Free volume particle

StoKes-Einstein and Free-Volume Theories The starting point for many correlations is the Stokes-Einstein equation. This equation is derived from continuum fluid mechanics and classical thermodynamics for the motion of large spherical particles in a liqmd. 

Limiting flow rates are hsted in Table 23-16. The residence times of the combined fluids are figured for 50 atm (735 psi), 400°C (752°F), and a fraction free volume between particles of 0.4. In a 20-m (66-ft) depth, accordingly, the contact times range from 6.9 to 960 s in commercial units. In pilot units the packing depth is reduced to make the contact times about the same. 

These two nomographs provide a convenient means of estimating the equivalent diameter of almost any type of particle Figure 1 of regular particles from their dimensions, and Figure 2 of irregular particles from fractional free volume, specific surface, and shape. 

Also, in cases where the dimensions of a regular particle vary throughout a bed of such particles or are not known, but where the fractional free volume and specific surface can be measured or calculated, the shape factor can be calculated and the equivalent diameter of the regular particle determined from Figure 2. 

Porosity constitutes a important criterion in a description based on straining. Porosity is determined by the formula V /Vc, in which V c is the total or apparent volume limitated by the filter wall and is the free volume between the particles. The porosity of a filter layer changes as a function of the operation time of the filters. The grains become thicker because of the adherence of material removed from the water, whether by straining or by some other fixative mechanism of particles on the filtering sand. Simultaneously the interstices between the grains diminish in size. This effect assists the filtration process, in particular for slow sand filters, where a deposit is formed as a skin or layer of slime that has settled on the... 

It was assumed that the volume V was reduced by the presence of other particles to the free volume V — Nvf) where N is the number of particles. In arriving at the binding energy effect the mean field approximation was used, which says that the soft (negative) part of the pair potential was sampled in an uncorrelated manner as if the system was an ideal gas. The corresponding free energy per particle in the bulk fluid is... 

The catalytic test of propane ODH reaction was performed in the 350-600°C range in a quartz fixed bed flow reactor with on line GC analysis. The free volume of the reactor after the catalyst bed was filled with quartz particles to minimize the homogeneous reactions. All the testing set was placed in a thermostat with heated lines to the gas chromatographs at about 100°C to prevent water condensation. The feed gas composition was C3H8/02/N2 = 20/10/70 vol.% at total gas flow 50 cm3 min-1. Catalyst fractions of 0.2-0.315 mm particle size and of 80 mg weight were loaded into the reactor. Before the reaction, the catalyst samples in the reactor were kept under airflow at 600°C for lh. 

The effect of an applied pressure on the UCFT has been investigated for polymer particles that are sterically stabilized by polyisobutylene and dispersed in 2-methy1-butane. It was observed that the UCFT was shifted to a higher temperature as the hydrostatic pressure applied to the system increased. There was also a qualitative correlation between the UCFT as a function of applied pressure and the 6 conditions of PIB + 2-methylbutane in (P,T) space. These results can be rationalized by considering the effect of pressure on the free volume dissimilarity contribution to the free energy of close approach of interacting particles. Application of corresponding states concepts to the theory of steric stabilization enables a qualitative prediction of the observed stability behaviour as a function of temperature and pressure. 

It was suggested in a previous publication (9) that flocculation at the UCFT can be ascribed to the free volume dissimilarity between the polymer stabilizing the particle and the low molecular weight dispersion medium. Incorporating this idea in a quantitative way into the theory of steric stabilization allowed for a qualitative interpretation of the experimental data. This idea is further extended to include the effect of pressure on the critical flocculation conditions. 

As an example of composite core/shell submicron particles, we made colloidal spheres with a polystyrene core and a silica shell. The polar vapors preferentially affect the silica shell of the composite nanospheres by sorbing into the mesoscale pores of the shell surface. This vapor sorption follows two mechanisms physical adsorption and capillary condensation of condensable vapors17. Similar vapor adsorption mechanisms have been observed in porous silicon20 and colloidal crystal films fabricated from silica submicron particles32, however, with lack of selectivity in vapor response. The nonpolar vapors preferentially affect the properties of the polystyrene core. Sorption of vapors of good solvents for a glassy polymer leads to the increase in polymer free volume and polymer plasticization32. 

A packed bed of particles of diameter dp and fractional free volume c is modelled as a group of parallel capilaries with a perimeter... 

Figure 2.5 A broken, spherical silica particle entrapping an API has 85% free volume. Such particles are used in formulations such as Eusolex UV-Pearls that reduce dermal uptake compared to free UV filters thus they do not irritate the skin while they make new application possibilities for hydrophobic UV filters. (Photo courtesy of Sol-Gel Technologies Ltd.)... 

Although, the true density of solid phase p=m/Vp (e.g., g/cm3) is defined by an atomic-molecular structure (/ ), it has become fundamental to the definition of many texture parameters. In the case of porous solids, the volume of solid phase Vp is equal to the volume of all nonporous components (particles, fibers, etc.) of a PS. That is, Vp excludes all pores that may be present in the particles and the interparticular space. The PS shown in Figure 9.17a is formed from nonporous particles that form porous aggregates, which, in turn, form a macroscopic granule of a catalyst. In this case, the volume Vp is equal to the total volume of all nonporous primary particles, and the free volume between and inside the aggregates (secondary particles) is not included. 

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