Self-similar Particle Size Distribution Research Articles

Teaching old laws new tricks: Specific energy requirements for breakage based on modern kernels and a population balance model

A general expression for specific energy requirements in particle breakage is derived from a population balance model. The model employs power-law rate kernels, self-similar fragment distributions and an energy dependence based on the Vogel-Peukert master curve. This model connects the empirical constants and exponents of size in older empirical energy laws to the parameters of the population balance model, the nature of the particles, and certain moments of the self-similar particle size distribution predicted by the model. The resulting model is used to extract values of the order of the power-law rate kernel from literature data. The extracted order values are consistent with known expressions for individual particle breakage. The model developed here may be coupled with experimental determination of the material parameter needed in the Vogel-Peukert master curve to enable extraction of energy utilization efficiency values from experimentally determined values of the size-independent portion of the selection function.

An entropy-like parameter of particle size distributions as packing density index in complex granular media

Information entropy dimension (ED) has been used earlier in the characterization of the particle size distribution (PSD) in complex multi-particle granular media. In this work the ED is first proposed as an indicator of the packing density on the basis of the theoretical based interpretation of the ED in the granular media PSD context. A one-parametric exact self-similar PSD model, where the information ED is known, together with a 2-D computational random packing algorithm, are used to test the ability of the ED as an indicator of packing density. It is found that the packing density increases when the ED does. Moreover, results show a strong linear dependence between packing density and the ED. Empirical results from a large soil data base also reinforce the computational results. As ED may be estimated from field or laboratory data, the above mentioned results suggest its use as an indicator of packing density in complex granular media and material science in order to predict their properties.

Influence of non-linear breakage kinetics on the attainment of self-similarity for dry milling processes

The attainment of a self-similar particle size distribution (PSD) is an interesting feature of many particulate processes such as milling. While it is well-known that the linear time-invariant population balance model (LTINV-PBM) admits a self-similar solution, the attainment of self-similarity is not well-understood within the context of non-linear breakage kinetics. Non-linear breakage kinetics emanating from mechanical multi-particle interactions is not uncommon in dense-phase dry milling processes and has been accounted for by the non-linear PBM (NL-PBM). Here, we investigate the dynamics and attainment of a self-similar PSD considering first-order (LTINV-PBM), non-uniform non-linear (NL-PBM Model B), and uniform non-linear (NL-PBM Model D) breakage kinetics. The PBMs were first fit to experimental data from the batch dry milling of quartz in a tumbling ball mill. Using the parameters estimated and simulating the impact of different extents of non-linear multi-particle interactions, we elucidated the impact of non-linear breakage kinetics on the evolution of the PSD to a self-similar one. In order to quantitatively assess self-similarity and thoroughly investigate the approach to a self-similar PSD during milling, we have introduced the use of the distance metric Canberra Distance. The simulations suggest that the retardation of the breakage of coarser particles due to their interactions with the finer particles can significantly delay the attainment of self-similarity and broaden the self-similar PSD attained. Both non-linear models predict attainment of self-similar PSDs, albeit slightly different, under cases of mild non-linear effects. However, when non-linear effects are severe, self-similarity can be delayed to prolonged milling in contrast to LTINV-PBM, which predicts the attainment of self-similarity shortly after the onset of milling. While LTINV-PBM and NL-PBMs can be easily discriminated via a single milling experiment, simulations suggest that only experiments utilizing various feed particle size distributions, preferentially including a bi-modal distribution, could discriminate between the non-uniform non-linear kinetics (Model B) and uniform non-linear kinetics (Model D).

The equal-size binary breakage problem: Evolution toward a steady shape or periodic behavior?

During the last few years, the self-similar particle size distribution for a particle population undergoing breakage in equal size fragments has been derived using approximating, numerical, and analytical means. But very recently it was shown [N.V. Mantzaris, J. Phys. A Math. Gen. 38 (2005) 5111] through transient simulation of the breakage process that the particle size distribution in case of breakage in two equal fragments, never attains a steady shape, i.e., a self-similar form. The new results give rise to questions about the real meaning and utility of the previously derived self-similar distributions for these systems. The scope of the present work is to answer these questions and it is attempted using only analytical (exact) means for the solution of the transient breakage problem. In doing so, the very interesting and rich underlying structure and properties of the solutions of the equal size breakage problem (seemingly, very simple) are revealed. It appears that the utility of the known self-similar distributions for this particular problem has to be redefined but yet not entirely abandoned.

On the Evolution of Particle Size Distribution in Pipe Flow of Dispersions Undergoing Breakage

The problem of a flowing dispersion in a pipe with particles undergoing breakage and radial diffusion is considered in this note. This problem has been addressed in an earlier work [Kostoglou and Karabelas, AIChE J., 2004, 50, 1746] where it is solved analytically for the case of radially uniform parameters, while the case of turbulent flow (with radially dependent parameters) was treated by other researchers [Nere and Ramkrishna, Ind. Eng. Chem. Res. 2005, 44, 1187]. Here, simplified versions of the extended mathematical problem in the limits of small and large diffusion influence are presented. The conditions under which self-similar particle size distributions can be developed are determined. Finally, a closed-form solution for the total particle number concentration for linear breakage rate with radial dependence and radial uniform velocity and diffusivity is derived.

On the self-similar solution of fragmentation equation: Numerical evaluation with implications for the inverse problem

It is well known that the fragmentation equation admits self-similar solutions for evolving particle-size distributions (PSD); i.e., if the shape of PSD is independent of time after an initial transient period. Although an analytical derivations of the self-similar PSD cases have been studied extensively, results for cases requiring numerical solutions are rare. The aim of the present work is to fill this gap for the case of homogeneous breakage functions. The known analytical and approximate solutions for the self-similar PSD are reviewed and a general algorithm for the numerical solution is proposed. Results for a broad range of breakage functions (kernel and rate) are presented. Further, the work is focused on the sensitivity of the relation between self-similar PSD and breakage kernel and its influence on the inverse breakage problem, i.e., that of estimating the breakage kernel from experimental self-similar PSDs. Useful suggestions are made for tackling the inverse problem.

Improved permeate flux by flocculation of biological feeds: comparison between theory and experiment

Deposition of the rejected species on the membrane surface during tangential flow microfiltration is often observed. This cake layer can lead to lower permeate fluxes and partial rejection of dissolved solutes. Flocculation of the feed stream prior to microfiltration has been shown to increase the volume-based modal particle size which in turn leads to increased permeate fluxes. Relating the properties of the flocculated feed stream to the permeate flux is important if the effectiveness of a flocculant is to be assessed. Experimental observations reported here indicate that flocculation of yeast and CHO cell feed streams result in a self-similar particle size distribution. Further, though flocculation leads to an increase in the volume-based modal particle size, the number-based modal particle size decreases. Therefore after flocculation there is a large fraction of smaller flocs. Observations of deposition of flocculated feed streams during microfiltration indicate that these smaller particles preferentially deposit on the membrane surface. Consequently the expected permeate flux based on the volume-based average particle size after flocculation is significantly higher than observed. Further rearrangement of the deposited particles in the cake during microfiltration often leads to a reduced permeability of the cake resulting in lower permeate fluxes.

Modeling and simulation for phase coarsening: a comparison with experiment.

The phase coarsening of precipitates is modeled in the framework of Debye-Hückel theory. The interactions observed among a population of precipitates dispersed throughout a matrix can be described by diffusion screening. The relationship between the maximum particle radius and the volume fraction of the phases is established, and the rate of coarsening is related to the volume fraction and the self-similar particle size distribution. We simulated the dynamics of late-stage phase separation using multiparticle diffusion methods. Experimental measurements on the rates of coarsening of delta(') ( Al3 Li) precipitates in binary Al-Li alloys are compared with our results using modeling and simulation. The theoretically predicted particle size distributions and the maximum radius expected for particles in the microstructure agree well with recent experimental results.

Thermal analysis of powdered alumina materials

How a DSC result is influenced by the particle size distribution of a powder sample is shown, and a simple and optimal method to be included in a routine DSC analysis (e.g., purity determination) to improve the reliability of the analysis is proposed. In case ofα-Al2O3 powder, most reliable heat capacity data can be obtained by preparing a powder with a self-similar particle size distribution with a distribution constant of 0.7, and by compressing it under a pressure of 1.5 MPa for a duration of 5 min or longer.

Batch grinding simulation—population balance models and self-similar size distributions

It has been observed that batch grinding of ore samples for different grinding times will sometimes produce a series of products that have approximately self-similar size distributions. Furthermore, it has been claimed that self-similar particle size distributions cannot be generated by the population balance method (PBM) of simulating ore grinding processes. That claim is not valid. Several methods for generating self-similar size distributions to any degree of precision using PBM's are presented. Firstly it is shown that a single self-similar product size distribution can be generated from any set of breakage parameters by an appropriate choice of feed size distribution, and that with some restrictions on the breakage parameters multiple self-similar product size distributions can be generated. It is also shown that for any feed size distribution which is a straight line when plotted on log scales, it is possible to find breakage parameters which generate multiple self-similar product size distributions. A practical example is solved to illustrate the method.

Thermal analysis and self-similarity law in particle size distribution of powder samples. Part 1

Based on the self-similarity law (or fractal nature) in particle size distribution introduced recently by Ochiai et al. (Proc. Inst. Stat. Math., 38 (1990) 257) the DTA and TG curves for the decomposition reaction of dolomite are interpreted. The phenomenologically well-known Rosin-Rammler and Gaudin-Schuhmann functions are derived by a statistical approach, and the functions thus re-written (with a particle size scaled with the absolute size constant) obey a power law. The particles of a powder obtained by a mechanical size reduction can be seen therefore to be distributed in a self-similar manner. This also states that the majority of powders obtained by comminution have a distribution of a fractal nature. Dolomite samples composed of particles whose size distributions obey the self-similarity law all undergo a similar decomposition reaction as observed in the TG-DTA curves. That is, they all appear to react smoothly and swiftly. Those having narrow particle size distibution, and to which the self-similar particle size distribution does not apply, appear to undergo a much delayed reaction; they yield irregular DTA peaks accompanied by small additional peaks or shoulders. This indicates that powder samples of dolomite having a fractal particle size distribution undergo an apparently regular and favorable reaction, and that the TG-DTA curves, though not strictly qualitative, provide speedy and useful information on the powder characteristics.

Self-similar particle-size distributions during coagulation: theory and experimental verification

A quantitative theory for particle coagulation in continuous particle size distributions is presented and experimentally verified. The analysis, following Friedlander (1960a, b), assumes a local equilibrium in the size distribution maintained by a particle flux through the size distribution, Only particle collisions caused by Brownian motion, fluid shear and differences in settling velocities are considered. For intervals of particle size where only one coagulation mechanism is dominant, dimensional analysis predicts self-similar size distributions that contain only one dimensionless constant for each mechanism. Experiments were designed to test these predictions with clay particles in artificial seawater sheared in the gap between concentric rotating cylinders. Particle-size distributions measured over time were self-similar in shape and agreed with the Brownian- and shear-coagulation prediction in terms of shape and dependence on fluid shear rate and particle volume flux through the size distribution.