ARR - Annual Report

Executive Summary

A first order model of the dynamics of pyrogenous fumes, based upon experimental and theoretical studies of the kinetics of particle growth and the structural rearrangements that occur following coagulation, suggests that the growth of nonagglomerated particles in aerosol reactors is best accomplished by dropping the temperature rapidly to quench coalescence before significant agglomeration occurs. Because of their larger collision cross sections, agglomerate particles coagulate more rapidly than do spheres of equal mass. As a result, once agglomerates begin to form, particle growth accelerates dramatically. Although agglomerate particles can be densified, the temperature would have to be increased significantly to do this, instead of decreasing continuously as occurs in most practical reactors.

Notably, the model predictions suggest that, for a given cooling rate, a high initial temperature will more effectively limit neck growth than will a lower one. The rate of decrease of the coalescence rate following the onset of agglomeration depends on the cooling rate and on the initial growth temperature. If the initial operating temperature is low enough that the coalescence time is comparable to the coagulation time, the coalescence time will increase only slightly faster than that for coagulation. Strong neck formation and hard agglomerates can then be expected. On the other hand, if the initial operating temperature is much higher so that coalescence is initially very rapid, the transition will occur much more abruptly. Neck growth within the first agglomerates to form will be reduced, and agglomerates will be more amenable to dispersion. These inferences require experimental testing. The proposed continuation of this project will focus on experimental definition of the bound between dense particle growth and agglomerate formation, and on quantifying the quench rate that is required to inhibit the formation of sintered agglomerates.

The classical model of neck development during sintering has been extended beyond the early stage of neck growth. Neck growth predicted using our model deviates significantly from that of the early-stage sintering model, raising serious questions about efforts to model the evolution of particle structure with sintering rates based on the classical model. Although this means that the model developed in this report can only be assumed to provide a qualitative picture of the transition, importance of the coupling of coagulation with coalescence is clear. Attempts at quantitative prediction of the structural evolution of pyrogenous fumes based upon the classical sintering time scales are, however, premature. Trace oxygen contamination of the silicon we have studied led to anomalous sintering behavior. Oxidation of the silicon surface would be expected to dramatically reduce the rate of surface diffusion. Neck growth predictions based upon the transport properties of pure silicon, but with surface diffusion eliminated, agree well with experimental observations. Thus, future work on the structural evolution of pyrogenous fumes must take the reaction atmosphere into account.

This report focuses on the theoretical interpretation of the results of our experimental program, and on defining the direction for future work. Experimental investigations aimed at obtaining quantitative measurements of the sintering of model agglomerates and at validating the models of the coagulation kinetics of agglomerate particles are near completion, and analysis of those experimental results is underway.

The work performed by our group during the previous IFPRI contract period focussed on fully developed, turbulent flow of gas-particle mixtures in vertical pipes, A K-E model for the fully developed flow was formulated and used to explain the mechanism responsible for the nonuniform distribution of particles over the pipe cross section (please refer to last year’s annual report, FRR 09-15, for a detailed description). In the present contract period is work will be extended to developing flow problems in an effort to understand (1994-97), th’ how the flow evolves in risers and standpipes, and get an appreciation for the length scales associated with various regimes of flow development.

We first derive the time-smoothed equations for a 2D, developing, turbulent gas- particle flow. The coordinate system used for this purpose is shown in Figure 1: x and y denote the axial and transverse coordinates, respectively. A Cartesian coordinate system is used rather than a cylindrical one so that asymmetric inlet conditions can eventually be analyzed. The vertically upward flow in a slit (Figure la) should have characteristics similar to those of flow in a riser so, henceforth, this geometry will be referred to as riser flow. By a similar token, the flow corresponding to Figure lb will be called a standpipe flow. The equations take the form of eight coupled partial differential equations ‘representing two mass balances, four momentum balances (two axial, two transverse) and balances for turbulent kinetic energy and its dissipation rate per unit volume of suspension. As in the case of the fully developed flow equations derived in FRR 09-15, separate transport equations for K and E are not required for the two phases, since, for the solid mass fluxes being considered, the inertia associated with the gas is negligible when compared to that of the solids. A full fledged analysis of developing flow in vertical ducts on the basis of this K-E model, taking into account the entrance and exit effects and the possibility of internal recirculation of particles and gas, is a very large-scale computational problem.

The axial development of the flow of gas-particle suspension in vertical ducts is characterized by a number of distinct zones. The pariticles are initially accelerated by a high-velocity gas stream (acceleration zone), where large changes in the particle concentration, pressure gradient, etc. take place over very small axial distances. Accurate numerical computations in this zone will necessarily require a very fibe axial mesh. Following the acceleration, the particle concentration and velocity fields slowly evolve to their fully developed states over a relatively larger distances (transition zone). This is followed by a fully-developed zone, after which is an end zone where the flow patterns change again to conform to the exit geometry. If the riser is not “sufficiently tall,” the fully developed zone may be absent and exit and entrance zones will interact strongly. A general purpose computational code that allows for all these regions will require large computer resources. It is therefore of interest to identify simplifications which will make the problem computationally more tractable.

In the present annual report, we have derived the time-averaged equations of motion for steady developing flow and simplified the resulting system of equations through a scaling 1 analysis (see Section 2). This analysis led to a system of equations containing only the first derivatives of dependent variables in the axial (vertical) direction, and the first and second derivatives in the transverse direction, This form allows us to view the developing flow problem as an initial value problem (as opposed to a boundary value problem) and compute the solution by marching form the inlet to the exit. It should be noted that the existence of a solution for such an initial value problem is not guaranteed. For example, when the flow develops an internal recirculation, such an approach will necessarily fail; however, in case no recirculation develops, this approach will yield the entire solution. Nevertheless, it is useful to solve the initial value problem and get a feel for the developing flow, largely because of the tremendous computational advantages it offers over the boundary value problem. With this in mind, we performed a number of calculations and these are described in Sections 3 and 4.

Section 3

which is devoted to a laminar flow model, that neglects the effect of tur-bulent fluctuations, shows that the entrance length in two-phase flow is considerably larger than that in a comparable single-phase flow, and this is a consequence of the particle seg- regation in the acceleration zone.

Section 4

results based on laminar and turbulent flow models are compared to demonstrate that the initial segregation of particles to the tube wall in the acceleration zone in purely a continuity effect, and the turbulent fluctuations, which come into play only in the transition zone, are solely responsible for causing segregation of particles in fully developed flow The main findings of the study are summarized in Section 5.

Abstract

Data from a range of filtration experiments on dilute particle fluid mixtures are used to determine the parameters that describe the physics of suspension flow in compaction. The range of solids volume fractions used is 0.00001<~0.1; <-potentials vary between O-50 mV. The relevant physical data are extracted from an analysis of the initial stages of a range of experiments at various - and - Theoretical considerations on suspension flow are presented to argue that the physical character of the flow at relatively dense, strongly interacting conditions is significantly different than that of quite dilute systems. The latter are dominated by fluctuations in the particle velocity near the septum to give gas-type diffusive btthaviour, while in the former the particles are more or less locahzed This observation has implications for the diffusion coefficient, which is predicted to behave quadratic in the filtration pressure for very dilute media and which is roughly independent of this quantity for mixtures containing strongly interacting particles. Experiments are described and analyzed to establish this behaviour and the experimental trends that are obtained bear out the main theoretical insights.

The general aim of the work is to elucidate the mechanisms of attrition of particulate solids. The specific objective of the current work is to investigate various types of damage under impact and sliding conditions. In particular, the transition velocities involved in impact breakage, the relative importance of normal and tangential stresses, the size distribution of the impact product and the effect of load and displacement on the material removal in surface wear have been investigated.

High-speed digital video recording was used to observe fracture patterns of a range of materials with diverse properties and structures as a function of impact velocity. The video recordings clearly show the existence of three identifiable velocity ranges where materials exhibit plastic deformation only, chipping, and a combination of chipping and fragmentation. This information is essential for developing realistic models of particle breakage.

Single particle impact tests of porous silica particles were carried out to investigate the dependence of attrition rate on impact velocity and impact angle. There is a significant increase of the attrition rate with impact velocity, with a maximum level of approximately 4.5% at 20 m s-l, for the size range 2.00-2.36 mm. The attrition behaviour of the samples is relatively insensitive to the impact angle in the range 25”-65’. However, preliminary impact tests with silica particles of different shape and porosities show that there is a dependence of the attrition rate on impact angle, depending on particle structure. Further work is on-going in this area.

A full size analysis of the impact products of PMMA and porous silica particles after a single impact was carried out in order to investigate the change of the size distribution with impact velocity. The Gates-Gaudin-Schumann distribution describes very well the size distributions of the mother particles and debris. The power index of the distribution is nearly constant for PMMA, but it varies significantly for porous silica. The cause of this variation is currently under investigation.

Single particle wear tests were also performed with the aim of elucidating the mechanisms of particle failure under sliding conditions. This occurs by abrasive wear at relatively low loads and long sliding distances, by chipping at slightly higher loads and short sliding distances, and by fragmentation at high loads. The data in the abrasive-wear regime of silica particles with high porosity, and hence low strength, show that the material loss is linearly proportional to the sliding distance and hence corroborate the predictions of the model developed by Ghadiri et al. (1995) for the semi-brittle failure mode. However, the wear test method developed here can provide quantitative information only about the abrasive wear rate, and the impact testing method has to be used for investigating the chipping rate of particulate solids.

Our objective is a robust and fundamental theory to predict the structure and dynamics of concentrated colloidal dispersions, including the shear viscosity, linear viscoelastic properties, and self diffusion coefficients. To achieve such the approach must handle three-body couplings that arise with pairwise additive interparticle potentials and many-body hydrodynamics. Our treatment is based on the classical configuration space, or Smoluchoski, approach which comprises a rigorous description of dynamics on the diffusion time scale. The couplings through the interparticle potential are approximated via nonequilibrium closures based on diagrammatic expansions and analogous to well-established equilibrium closures. Hydrodynamic interactions are embedded in mean-field approximations that interpolate between physically valid near- and far- field limits, incorporating results for the short-time self-diffusion coefficient and the high frequency limiting dynamic viscosity, In particular, the theory calculates the non-equilibrium structure from a two-particle Smoluchoski equation with a hypemetted chain closure to account for three-body couplings through a pairwise additive potential and three different mean-field approximations for the hydrodynamics. Substitution of the structure into conventional expressions for the stresses and fluxes determines the transport coefficients.

The accomplishments to date include

• extensive comparison of predictions without hydrodynamic interactions with results for the low shear viscosity and long-time self-diffusion coefficient from Brownian dynamics simulations for soft spheres;
• comparison of predictions with hydrodynamic interactions with experimental data for the low shear viscosity, nonequilibrium structure, long-time self-diffusion coefficient, and high frequency shear modulus for hard spheres;
• calculations of the linear viscoelastic spectra of concentrated dispersions of hard and soft spheres with and without hydrodynamic interactions; and
• calculations for shear thinning in the dilute limit for hard spheres. The accumulated results show quantitative or, at least, semi-quantitative agreement with data and simulations, suggesting success for the mean-field hydrodynamics but a tendency of the thermodynamic closure to over-estimate the total interparticle force attempting to restore equilibrium at high concentrations. The sensitivity of the response of hard spheres to the magnitude of the perturbation near contact implies that improvements in the hydrodynamics and the thermodynamic closures must be closely coupled. This work comprises the PhD dissertation which Robert A. Lionberger defended in December and is reported in this and pas1 Annual Reports as well as the papers listed below.

In addition we have either completed or made significant progress toward

• extracting a more “user friendly”, approximate form of the theory that distributes the contribution from the thermodynamic closure between diffusion and interparticle force terms,
• developing analogous approximations for the hydrodynamics in the presence of grafted polymer layers and short range attractions,
• implementing Monte Carlo simulations to generate accurate equilibrium structures as the input for calculations with more complex pair potentials, and
• calculating the non-equilibrium structure, long-time self-diffusion coefficient, low shear viscosity, and shear modulus for polymerically stabilized spheres. These objectives are being pursued by Stacey L. Elliott, with initial results to be reported at the International Congress on Rheology in August and in the next Annual Report, Our further goals of calculating the non-equilibrium structure, long-time self-diffusion coefficient, low shear viscosity, and shear modulus for adhesive hard spheres and addressing polydispersity through selected calculations for binary mixtures and appropriate pre-averaging (over the size distribution) of the conservation equation governing the non-equilibrium structure will be undertaken after establishing a simpler approximate form of the theory.

This and the 1994 Progress Report focus on comparisons of the predictions with experimental data from hard sphere colloidal dispersions, which reveal the following: finite ------- with the lubrication approximation and --------- with the discontinuous approximation, with both in agreement with definitive sets of experimental data, ------ with either the lubrication or discontinuous approximation that conform within 20-30s with the body of experimental data and qualitatively captures the divergence as ----------- with the lubrication or discontinuous approximations that conform to the body of data for 4 I 0.45 and qualitatively captures the zero as ---------- non-equilibrium static structure factors, i.e. Fourier transforms of the non-equilibrium structure, for weak steady shear that exhibit the proper dependence on wave number but too small a magnitude with the lubrication or discontinuous approximations, and stress-optical coefficients in the low shear limit with the proper dependence on volume fraction and roughly the right magnitude but a size dependence inconsistent with the only set of data.

Underprediction of the non-equilibrium structure for hard spheres with hydrodynamic interactions and the low shear viscosities for soft spheres without hydrodynamic interactions suggests that the HNC closure over-estimates the interparticle forces driving the structure towards equilibrium. If corrected, the viscosities/self-diffusion coefficients predicted for hard spheres with the discontinuous and lubrication approximations for the hydrodynamic interactions would undoubtedly be too large/small. However, the physically more reasonable interpolation constructed with the ADA then might prove more accurate. The predictions of the high frequency limiting viscosities lend considerable credibility to the simple, mean-field hydrodynamic approximations. None the less, deficiencies in the hydrodynamic models undoubtedly still exist, e.g. direct interactions with a third particle neglected in the conservation equation itself and the approximation of the conditionally averaged divergence of the relative velocity in the Brownian stress. These neglected terms could be estimated through Stokesian dynamics simulations.

Though imperfect the theory clearly provides robust semi-quantitative predictions for the structure and dynamics of concentrated dispersions with a variety of repulsive interparticle potentials. At present the only alternative is Brady’s approach, Though based on more serious, ad hat approximations, his theory is appealingly simple and, thus far, provides comparably accurate predictions for the transport coefficients, despite erring qualitatively and quantitatively on the non-equilibrium structure. We proceed now with reasonable confidence that the current formulation and, perhaps, that of Brady’s should suffice for mechanistic studies involving complexities such as polydispersity, adsorbed or grafted polymer, or attractive interactions.

Publications R.A. Lionberger and W.B. Russel, “High frequency modulus of hard sphere colloids”, J. Rheology 38 1885 1908 ( 1994). R.A, Lionberger and W.B. Russel, “Effectiveness of closures for many-body forces in concentrated colloidal dispersions”, J. Chem. Phys, (submitted). R.A. Lionberger, Rheology, Structure, and Diffusion in Concentrated Colloidal Dispersions, PhD Dissertation, Department of Chemical Engineering, Princeton University, December 1995. A.A. Potanin and W.B. Russel, “Hydrodynamic interaction of particles with grafted polymer brushes and applications to rheology of colloidal dispersions”, Phys. Rev. E 52 730-7 (1995).

This report presents the work on cornmunition under the IFPRI grant since 1993. A literature review has already been produced together with a detailed presentation of the experimental methods used (see report AR 27.02). Complementary references can be obtained from the literature review on communition research at the university of Karlsruhe (report SAR 012-06).

The objective of this research is to advance understanding of the fundamentals of fragmentation behaviour and develop experimental techniques to predict communition behaviour from a universal test. An experimental rig has been built to reproduce single particle impacts on a target and study the influence of the material properties on breakdown in ultra-fme grinding. Another experimental rig has also been constructed to study the impact of two jets of particles, as in an air-jet mill. Further experiments concern grinding experiments in laboratory scale equipment in particular wet grinding with a stirred bead mill and dry grinding with an air jet mill. The develoment of methods for characterizing debris from fragmentation forms a link between these two experimental studies.

Part A

Part A of this report presents the results of the experiments performed with the single jet apparatus. Seven kinds of particles have been used showing different behaviour depending on the type material and the processing involved in its manufacture.

Part B

Part B gives the results of the study with the opposed jet rig. Three diffent powders have been used so far.

Part C

Part C presents the methods being developped for morphological analysis.

Part D

Part D describes a method developed for analysing fine grinding kinetics for determining breakage and selection functions.

Executive Summary

The primary project aims are to develop relationships which predict the wet massing behaviour of particulate solids granulated with binders by mechanical agitation, and to apply such findings in probing scale-up factors.

Work with model substrates: polymer binder systems has previously shown the critical estimate from surface free energy measurements, in determining the wet-massing and rheological character of the role of solid:liquid interfacial phenomena, particulate systems during granulation. This approach has been employed to predict the wet-massing behaviour of four representative powder substrates - two microcrystalline celluloses, calcium carbonate and griseofulvin - granulated with two aqueous polymer binders - polyvinylpyrrolidone and hydroxypropylmethylcellulose. The findings have been tested with a new mixer torque rheometer, which has been shown to provide data which can be directly related to the theological terms yield stress (T), kinematic viscosity (7) and degree of non-Newtonian rheological behaviour (n).

In general, the rheological behaviour of the various substrate:binder systems was consistent with the predictions made from surface free energy calculations, and followed similar patterns to those observed for model substrates. The spreading of substrate and binder components were critical factors in influencing the stability of the wet masses in the domains where, in industrial processes, many granules are prepared. Preliminary observations with mixed powder substrates suggest that topographical features of particulates also play an important role in determining rheological behaviour during granulation.

In the scale-up studies, a modified power number/Reynolds number relationship has been developed and successfully applied to large scale (up to 600L) mixer granulators. This approach has shown that, via measurements of wet mass rheology by mixer torque rheometry, a master surve for a specific formulation can be prepared using laboratory scale equipment which allows prediction of optimal granulation end-point conditions for large scale production equipment.

This report concerns new work using a computer simulation of solid fracture. The details of the simulation technique were described in previous reports and wiIl not be repeated in detail here. For this model, rigid elements are assembled into a simulated elastic solid by “gluing” the elements together with compliant boundaries. The joints ticture when the tensile strength of the glued joints is exceeded, permitting a crack to propagate across the simulated solid. The great value of a computer simulation is that literally everything is known about the simulated system and is accessible to the computer experimenter. In addition the simulation allows the independent variation of material properties such as Young’s modulus, Poisson’s ratio and work of fracture - flexibility, that in the laboratory, is limited to the properties of available test materials. Consequently, a computer simulation offers the abiity to make investigations with a detail that is unthinkable to duplicate experimentally. As such simulation techniques are valuable is areas such as comminution and attrition, for which the actual problem is so complicated and so many events happen in so short a time, that experiments are historically limited to performing post-mortems on the fragments.

The first topic to be addresses in this year’s report finishes an investigation started in last year’s report. There we identified two breakage mechanisms that are responsible for the 16nal crack patterns observed in impact breakage. The frrst mechanism (Mechanism I) can be attributed to the stresses that develop in unbroken particles. These stresses are oriented azimuthally about the contact point and produces a series of fanlike cracks issuing outwards from the contact point. This is the only type of breakage occurs between the time that the particle fist makes contact with the wall and the time that the contact force has brought the center of mass to a halt (which is also roughly the point of maximum compression at the contact point). This type of cracking will continue as the particle rebounds, but will also develop cracks that are oriented perpendicular to the fanlike cracks. Such cracking cannot be accounted for by the stresses that develop within an unbroken particle and must be attributed to some other mechanism (Mechanism II) that in turn must be a byproduct of the Mechanism I cracks that have already developed in the particle. Making full use of the powers of a computer simulation, we were able to determine that buckling of the Mechanism I fragments was ultimately responsible for the Mechanism II breakage. From the way that this investigation utilizes the abilities of a computer simulation to control the simulated system and thus reveal the bending stresses that bring about the Mechanism II breakage, this investigation is an unusually good example of the utility of a computer simulation for studying this type of problem.

For the next topic, we returned to the Ball Drop simulations that were created long ago as examples of the simulation in action. These simulations were based on the IFPFU supported Ultra-Fast Load Cell experiments performed at the University of Utah. These are approximate experiments related to ball-milling that are tractable (i.e. involve relatively few particles) by the simulation (i.e. involve relatively few particles). Essentially they are 2-D simulations of a single large grinding ball dropped onto a static bed of breakable particles. As a result the particles in the bed are broken and/or scattered away from the grinding ball as it falls. Simulations were performed for 4 different bed depths and 3 diierent frictions. Generally, the deeper the bed, the less the total amount of breakage. The results show also show the dual role that friction plays in the breakage process. On the one hand, the majority of the grinding ball’s energy is lost to friction so that increasing the friction increases this energy loss. On the other hand, the friction holds the bed together and the larger the friction, i the longer the bed stays in place for the grinding ball to do its work. It appears that this latter effect is the stronger of the two in that increasing the particles’ coefficients of surface friction greatly increases the amount of breakage; that extra energy for breakage appears to come from a reduced kinetic energy of the scattered fragments.

Fiiy, to help address the interest expressed in the breakage of porous particles, we have performed some simulations of the effect of internal defects in the form of circular holes. The simulations show that the holes have two effects. First of all, they concentrate stress internally and are thus become the source of many of the cracks that form. Secondly, they act as internal free surfaces and thus attract propagating cracks. As a result, if the internal holes are regularly spaced the particle will break with a large number of fragments with sizes of the order of the hole spacing. Consequently, it were possible to create particles containing such holes, it would be possible to gain some control on the particle sizes generated by impact (and, as the mechanisms are much the same, presumably by crushing as well).

Executive Summary

Although compaction has been the object of considerable study, questions remain about compaction. These questions exist due to the difficulties with obtaining in-situ information about powder rearrangement/breakage during compaction. In general, compaction work has primarily studied stress-strain curves. Depending upon the relative density and the change of density with pressure, one tries to infer the mechanism(s) of compaction.

In this work, we address the development of in-situ techniques to tlrobe comtlaction mechanisms. Instead of simply measuring density variation and attempting to infer compaction mechanisms, we employ scattering to study compaction. In order to demonstrate the utility of these in-situ techniques, we have:

1. performed preliminary experiments to demonstrate the sensitivity of the technique to a dilute second phase (sensitive to l-5% depending upon the powder size, morphology, and electron density),
2. used compaction with ex-situ scattering to demonstrate how compaction mechanisms can be directly observed and
3. employed fluids with electron density matched to that of the solid phase (or one powder if a mixture) to selectively study the second powder.

In one example, we studied changes of a silica compact as it was compacted isostatically or by drying capillary pressure. By assessing the variation in the scattering intensity associated with different length scales as well as the change in the hydraulic radius with pressure/density, pores around the agglomerates were observed to disappear due to agglomerate breakage and compaction. This result could not have been obtained by simply measuring the stress-strain of the compact as is normally done in compaction studies. For another example, dibromomethane was impreganted into a silica-titania mixture and scattering was performed. This shows that contrast matching and small-angle x-ray scattering may be employed to selectively “illuminate” a few hard aggiomerates of a second material type during compaction.

Optical rheometric methods are being used in this project to characterize the structure and dynamics of suspensions during flow. The methods being employed are primarily scattering methods: scattering dichroism and small angle light scattering. These are being applied to a wide range of problems that include:

1. determinations of the aspect ratio of colloidal particles,
2. flow induced structure in strongly flocculated suspensions,
3. orientation dynamics of a concentrated nematic suspension, and
4. the influence of interparticle interactions on the microrheology of dense suspensions.

The platform for these measurements is a newly developed facility based upon the principle of light transmission at oblique angles of incidence within a commercial stress rheometer.