operating bulk solids such as flow obstructions and discontinuous flow resulting in doming and piping in hoppers. This problem is of interest to a variety of industries such as chemical processing, food, detergents, ceramics and pharmaceuticals. An objective of this work is to use finite element analysis to produce results that could easily be incorporated into the industry and be used by the design engineers. A complete study includes both a numerical approach, with an appropriate material response model and an experimental procedure to be implemented on the bulk material of interest. Results from the analysis are to be used to relate the slopes of the channels and the size of the outlets. Results from the FEM analysis are to be verified using measured stresses from a Diamondback hopper.

A finite element approach to solve for displacements, velocities and stresses is presented. The stress distribution is to be determined during storage and discharge. For bulk material a highly nonlinear constitutive model needs to be used that describes the solid-like behavior of the material during small deformation rates as well as the fluid-like behavior during flow conditions. A 3-D general elastic/viscoplastic constitutive equation with non-associated flow rule is used. This model captures all major features of powder mechanical response and describes the evolution of deformation and volume change. This model is reduced into a FORTRAN code that will is incorporated into the commercially available software ABAQUS to describe the material response. Other models such as the Capped Drucker-Prager model are being investigated. It is used to measure frictional materials that exhibit pressure dependent yield. The material properties for calibration of the models are determined experimentally using both direct shear testers like the Schulze and the Jenike testers and indirect shear testers like the Triaxial Compression tester. The direct testers assume that the bulk material is rigid perfectly plastic and help determine the strength of the material including the yield strength, angle of friction and cohesion. The calibration of the viscoplastic model however is more involved and requires determination of parameters from tests where the loading path is controlled and the stress-strain history is determined. This is achieved by conducting triaxial tests. Wall stresses in the hopper are measured using pressure sensitive pads made by TekScan. The pads consist of two sheets that when in contact produce a contact resistance that varies when normal stress is applied. These pads were glued to the inside surface of the round-to-oval hopper.

Shear tests were conducted on Silica powder with particle size of about 50 microns. Values of cohesion were determined to be between 0.2Kpa and 1.2Kpa. The angle of internal friction was determined to be around 32 deg. The wall friction angle used was 22 deg. The wall stresses generally decrease around the perimeter and are smaller at greater material depths. They also show an oscillatory pattern. The calculations of wall stresses using the Janssen equation produce results that capture the magnitude and decreasing stress values but fails to predict the oscillations. FEM analysis in ABAQUS was conducted using both the capped Drucker-Prager model and the visco/plastic model. The results show the decreasing pattern of the wall stress similar to the measured results capturing some of the oscillatory pattern.

It is concluded from the results that wall stresses inside a diamondback hopper can be successfully measured using pressure sensitive pads. FEM simulations with the right constitutive model are a powerful tool in predicting the stress distribution in a hopper. Using the Drucker-Prager model produces results that have similarities to measured results but yet have significant variations. A more complex model such as the 3D viscoplastic model is needed to produce better predictions.