(276d) Purpose-Filled-Design for Handling Systems Using a New Radial Stress Theory to Predict Segregation and Vibration Flow in Feed Systems

At the heart of designing handling systems for bulk solids is the radial stress theory. This theory was developed in the 1960’s by Andrew Jenike and Jerry Johanson as they worked at the University of Utah. The original intent of the work was to help understand the complex stresses that act in typical process vessels. However, when the radial stress theory was coupled with the concept of a radial velocity the theory was then able to predict stresses, as well as the expected velocity profiles, in these pieces of process equipment. Because of the antiquated computing capability of that day, these theories were explained and developed with the concept of limiting behavior. It became possible and easy to measure a wall friction angle and an effective angle of internal friction as well as predict a flow or no-flow criteria. Mass-flow and funnel-flow were terms coined to describe the fact that under some conditions solids may flow in only a central flow channel surrounded by stagnant zones of material. Pseudo concepts like first-in-and-last-out were used to determine process behavior and make decisions about segregation in process vessels or residence time distributions or mixing in process vessels. This approach was not very quantitative and engineers were left having to rely on pilot-scale experiments to quantify these behaviors. Inherent in the original radial stress theory were a set of simplifying assumptions that made the calculations easy to perform, but limited the usefulness of the theory to just a subset of some process vessels. Another issue about the development of the radial stress theory is that the theory was developed first and then applied to process conditions in an effort to make all observations fit the theory.

The first goal of this work is to relax these assumptions in a systematic way to extend the usefulness of the work to more extensive process conditions. The second goal of this work is to redefine what the theory provides as an output and to express this in terms of specifics of the product to be handled. Modeling and measurement go hand in hand and sometimes it is necessary to take a look back and ask the two questions:

• Given the operation protocols and geometry of the process along with the product specifications, is there a way that the theory at hand can be applied to predict something about my end product leaving my process?
• If that is possible, what must be used as a metric to describe a good product and can the theory at hand help me to predict that metric so I can make an informed judgment call on the design of my process or product?

For example, in packing I am interested in controlling the weight of material as well as the concentration of key components introduced into the package, so that I can define a metric that tells me if my product is good. I define one of those metrics by stating that the content uniformity of a product leaving the handling system must always be greater than the lower limit of allowed content uniformity and less than the upper limit of content uniformity as dictated by the customer's needs. With that definition in mind, I can use the theory at hand to compute the velocity profile in all the pieces of process equipment and then use the segregation profile to estimate what leaves the process as a function of time. I may also need to compute the residence time distribution of some parts of the process from the theory at hand and do a mixing analysis to estimate the remixing that may occur in the handling process. But, I can use the theory to compute the content uniformity leaving the process and then create a plot of content uniformity as a function of key material flow properties and process variables. Such as, the radial stress theory gives a relationship between the wall friction angle and hopper slope which provides a limiting line that determines if the material is in funnel flow or mass flow. This does not really help with the metric I have chosen to determine good or bad product, but I can use the radial stress theory coupled with the segregation profile and process description to generate a plot of expected content uniformities leaving the process as a function of the wall friction and hopper slope angle. Now we have an intelligent design tool that allows me to design the process to fit product needs. A similar thing can be done with the consistent weight metric required for the robust packing of my product. The theory, whatever it is, should be used in such a way that engineers can easily design with a purpose in mind (Purpose-Filled-Design).

All theories are based on a set of simplifications that make it easier to relate key properties to theoretical outcomes. If possible, that relationship needs to be straightforward. But the simplifying assumptions cannot be so extensive that the theory does not predict most of the real-world behavior. Therefore, it stands to reason that whatever theory is used the assumptions should be able to cover the real-world operations. The radial stress theory as originally introduced had a fairly extensive list of assumptions::

1. Limited or small accelerations of the bulk material (i.e. quasi-static flow)
2. Axisymmetric stress application and axisymmetric boundary conditions
3. A linear relationship between the stress in the process vessel and the vessel diameter
4. No external force gradients acting on the bulk material except gravity which acts in the direction of flow
5. The stress and strain rate tensors are aligned in principal directions
6. No pseudo-cohesive effects (i.e. continual deformation without volume change – flow happens without induced cohesion effects)

If flow properties are measured, then mathematical theories such as the radial stress theory can be employed to estimate the velocities in process equipment and compute residence time distributions and segregation emptying profiles for fill-then-empty processes – as well as semi-continuous processes. However, the basic assumptions of the radial stress theory are limited and many real processes do not conform to the assumptions behind the standard theory. Therefore, the solids velocity profiles computed from the radial stress theory may not be sufficient to describe segregation profiles and residence time distributions for material leaving the process.

This work examines the relationship that external gradients such as non-radial stress gradients or local gas pressure gradients in the process might have on the solids velocity profiles in process equipment. The work outlines calculations to compute velocities in bulk handling process equipment subject to these gradients and then applies those velocities to the calculation of segregation profiles. The ultimate goal and focus of the work is to relate the external force gradients to product consistency. A new radial stress theory to include non-radial stress and gas pressure gradients will be presented.

Finally, the mass flow plots created by Andrew Jenike and Jerry Johanson were used to determine velocity ratios in process equipment, feeders stress loads, and even flow factors to describe arching effects. However, they do not contain a segregation index that could be used to determine what type of mass flow will provide a certain degree of segregation prevention. The plots also do not contain a flow aid vibration index to prevent or modify funnel flow behavior. Therefore, as part of this work new mass flow plots containing segregation intensity numbers and vibration intensity numbers are generated for various non-radial solid stress gradients, local gas pressure gradients, effective internal friction angles, hopper angles, wall friction angles, and segregation patterns. These tools can assist practicing engineers in designing process equipment to more easily solve segregation problems or solve problems where the vibration of the mass is required to induce the proper flow. Depending on the direction of the excess stress gradients, gas pressure gradient, or external acceleration force, applying the radial stress theory can predict a change in mass flowability to help or hinder the flow and segregation prevention ability of the handling system.