Particle sizes prediction

As shown in Fig. 8.26, the quadratic interactions improved the accuracy of the model for particle size prediction. For powder BD, the optimization study confirmed the suitability of a linear model (only function of Pout)-... 

No reliable model was obtained for particle size prediction (a desired outcome, considering the nature of this study), which is in agreement with the small response of Dv50 observed in Fig. 8.28 (between 44 and 48 p,m). For BD, however, it was still... 

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]. 

Altematively, tire polymer layers may overlap, which increases tire local polymer segment density, also resulting in a repulsive interaction. Particularly on close approach, r < d + L, a steep repulsion is predicted to occur. Wlren a relatively low molecular weight polymer is used, tire repulsive interactions are ratlier short-ranged (compared to tire particle size) and the particles display near hard-sphere behaviour (e.g., [11]). 

A combination of equation (C2.6.13), equation (C2.6.14), equation (C2.6.15), equation (C2.6.16), equation (C2.6.17), equation (C2.6.18) and equation (C2.6.19) tlien allows us to estimate how low the electrolyte concentration needs to be to provide kinetic stability for a desired lengtli of time. This tlieory successfully accounts for a number of observations on slowly aggregating systems, but two discrepancies are found (see, for instance, [33]). First, tire observed dependence of stability ratio on salt concentration tends to be much weaker tlian predicted. Second, tire variation of tire stability ratio witli particle size is not reproduced experimentally. Recently, however, it was reported that for model particles witli a low surface charge, where tire DL VO tlieory is expected to hold, tire aggregation kinetics do agree witli tire tlieoretical predictions (see [60], and references tlierein). 

At equilibrium, in order to achieve equality of chemical potentials, not only tire colloid but also tire polymer concentrations in tire different phases are different. We focus here on a theory tliat allows for tliis polymer partitioning [99]. Predictions for two polymer/colloid size ratios are shown in figure C2.6.10. A liquid phase is predicted to occur only when tire range of attractions is not too small compared to tire particle size, 5/a > 0.3. Under tliese conditions a phase behaviour is obtained tliat is similar to tliat of simple liquids, such as argon. Because of tire polymer partitioning, however, tliere is a tliree-phase triangle (ratlier tlian a triple point). For smaller polymer (narrower attractions), tire gas-liquid transition becomes metastable witli respect to tire fluid-crystal transition. These predictions were confinned experimentally [100]. The phase boundaries were predicted semi-quantitatively. 

Figure C2.17.12. Exciton energy shift witli particle size. The lowest exciton energy is measured by optical absorjDtion for a number of different CdSe nanocrystal samples, and plotted against tire mean nanocrystal radius. The mean particle radii have been detennined using eitlier small-angle x-ray scattering (open circles) or TEM (squares). The solid curve is tire predicted exciton energy from tire Bms fonnula.

A PVDF membrane filter has been shown to remove >10 particles of vims for vimses >50 nm independent of fluid type (8). Vimses smaller than 50 nm are not removed as efficientiy but are removed in a predictable manner which correlates to the vims particle size. The chemistry of the suspending fluid affects titer reduction for vimses <50 nm owing to other removal mechanisms, such as adsorption, coming into play. The effects of these other mechanisms can be minimized by using filtration conditions that minimize adsorption. 

Cyclone Efficiency. Most cyclone manufacturers provide grade-efficiency curves to predict overall collection efficiency of a dust stream in a particular cyclone. Many investigators have attempted to develop a generalized grade-efficiency curve for cyclones, eg, see (159). One problem is that a cyclone s efficiency is affected by its geometric design. Equation 15 was proposed to calculate the smallest particle size collectable in a cyclone with 100% efficiency (157). 

The term essentially a drag coefficient for the dust cake particles, should be a function of the median particle size and particle size distribution, the particle shape, and the packing density. Experimental data are the only reflable source for predicting cake resistance to flow. Bag filters are often selected for some desired maximum pressure drop (500—1750 Pa = 3.75-13 mm Hg) and the cleaning interval is then set to limit pressure drop to a chosen maximum value. 

The heat-transfer coefficient depends on particle size distribution, bed voidage, tube size, etc. Thus a universal correlation to predict heat-transfer coefficients is not available. However, the correlation of Andeen and Ghcksman (22) is adequate for approximate predictions ... 

Many attempts have been made to develop models which predict the behavior of materials undergoing size reduction. One proposal is that the energy expended in size reduction is proportional to the new surface formed (5). Another theory is that the energy required to produce a given reduction ratio (feed size product size) is constant, regardless of initial feed particle size (6). Practical results show, however, that both these theories are limited in their usehilness. 

The summation term is the mass broken into size interval / from all size intervals between j and /, and S is the mass broken from size internal i. Thus for a given feed material the product size distribution after a given time in a mill may be deterrnined. In practice however, both S and b are dependent on particle size, material, and the machine utilized. It is also expected that specific rate of breakage should decrease with decreasing particle size, and this is found to be tme. Such an approach has been shown to give reasonably accurate predictions when all conditions are known however, in practical appHcations severe limitations are met owing to inadequate data and scale-up uncertainties. Hence it is stiH the usual practice to carry out tests on equipment to be sure of predictions. 

The two steps in the removal of a particle from the Hquid phase by the filter medium are the transport of the suspended particle to the surface of the medium and interaction with the surface to form a bond strong enough to withstand the hydraulic stresses imposed on it by the passage of water over the surface. The transport step is influenced by such physical factors as concentration of the suspension, medium particle size, medium particle-size distribution, temperature, flow rate, and flow time. These parameters have been considered in various empirical relationships that help predict filter performance based on physical factors only (8,9). Attention has also been placed on the interaction between the particles and the filter surface. The mechanisms postulated are based on adsorption (qv) or specific chemical interactions (10). 

One manner in which size may be computed, for estimating purposes, is by employing a volumetric heat-transfer concept as used for rotary diyers. It it is assumed that contacting efficiency is in the same order as that provided by efficient lifters in a rotaiy dryer and that the velocity difference between gas and solids controls, Eq. (12-52) may be employed to estimate a volumetric heat-transfer coefficient. By assuming a duct diameter of 0.3 m (D) and a gas velocity of 23 m/s, if the solids velocity is taken as 80 percent of this speed, the velocity difference between the two would be 4.6 m/s. If the exit gas has a density of 1 kg/m, the relative mass flow rate of the gas G becomes 4.8 kg/(s m the volumetric heat-transfer coefficient is 2235 J/(m s K). This is not far different from many coefficients found in commercial installations however, it is usually not possible to predict accurately the acdual difference in velocity between gas and soRds. Furthermore, the coefficient is influenced by the sohds-to-gas loading and particle size, which control the total solids surface exposed to the gas. Therefore, the figure given is only an approximation. 

The dust is composed of fine particles but is highly flocculated or tends to flocculate in preceding equipment and in the cyclones themselves. Efficiencies predicted on the basis of ultimate particle size will be highly conservative. 

Fig. 24. The effect of particle size on the predicted tensile failure distribution of grade H-451 graphite.

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