Powder Flow

A theory is presented for the fully-developed flow of gas and particles in a vertical pipe. The relation between gas pressure gradient and the flow rates of the two phases is predicted, over the whole range of cocurrent and countercurrent flows, together with velocity profiles for both phases and the radial concentration profile for the particles. The gas and the particles interact through a drag force depending on their relative velocity, and there are mutual interactions between pairs of particles through inelastic collisions. This model is shown to account for marked segregation of gas and particles in the radial direction, and the predicted relation between the pressure gradient and the flow rates of the two phases is surprisingly complex.

Introduction

There are many important problems on the conduction of heat in granular powder beds, i.e. silo beds, coal beds, high furnace in steel making, compaction of powder in ceramics industry, etc. The mathematical treatise on these problems, however, is not necessarily clear, because the correct physical model is not considered. In this report it is theoretically considered.

SUMMARY

This report covers research under the auspices of IFPRI during the period January 1986 - December 1989. Mr. M. C. Turner has continued in the appointment of a Research Studentship throughout this period of the IFPRI research project.

Results are reported of investigations into the flow properties of idealised powders (monodisperse spherical particles) using experimental techniques, computer simulations of idealised models, and fundamental theoretical studies. The objective of the research is to predict constitutive rheological relations for use in fluid mechanics calculations of the flow of powders in given geometric devices.

Experimental studies have been based on measurements of the behaviour of well-characterised powders of monodisperse spheres in a rotating fluidised bed, and also on direct laboratory measurements of the coefficient of restitution for glass ballotini spherical beads. Results are reported for glass ballotini particles in the size range from 10m4 to 10-3 m. Experimental measurements of the properties of *ideal powders* are required to test the accuracy of computer simulation models alongside the development of the computational approach.

Computer simulation results are reported for four different types of model system.

The first approach was to set up a computer simulation model of chute flow closely resembling the simple experimental geometry. This gives information on boundary effects but it realised early on that, whilst this may also give some information on the constitutive rheology, "particulate fluid mechanics" on relatively tiny numbers of particles is not the way * forward.

Computational rheology and computational fluid mechanics must be treated separately.

The calculations of the constitutive rheology of an idealised powder require the use of homogenous periodic boundary systems with well-defined particle and system state variables similar to non-equilibrium molecular dynamics used to determine transport properties of molecular systems. In this case, the computations are properly described as steady-state granular dynamics.

The three types of system for which we report results are:

1. An isokinetic system of the ideal powder of frictionless, monodisperse, elastic hard spheres where the *granular temperature* (total kinetic energy) is held constant by uniform continuous velocity renormalisation. This artificial system has no direct experimental counterpart but it relates to real-granular systems by analytical scaling laws which we have developed.
2. The direct simulation of flowing systems of inelastic frictionless spheres with a constant coefficient of restitution for comparisons with available theoretical and experimental results. These are essentially exact computations of the constitutive rheology of that model and can be used to test the approximations in the kinetic theory approach previously advocated by other researchers in this area.
3. The steady-state granular dynamics simulations have been extended to incorporate surface friction. By comparing the results for systems with and without surface friction we are able to estimate its effect on the constitutive rheology and examine means of incorporating surface friction besides inelasticity into simple analytic forms for the rheology using scaling prediction methods.

The scaling laws which we report have been developed to predict the dependence of the pressure tensor initially in the region of rapid granular flow, on the rate of strain deformation. Using known properties of the thermal equilibrium hard-sphere fluid and its steady-state isokinetic flow curves, these scaling laws enable the constitutive rheology for systems over a wide range of particle and state variables to be presented analytically. Results have been obtained for various forms of the coefficient-of-restitution, since this is not known experimentally for even the simplest real powders, to give some insight into how it affects the rheology in the rapid flow domain. The scaling predictions have been compared with both experimental results and predictions of kinetic theory.

The scaling laws also predict the shear-rate dependent granular *temperatures* (particle kinetic energies), kinetic conductivities and particle diffusivities from available hard-sphere fluid transport data. This produces all the input data necessary to proceed with a finite difference or finite element fluid mechanics prediction, either transient or steady, of laminar shear flow. The methods of predicting the constitutive rheology are easily extended to elongational and bulk deformations for more general flows.

ABSTRACT

This study examines the transition from fluid behavior to solid behavior that too often occurs in granular flow and brings with it such unwelcome events as funnel flows in hoppers and other clogging of material handling devices. This situation is studied using a discrete particle computer simulation of a Couette flow with gravity. This simulation exhibits the full range of granular flow behavior, from a stagnant solid-like material, through a quasistatic transition zone, to a rapid granular flow. The most important result is that the first motion in the material just above the static bed, occurs in a quasistatic mode at a fixed value of the stress ratio rxy/ryy. Thus it appears that the location of the transition from solid to fluid behavior can indeed be described by a Mohr-Coulomb failure criterion.

In order to study the mechanisms of fines ejection at the bed surface and the mixing of fines, two sets of experiments were carried out.

Cracking catalyst, mean size 58 um, was continuously injected into a bed containing sand (450 um). The axial and radial fines concentration in the bed and the elutriation rate were measured. If diffusional mixing process is operative, then for continuous injection of fines to the bottom of the bed, the fines concentration would be higher at the bottom and lower at the top. However, experiment showed an inverse concentration gradient of fines. An expression is developed to predict this profile by assuming that the fines migrate upwards in the bubble wake and downwards by eddy diffusion. The radial concentration profile is zero. Size analysis of the fines in the bed indicated the existence of fines stratification according to size.

Single bubble injection experiments were carried out by injecting a single bubble into a bed fluidised at Umfc. The quantity of fines ejected at the bed surface per bubble was determined. The experimental results are compared with the predictions from the bubble wake and the bubble roof ejection mechanisms. Another fines ejection or migration mechanism is suggested - the ejection of fines dispersed &thin the bubble void. This mechanism has not been previously suggested. There is reasonably good agreement between theory and experimental results.

Summary

The object of this work is to develop methods for the quantitative prediction of all the major features of flow of a gas, together with solid particulate material, through a duct of arbitrary size and inclination. Flows of this sort are of great technical importance in pneumatic transport of particulate material, and in the circulation of particulate materials within chemical processes. Examples of the latter type include the riser reactors and standpipes which form components of the catalyst circulation loop in catalytic crackers, used in the refining of oil, and the long standpipes used in certain coal liquefaction plants. In all these systems the particles tend to distribute themselves over the cross section of the duct in a markedly non-uniform way, making it very difficult to predict the hold up of solid material and the pressure drop along the duct, or even to extrapolate these quantities from measurements made with the same materials in ducts of other sizes. In addition, the crowding of the particles into limited parts of the cross section can lead to undesirable effects, such as recirculation of the solid material against the direction of the main flow.

The key to making useful predictions for these systems is to understand and quantify the mechanism that determines the distribution of particle concentration over the cross section. In many situations of practical interest the gas flow is highly turbulent, and it is tempting to attribute the observed distribution of the particles to their interaction with turbulent eddies. However, a closer examination shows that this could produce the observed effects only in quite restrictive circumstances, where the particles are almost, but not quite light enough to follow the gas motion exactly. In this work we investigate an alternative mechanism, which attributes the stratification of the concentration distribution to collisions between particles. We have derived a criterion (presented in this report) to judge whether this mechanism is likely to be important in any given system, and have developed a complete mathematical model to predict the distribution of particle concentration and the velocity profiles for both gas and particles in steady flows of this kind. A computer program has been constructed to solve the model equations, and extensive sets of solutions have been found for ducts of different sizes and inclinations.

The solutions obtained appear to simulate most of the characteristic observed properties of flows of this sort, including the undesirable recirculation patterns referred to above. There is a dearth of good quantitative experimental data covering ranges of design and operating conditions broad enough to provide a searching test of the theory, but we intend to seek out what is available for comparison with our predictions. For systems where collisions between particles mediate the pattern of flow our program provides, for the first time, a rational basis for the design of particle-gas transport systems and, perhaps of equal importance, for identifying those circumstances in which their performance is likely to be unsatisfactory. It is not a very large step to extend the method to include developing flows, where the particles are accelerating under the influence of forces exerted on them by the gas stream. To the extent that turbulence can be modelled, it is also possible to introduce into the model some effects of turbulent fluctuations in the gas velocity.

This proposal originally addressed the issue of why stagnant zones, such as funnel flows in hoppers, appear in particle flows. To that end, we studied computer simulations of a Couette flow with gravity acting on a system of two-dimensional discs. These were largely described in last year’s report. In those simulations, gravity acted to force a stagnant zone of material to form, so that the conditions that led to the transition from fluid-like to solid-like behavior could be observed and studied. Much to our surprise, the initial motion of the layer occurred in a quasistatic manner with the location of the interface coinciding with a constant value of the ratio of shear to normal stress. We have continued this work in three directions.

1. Extension of the model from two to three dimensions. This phase is almost complete.
2. Shear cell tests on our simulated material. As we have shown that the yielding appears to follow a Mohr-Coulomb failure criterion, similar to that used in plasticity models, why then, did plasticity models fail in predicting and describing funnel flows in hoppers? One answer is that the plasticity models depended on friction angles measured in Jenike or other, similar, shear cells and, because of the different flow conditions, the observed yielding behavior may be different. To test this hypothesis, we created a simulation of a shear tester to perform the equivalent of a shear test on our simulated material and, indeed, in preliminary results, it appears that the stress ratios measured in the shear cell are larger than those that govern the location of the yield interface in the Couctte flow with gravity simulation.
3. Inclined chute flows: Chute flows have an odd yielding behavior in which they often move largely as a solid block over a thin shearing basal layer. At first glance, this seemed to be in contradiction to the Couette flow results. However, simulations show that the two results arc indeed consistent. But at the same time, they pose other questions. In the Couette flow situation, the location of yielding could be determined more or less on the basis of a frictional criterion. In the chute flow simulation, while the stress ratio at the yield point wus numerically similar to that observed in the Couette flow simulation, one could not determine its location baaed on such a criterion. Consequently, there must be some unexplained connection between the location of the interface and the flow mechanics. As it probably has implications in other flow situations, this is a problem worthy of further study.

At the Teaneck meeting in 1989, Gordon Butters asked me on behalf of the TC, to see if my work could shed some light on the fracture problem. On further consultation with Paul Isherwood, I learned that there was a general lack of information about the forces that are exerted by the flow induced particle collisions. While the simulations have been previously made stress tensor measurements and thus determined averaged forces applied to particles, these are generally irrelevant to the fracture problem as the most damage will be caused by the maximum and not the average forces. Thus, I conducted a series of simulations to determine the maximum collisional impulses that the particles experience in a simple shear flow and their dependence on particle properties and solids concentration. The impulses are divided into their components normal and tangential to the particle surface as it was felt that the two might contribute to different attrition characteristics. The normal impulses - which might lead to large scale particle fracture - was always significantly larger than their tangential counterparts - which would tend to shear off the microroughness that lead to the interparticle surface friction. Along the way, histograms of the distribution of collision impulses as well as their geometric distribution over the surface of the particle were recorded.

Also, at the Teaneck meeting, Hans Buggish, not on behalf of anybody but himself, suggested that I might be able to contribute to his IFPRI sponsoned work on the flow induced mixing of particles in his granular shear cells. Be had observed that the mixing might be modeled as a diffusion process, similar to that of molecules in a gas or liquid. The use of a computer simulation was particularly attractive in such a study as his experimental technique was limited to measuring the diffusion of particles in only the direction parallel to the velocity gradient, while the computer simulation could measure the diffusion in all directions. The results show that the particles do mix by diffusion except at the highest concentrations when the particles become tightly packed in a crystalline microstructure and unable to move relative to their neighbors. However, the diffusion in a shear flow is not isotropic and is only appropriately modeled as a tensor of diffusion coefficients. By far, the largest mixing occurring in the direction of flow. The components of the diffusion tensor were measured both by particle tracking and by a statistical technique developed by Taylor (1922). Furthermore, it showed that the mixing in a granular flow flow was an example of Taylor diffusion by which the diffusion of particles in the direction of the velocity gradient greatly enhanced their mixing.

Finally, this report describes a preliminary attempt to model the flow through pinmills. Like the impulse strength studies mentioned above, this work done pursuant to Gordon Butters’ request that 1 attempt to apply computer simulations to fracture problems. This work was intended to study a situation of more direct interest to industry than a simple shear flow. However, the project has been abandoned due to a general lack of interest in pinmills within the industrial community.