FRR - Final Report

EXECUTIVE SUMMARY

IFPRI funding enabled us to consolidate and complete work on the rheology of stable, phase separated, and flocculated dispersions initiated with partial funding from several other sources. While not attaining the predictive level initially anticipated, the effort has furthered the qualitative understanding connecting interparticle forces to specific rheological responses.

The work has the following components:

1. The development of a non-equilibrium statistical mechanics approach for stable dispersions provides the framework for quantitative predictions of the low shear viscosity and linear viscoelastic properties as functions of the particle size and volume fraction and the interparticle potential. Though potentially powerful, the approach is currently limited by the lack of a tractable approximation for many-body hydrodynamic interactions.
2. The effective medium and self-consistent field approximations for the elasticity and plasticity, respectively, of flocculated suspensions capture the sensitivity to volume fraction due to the network structure and the breakup of the network into floes under shear. The predictions should be useful in correlating data, but are not yet fully predictive.
3. The correlation of data for model systems with well characterized particle sizes and interaction potentials demonstrates the value the qualitative understanding gained from fundamental approaches even when not completely predictive.

Executive Summary

The attrition of particles of known properties has been studied in a number of pieces of testing equipment. The novel and most flexible of these, a cone cell, has permitted the breakage of particles in gaps of known width to be studied in a rational manner for the first time. It is found that a close-sized feed can suffer high breakage rates at two gap openings, with a minimum rate between these at an intermediate gap size. Negligible breakage rates are found for low, less than one half a particle diameter, and for high gap widths, say exceeding three particle diameters. For mixed particle feeds, segregation of material into the gap is of crucial significance. These findings are of broad relevance to a wide range of solids processing equipment.

Studies on an annular cell, used to simulate breakage in a failure zone within the bulk of a material, showed that the Gwyn kinetics applied over three orders of magnitude of stress. It was shown by use of a well characterised and specially prepared extrudate that there was a shift from bodily failure at high stresses to surface abrasion at low stresses. These extrudates could also be formed into particles of different shapes; it was found that one of the Gwyn parameters was independent of shape and size whereas the other was dependent upon both.

The potential to modify the annular cell to establish links with the results of the cone cell was demonstrated but much remains to be done to cement these. Nonetheless, a complete understanding of attrition due to mechanical means now seems to be within our grasp. An aim of future work is to form estimates of product size distribution from a knowledge of the breakage of single particles under a number of loading conditions and an appreciation of packing theory.

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.

EXECUTIVE SUMMARY

This report brings to completion our work on the kinetics and structure of floes grown in shear flows at dilute concentrations. The first phase of the work consisted of detailed measurements of floe structure and growth kinetics for rapid, irreversible flocculation in a simple shear flow with minimal effect of Brownian motion. We employed polystyrene latices of 0.1um diameter in glycerol-water mixtures at 1.0 M NaCl at a volume fraction of lo4 with dynamic light scattering detecting the hydrodynamic radius and static light scattering probing the internal structure of the floes.

Comparison of the results for sheared dispersions with data for Brownian flocculation revealed a similar structure, i.e. floes having characteristics of fractals with dimension d=1.8+0.1 and an equal number of nearest neighbors. Of course, the kinetics differ substantially with shear accelerating the rate in proportion to the Peclet number, which gauges the ratio of shear to Brownian collisions. As the floe size approached lpm, however, the growth rate decreased significantly, suggesting that viscous forces impose a maximum for these tenuous structures. Straightforward calculations -- assuming Smoluchowski kinetics with weak hydrodynamic interactions, adhesion of particles upon contact, and a maximum size estimated by comparing the dispersion attraction to the viscous force -- reproduced the data within the experimental uncertainty.

In the second phase we addressed the evolution of the structure, seeking to understand the similarity between the results for the shear and Brownian modes. This involved hierarchical simulations performed by combining N particles into N/2 doublets, colliding those doublets to form quadruplets, etc., until only a single N-particle floe remained. At each step two aggregates at randomly chosen initial positions and orientations were translated along streamlines of the undisturbed velocity field and rotated with the local vorticity until two particles made contact. There the particles were assumed to stick, forming rigid bonds. The structure was characterized statistically through particle-particle correlation functions within the floes, the variation of the radius of gyration with number of particles, the asymmetry of the shape, and the corresponding light scattering spectrum.

Remarkably, the simulations produce floes with light scattering spectra indistinguishable from those studied experimentally and with no detectable difference between Brownian, shear, and extensional collisions processes. Hence, we conclude that irreversible flocculation, with no subsequent rearrangement or breakup, generates fractals of low dimension (d= 1.8) essentially independent of the kinematics of the collision process. A corollary is that creating compact, uniform floes with d=3.0 must require substantial rearrangement and/or breakup, processes that probably depend on many collisions. Thus future work along these lines must deal with concentrated dispersions and longer shearing times than examined here.

EXECUTIVE SUMMARY

Agglomeration is an inherent problem in almost all industrially relevant powder processing techniques. Successful dispersion of powders often requires the total elimination of agglomerates. To achieve this, it is important to understand the nature of, and to ascertain the properties of, these agglomerates. Of particular importance is the strength of the interparticle bonds, (between primary particle units forming the agglomerate), in relation to the powder processing procedure used for their generation.

In this IFPRI-funded project, a characterization technique has been developed to quantitatively determine agglomerate strength distributions in bulk powders by exposing a sample dispersion to a calibrated ultrasonic field and following the ensuing changes via particle size analysis. This technique may be used throughout a powder processing procedure to determine the source (onset) of agglomeration, as well as an aid in the elimination of these agglomerates by helping the user to understand the nature of the interparticle bonds.

The primary advantage of this technique (over other agglomerate strength determination techniques) is that it classifies agglomerate strength with a distribution rather than a single value, giving the user a more complete understanding of the system under consideration, as single value agglomerate strengths describe only the magnitude of agglomeration and not the extent. Furthermore, this is a ‘wet’ testing technique that allows agglomerate strengths to be determined under conditions similar to process conditions, which can be of great significance as powders often display different physical and/or chemical properties in and out of a liquid. Also, the simplicity of the testing procedure lends itself to automated, in-situ analysis (currently under development), whereby agglomeration in a system can be continuously monitored. Thus changes in the extent/magnitude of agglomeration (in response to modifications in the processing procedure) can be observed, and also monitored in a time dependent fashion.

The validity of this approach has been demonstrated for model silica agglomerates of known bond strength distribution and primary particle size prepared by heat treating ordered, sub-micron silica spheres at various temperatures. The use of this approach to find at what step agglomerates form in industrial processing is illustrated by measuring agglomerate strength during various titania processing schemes. As an example, the type of washing (ethanol versus water) and the drying conditions (temperature) caused large differences in the quantity and strength of hard agglomerates formed from the same titania precursor powder.

A numerical scheme has been developed for analyzing the change in the particle size distribution of a powder dispersion during ultrasonic breakdown and determining the actual mechanism of particle breakdown (erosion vs. fracture). This type of analysis also allows a specific particle size group (and thus particle strength group) to be followed throughout the degradation process. When applied to industrial processes, this approach should provide design information to minimize agglomerate formation during processing and for selecting successful dispersion and size reduction strategies.

EXECUTIVE SUMMARY

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. 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. This work has been performed for both two-dimensional simulations of disc flows and three dimensional simulations of spheres. Large portions of this project involved the development of soft particle simulation models that can be applied to this, as well as other, studies.

The results indicate that this is a problem that runs the gamut of granular flow regimes, from the molecular like, rapid flow regime, to the slow quasistatic regime. As such, it covers the transition between the two limiting states, an area that has not been tackled theoretically. It was observed that, while a region demonstrating molecular-like behavior may exist, the transition to solid behavior is not, as was originally hoped, an analog to a phase change in real molecular systems. Instead, the first movement of the material occurs as a quasistatic yielding. The results are beginning to shed some light on the transition process. The main difference between macroscopic particles and molecules is that particles can sustain long duration contact with their neighbors (this is what makes quasistatic behavior possible) and this permits the material to push-through the phase change.

The scope of this project has been extended beyond its original proposal to include more general problems of the computer simulation of powder flows. 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 used to make 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 force. 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 sponsored work on the flow induced mixing of particle in his granular shear cells. He had observed that the mixing might be modeled as a diffusion process, similar to that of molecule 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 tear 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 was an example of Taylor diffusion by which the diffusion of particles in the direction of the velocity gradient greatly enhanced their mixing.

EXECUTIVE SUMMARY

The long term aim of the IFPRI projects on suspension flow remains: “to be able to predict and manipulate the flow behaviour of suspensions. Within this framework, the general objective of this project is the development of a qualitative insight and possibly quantitative predictions for the rheological behaviour of stable colloidal suspensions.

The present project is the final stage of a series of IFPRI projects which dealt with the same objectives. The general approach has been to generate experimental results on well defined model systems in order to identify and possibly quantify the contributions from various phenomena to the rheology of stable colloidal suspensions. Specifically attention has been concentrated on three aspects, for which no satisfactory solution was found in the previous work:

• the effect of particle interaction forces (the repulsion forces which keep the particles from aggregating);
• the effect of particle size distribution;
• the onset of shear thickening, i.e. the limiting conditions beyond which flow becomes practically impossible.

Systematic measurements were performed on model colloidal systems in steady state and oscillatory shear flow. These systems consisted of a hard core on which a soft polymeric layer was grafted to keep the particles from aggregating. For monodisperse systems of spherical particles general expressions can be found for:

• the concentration dependence of the low and high shear Newtonian viscosities;
• the shape of the viscosity shear rate curve.

The concentration dependence uses the maximum packing as the only empirical parameter. This packing fraction depends on the deformability of the stabilizer layer. It has been shown that this deformability can be related to the interparticle repulsion caused by the stabilizer layer. The repulsion potential can be derived from measurements of the storage moduli in oscillatory flow. From the potential an effective hard sphere radius can be calculated which can then be used to obtain the viscosities from the known results for suspensions of Brownian hard spheres.

It could also be shown that the shear thinning region shifts inversely proportional to the zero shear viscosity. The concentration dependence of the latter is closely related to that of the mobility and the diffusivity of the particles. For that reason it is also found to correlate with the relaxation times derived from oscillatory experiments. The various data reduction schemes and scaling principles which are now available make it possible to characterize a wide range of suspensions with a limited number of experiments. In addition the rheology can be related to colloidal properties of the systems under investigation.

Qualitative insight is provided for the possible effect of slight deviations from monodispersity. In order to understand large degrees of polydispersity, bimodal distributions containing large and small colloidal particles were prepared. A procedure is suggested to calculate the low and high shear Newtonian viscosities. It is based on the computation of an equivalent maximum packing which takes into account the polydispersity as well as absolute particle size effects caused by stabilizer deformability and Brownian motion. The procedure describes the available measurements well.

At high volume fractions the viscosity starts to rise at a given shear rate (shear thickening), limiting the application conditions for such materials. Two different kinds of shear thickening could be distinguished, respectively showing a gradual and a sudden increase in viscosity. From rheological and rheooptical measurements it is concluded that the sudden increase in viscosity is associated with the formation of large, hydrodynamic aggregates. Particle inertia, as expressed in the particle Reynolds number, does not provide a suitable scaling for shear thickening. The effect of volume fraction on the onset of this phenomenon is similar to that for the fluidity (inverse of viscosity). A reduction in particle size shifts the onset to higher shear rates.

In conclusion it can be said a consistent picture has been obtained of the rheology of polymerically stabilized colloidal suspensions. The available empirical relations and scaling principles make it possible to correlate various experiments and to predict viscosities of various materials from a small number of experiments. The correlation with colloidal properties provides the necessary insight in the contributions from various phenomena and provides a basis for a rational manipulation of the suspension rheology and for their formulation.

Executive Summary

Agglomeration or granulation, as the name implies, is a process by which larger (millimeter or fractions of millimeter in diameter) granules are produced from fine (micron sized) powders in a mechanical agitator such as a drum, pen or high shear mixer or in a fluidized bed. Particle growth in these devices is facilitated by the use of a binder, i.e., a sticky fluid, a solution or a melt which upon dispersion in the powder mass and subsequent solidification at the interstices between particles generates stable granules. While powder size increase (agglomeration or granulation) is a widely used unit operation, few underlying physical principles describing the phenomena have been drawn. Successful granulation operation is therefore a largely haphazard undertaking. The present research attempts to lay a rational foundation to describe the mechanics of granulation by examining the process at the level of particle-binder-particle interaction, at the so-called microlevel.

The ultimate goal of the present work was to build a granulation model to predict granule size and growth rates from first principles using the properties of the powder, the binder and the characteristics of the mixer. It was discovered early in the project that liquid (binder) bridges formed between moving solid particles are the key to understanding of the many different processes taking place during granulation. It was also found that the study of these bridges, although attempted as far as their behavior with regard to surface tension effects is concerned, is not sufficiently developed and hence, a basic study of viscous effects in moving liquid bridges was undertaken. Furthermore, the phenomenon of particle coalescence and growth was studied using the theory of viscous liquid bridges developed earlier and regimes of granulation were defined in which both the growth rate and the limiting particle (granule) size were calculated. Three such regimes were identified, each characterized by a different dependence of the growth rate on such parameters as particle size, binder viscosity and surface tension and other parameters of lesser importance. All these quantities were incorporated in a dimensionless so-called Stokes (or Reynolds) number, characteristic values of which in turn delimit the different regimes.

Finally, the existence of the different growth and granule consolidation regimes was tested by experiments specially designed to isolate the important phenomena in question for each regime. As it clearly appears from the present work, the theory of coalescence regimes as presented above is only a framework which provides us with some basic insight into the phenomena of granule growth and consolidation but is not in fact a comprehensive model of granulation (although an attempt was made, to incorporate the above findings into a theory of defluidization of fluidized beds). One of the major practical achievements of the present work was the development of an instrument to characterize binders used in granulation and to measure binder strengthening times. These measured characteristics were then used during pilot scale fluid bed and drum granulation experiments to predict limiting granule diameters.

The present work did not provide a final solution for granulation theory but rather opened the field, presented a preliminary general framework and established the important lines of inquiry to be followed in the future. First and most important, is the measurement and/or prediction of shear forces in a mixer. This is essential since the knowledge of these forces is key to developing a comprehensive granule growth model by equating the disruptive and cohesive forces in the device. Introduction of particle and granule breakage into the overall theory through fracture mechanics is also paramount especially in such devices in which some drying of the granules occurs such as a fluid bed granulator or where the disruptive forces are very high such as in high shear mixers.

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 100~ 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. This understanding must be based on equations of motion for the gas and the particles, so the object of the present work has been to propose such equations of motion and explore their solutions for flow through ducts. These solutions appear to simulate many of the characteristic observed properties of flows of this sort, including the undesirable recirculation patterns referred to above. However, they are unduly sensitive to the values of certain physical properties of the gas and the particles, indicating that turbulent flows must be considered to give a satisfactory account of the situation of greatest technical importance, where suspensions flow at high rates through large ducts. The modelling of turbulent flow of a suspension is difficult, but a start in this direction has been made.

EXECUTIVE SUMMARY

We have developed a model of impact attrition of particulate solids with a semi-brittle failure mode. Observations of the impact damage by high speed photography show that attrition is caused by platelets chipping off from faces adjacent to the impact site. Detailed examinations by optical as well as scanning microscopy, and more recently by confocal laser scanning microscopy show that the platelets are produced by propagation of sub-surface lateral cracks. Therefore, the analysis of impact attrition is based on the fracture mechanics of sub-surface lateral cracks. A dimensionless parameter, representing the volume fraction of materials lost from a single particle by the formation of such cracks, is derived:

where p is the density, 1 is the linear dimension, v is the velocity, H is the hardness, Kc is the critical stress intensity factor, and 4 is the constraint factor given by the ratio of the hardness to the plastic yield stress. This parameter quantifies the attrition propensity, and it includes all the relevant material properties. The fractional loss per impact is considered to be function of Y/. In the first instance the existence of a simple linear relationship has been explored.

A series of tests on a number of model materials were carried out to verify the theoretical predictions, in particular for the effect of velocity, particle size, and material properties. It is shown that, overall as a first attempt, the trend of the data agrees reasonably well with the theory. However, there are intricacies in the experimental data, the description of which requires further refinement of the theoretical model.