Particle size distribution theoretical predictions

The general form of the population balance including aggregation and rupture terms was solved numerically to model the experimental particle size distributions. While excellent agreement was obtained using semi-empirical two-particle aggregation and disruption models (see Figure 6.15), PSD predictions of theoretical models based on laminar and turbulent flow considerations... 

Particle Number Concentration and Size Distribution. The development of aerosol science to its present state has been directly tied to the available instrumentation. The introduction of the Aitken condensation nuclei counter in the late 1800s marks the beginning of aerosol science by the ability to measure number concentrations (4). Theoretical descriptions of the change in the number concentration by coagulation quickly followed. Particle size distribution measurements became possible when the cascade impactor was developed, and its development allowed the validation of predictions that could not previously be tested. The cascade impactor was originally introduced by May (5, 6), and a wide variety of impactors have since been used. Operated at atmospheric pressure and with jets fabricated by conventional machining, most impactors can only classify particles larger... 

Several publications in the literature address the particle size of the dmg substance and USP content uniformity from a theoretical and statistical basis. In 1972, Johnson2 established an equation that predicts the expected variation in a unit dose when the particle size distribution of dmg substance is analyzed. This theoretical calculation... 

Although the fly ash particle size distribution in the submicron regime is explained qualitatively by a vaporization/homogeneous nucleation mechanism, almost all of the available data indicate particles fewer in number and larger in size than predicted theoretically. Also, data on elemental size distributions in the submicron size mode are not consistent with the vapor-ization/condensation model. More nonvolatile refractory matrix elements such as A1 and Si are found in the submicron ash mode than predicted from a homogeneous nucleation mechanism. Additional research is needed to elucidate coal combustion aerosol formation mechanisms. 

Batch suspension reactors are, theoretically, the kinetic equivalent of water-cooled mass reactors. The major new problems are stabilization of the viscous polymer drops, prediction of particle size distribution, etc. Particle size distribution was found to be determined early in the polymerization by Hopff et al. (28, 29,40). Church and Shinnar (12) applied turbulence theory to explain the stabilization of suspension polymers by the combined action of protective colloids and turbulent flow forces. Suspension polymerization in a CSTR without coalescence is a prime example of the segregated CSTR treated by Tadmor and Biesenberger (51) and is discussed below. In a series of papers, Goldsmith and Amundson (23) and Luss and Amundson (39) studied the unique control and stability problems which arise from the existence of the two-phase reaction system. 

The originators of the Penn Kem system claim that it is capable of measuring particle size distributions in the size range 0.01 to 100 pm for slurry concentrations at volume concentrations as high as 50%. They report experimental work with an on-line system using titanium dioxide at volume concentrations from 3.5% to 42.3%. Quantitative comparison of data was carried out at eighteen frequencies and eleven concentrations by volume [248,249]. Theoretical work resulted in the development of a unified coupled phase model which successfully predicted the experimental data for suspensions, emulsions and aerosols [250]. 

With their DFT-based model for the number of active sites as a function of nanoparticle radius, the only experimental input Honkala et al. needed to compare their predictions with experiments was the particle size distribution of the experimental catalyst. The catalyst used in the experimental portion of this work was 0.2 g of an 11.1 wt% Ru/MgAl204 material. The particle size distribution was established by examining 1000 nanoparticles using TEM.35 With this information, Honkala et al. compared their DFT-based rate expression with experimental data over a range of operating conditions. It is fair to describe this comparison of theory and experiment as a first principles comparison, since no information from the catalyst under operating conditions was used to fit the theoretical data. Remarkably, the theory does an excellent job of predicting the ammonia reaction rate. The experimentally observed rate was underpredicted by a factor of 3 20.35... 

This chapter has presented a theoretical derivation of continuous particle size distributions for a coagulating and settling hydrosol. The assumptions required in the analysis are not overly severe and appear to hold true in oceanic waters with low biological productivity and in digested sewage sludge. Further support of this approach is the prediction of increased particle concentration at oceanic thermoclines, as has been observed. This analysis has possible applications to particle dynamics in more complex systems namely, estuaries and water and waste-water treatment processes. Experimental verification of the predicted size distribution is required, and the dimensionless coeflBcients must be evaluated before the theory can be applied quantitatively. 

Friedlander (11) has examined the effects of flocculation by Brownian diffusion and removal by sedimentation on the shape of the particle size distribution function as expressed by Equation 9. The examination is conceptual the predictions are consistent with some observations of atmospheric aerosols. For small particles, where flocculation by Brownian diffusion is predominant, p is predicted to be 2.5. For larger particles, where removal by settling occurs, p is predicted to be 4.75. Hunt (JO) has extended this analysis to include flocculation by fluid shear (velocity gradients) and by differential settling. For these processes, p is predicted to be 4 for flocculation by fluid shear and 4.5 when flocculation by differential settling predominates. These theoretical predictions are consistent with the range of values for p observed in aquatic systems. 

The effect of particles and matrix properties on the shear viscosity of LDPE/ GTR blend was also studied by developing a theoretical model to predict the viscosity of the composites as a function of the rheological properties of the matrix, solid concentration, particle size distribution, particle shape, and deformability (Bhattacharya and Sbarski 1998). The real viscosity measurements were found in good agreement with the values predicted below the maximum packing fraction. 

At the simplest level we use particle size measurements to monitor their concentration or to control the reproducibility of a product. Thus, we compare what we find with what we expect and if the two do not coincide we reject the product. The science of powder technology, however, is concerned to use the microscopic properties of the system, for example the particle size distribution, to interpret the bulk behaviour of the powder. If it is to be used in dilute circumstances, then the bulk behaviour can be derived by integrating the behaviour of the individual particles but usually this is not so and the relationship between the microscopic and macroscopic properties must take account of the particle interactions. By observing the difference in particle size distribution of samples which exhibit a different bulk behaviour, we begin to make a "correlation" between the two which, whether empirical or theoretical, quantitative or qualitative, involves interpretation of the mechanisms involved. Somewhere between these two purposes usually lies the purpose of a particle size measurement. There is, however, a far more ambitious level at which powder technology must eventually operate and, as yet, rarely does. That is to design the particles and the particle mixture to produce required properties, to use the relationships between microscopic and macroscopic properties in a predictive manner. It is the more rigorous use of particle size measurements which introduces the real diversity and which requires the measurements to be carefully matched to the problem. The increased diversity does not alter the basic needs which Heywood described. 

The imaging technique developed for the above work was compared with the use of Adobe Photoshop software to measure particle size and particle size distribution of the montmorillonite in the polymer composite. The Adobe Photoshop method calculated a smaller average length for the particles (more small particles were imaged by the Adobe Photoshop method). The calculated aspect ratios of both methods produced similar results and were commensurate with the experimental data and theoretical predictions. The results indicated that the efficiency in exfoliation of the organomontorillonite in nylon 6 was superior to that in nylon 6,6. The modulus values of the nylon 6,6-montmorillonite polymer nanocomposites were consistently inferior to those of nylon 6. The modulus values for nylon 6,6 were further compromised as a function of increased montmorillonite content in a nonlinear fashion. 

A characteristic feature of the size distribution function is the cut-off at > 1.5. The temporal changes in the particle size distribution of 1,2-dichloroethane-in-water emulsions were obtained by statistical analysis of the photomicrographs shown in Figure 9.1. The results are plotted as the size distribution function, and compared with the theoretical prediction based on LSW theory (Figure 9.4). Although small deviations from LSW theory were noted (i.e. the size distribution function broadens slightly), it is clear that the time-independent nature of the size... 

Relaxations in the double layers between two interacting particles can retard aggregation rates and cause them to be independent of particle size [101-103]. Discrepancies between theoretical predictions and experimental observations of heterocoagulation between polymer latices, silica particles, and ceria particles [104] have promptetl Mati-jevic and co-workers to propose that the charge on these particles may not be uniformly distributed over the surface [105, 106]. Similar behavior has been seen in the heterocoagulation of cationic and anionic polymer latices [107]. 

Lichti, G., R. G. Gilbert, and D. H. Napper, J. Polym. Sci. Polym. Chem. Ed., 18,1297 (1980) Theoretical Predictions of the Particle Size and Molecular Weight Distributions in Emulsion Polymerization, Chap. 3 in Emulsion Polymerization, I. Piirma, ed., Academic Press, New York, 1982. 

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