Interparticle

Marlow and Rowell discuss the deviation from Eq. V-47 when electrostatic and hydrodynamic interactions between the particles must be considered [78]. In a suspension of glass spheres, beyond a volume fraction of 0.018, these interparticle forces cause nonlinearities in Eq. V-47, diminishing the induced potential E. 

This description is traditional, and some further comment is in order. The flat region of the type I isotherm has never been observed up to pressures approaching this type typically is observed in chemisorption, at pressures far below P. Types II and III approach the line asymptotically experimentally, such behavior is observed for adsorption on powdered samples, and the approach toward infinite film thickness is actually due to interparticle condensation [36] (see Section X-6B), although such behavior is expected even for adsorption on a flat surface if bulk liquid adsorbate wets the adsorbent. Types FV and V specifically refer to porous solids. There is a need to recognize at least the two additional isotherm types shown in Fig. XVII-8. These are two simple types possible for adsorption on a flat surface for the case where bulk liquid adsorbate rests on the adsorbent with a finite contact angle [37, 38]. 

These are extrapolated values and undoubtedly contain an important contribution from interparticle condensation. 

The fimctiong(ri is central to the modem theory of liquids, since it can be measured experimentally using neutron or x-ray diffraction and can be related to the interparticle potential energy. Experimental data [1] for two liquids, water and argon (iso-electronic with water) are shown in figure A2.4.1 plotted as a fiinction ofR = R /a, where a is the effective diameter of the species, and is roughly the position of the first maximum in g (R). For water, a = 2.82 A,... 

The relationship between g(r) and the interparticle potential energy is most easily seen if we assume that the interparticle energy can be factorized into pairwise additive potentials as... 

The first tenn, P(q), represents the interferences within particles and its contribution is proportional to the number of particle, N. The second tenn, Q(q), involves interparticle interferences and is proportional to the... 

Hard-sphere models lack a characteristic energy scale and, hence, only entropic packing effects can be investigated. A more realistic modelling has to take hard-core-like repulsion at small distances and an attractive interaction at intennediate distances into account. In non-polar liquids the attraction is of the van der Waals type and decays with the sixth power of the interparticle distance r. It can be modelled in the fonn of a Leimard-Jones potential Fj j(r) between segments... 

The three internal coordinates aie expressed as combinations of squares of the interparticle distances ... 

In the region where > 1 then if the interparticle force is assumed to be constant over thi integration time step the following result is obtained [van Gunsteren et al. 1981] ... 

There is a very convenient way of writing the Hamiltonian operator for atomic and molecular systems. One simply writes a kinetic energy part — for each election and a Coulombic potential Z/r for each interparticle electrostatic interaction. In the Coulombic potential Z is the charge and r is the interparticle distance. The temi Z/r is also an operator signifying multiply by Z r . The sign is - - for repulsion and — for atPaction. 

Fig. 4. Scanning electron micrograph of 5-p.m diameter Zn powder. Neck formation from localized melting is caused by high-velocity interparticle coUisions. Similar micrographs and elemental composition maps (by Auger electron spectroscopy) of mixed metal coUisions have also been made.

Fig. 5. The effect of ultrasonic irradiation on the surface morphology and particle size ofNi powder. Initial particle diameters (a) before ultrasound were i 160 fim-, (b) after ultrasound, fim. High velocity interparticle coUisions caused by ultrasonic irradiation of slurries are responsible for the smoothing...

The basic concepts of a gas-fluidized bed are illustrated in Figure 1. Gas velocity in fluidized beds is normally expressed as a superficial velocity, U, the gas velocity through the vessel assuming that the vessel is empty. At a low gas velocity, the soHds do not move. This constitutes a packed bed. As the gas velocity is increased, the pressure drop increases until the drag plus the buoyancy forces on the particle overcome its weight and any interparticle forces. At this point, the bed is said to be minimally fluidized, and this gas velocity is termed the minimum fluidization velocity, The bed expands slightly at this condition, and the particles are free to move about (Fig. lb). As the velocity is increased further, bubbles can form. The soHds movement is more turbulent, and the bed expands to accommodate the volume of the bubbles. 

Equations 3 to 7 indicate the method by which terminal velocity may be calculated. Erom a hydrodynamic force balance that considers gravity, buoyancy, and drag, but neglects interparticle forces, the single particle terminal velocity is... 

This equation indicates that, for small particles, viscosity is the dorninant gas property and that for large particles density is more important. Both equations neglect interparticle forces. 

Interparticle Forces. Interparticle forces are often neglected in the fluidization Hterature, although in many cases these forces are stronger than the hydrodynamic ones used in most correlations. The most common interparticle forces encountered in gas fluidized beds are van der Waals, electrostatic, and capillary. 

Transport Disengaging Height. When the drag and buoyancy forces exerted by the gas on a particle exceed the gravitational and interparticle forces at the surface of the bed, particles ate thrown into the freeboard. The ejected particles can be coarser and more numerous than the saturation carrying capacity of the gas, and some coarse particles and clusters of fines particles fall back into the bed. Some particles also coUect near the wall and fall back into the fluidized bed. 

Once the precipitates grow beyond a critical size they lose coherency and then, in order for deformation to continue, dislocations must avoid the particles by a process known as Orowan bowing(23). This mechanism appHes also to alloys strengthened by inert dispersoids. In this case a dislocation bends between adjacent particles until the loop becomes unstable, at which point it is released for further plastic deformation, leaving a portion behind, looped around the particles. The smaller the interparticle spacing, the greater the strengthening. 

Chemical Grafting. Polymer chains which are soluble in the suspending Hquid may be grafted to the particle surface to provide steric stabilization. The most common technique is the reaction of an organic silyl chloride or an organic titanate with surface hydroxyl groups in a nonaqueous solvent. For typical interparticle potentials and a particle diameter of 10 p.m, steric stabilization can be provided by a soluble polymer layer having a thickness of - 10 nm. This can be provided by a polymer tail with a molar mass of 10 kg/mol (25) (see Dispersants). 

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