Spherical particles dispersions

Figure 3.18 The primary electroviscous effect estimated from Equation (3.59) for spherical particles dispersed in 10 2 M KCl...

Effective Conductivity of Dilute Mixtures, The simplest, best defined case, is a cluster of spherical particles dispersed in a liquid and located in a... 

Effective Conductivity of Dilute Mixtures. The simplest best-defined case is a cluster of spherical particles dispersed in a liquid and located in a uniform electrical field. If the particles have the same conductivity as the liquid, the potential around the particles will not be distorted, and the mixture conductivity is equal to that of the liquid. If the particles have a lower conductivity, the streamlines will diverge away from the particles, and the mixture conductivity will be lower than that of the liquid. If the particles have a higher conductivity, the streamlines will converge into the particle, and the mixture conductivity will be higher than that of the liquid. 

Flocculation rates. Stability ratios for spherical particles dispersed in polymer solution were calculated numerically by Feigin and Napper (1980b) from... 

The basis for relating the viscosity of a polymer solution to the structure of a dissolved polymer can be traced to Albert Einstein [1906, 1911], who showed, for spherical particles dispersed in a viscous medium, that the solution viscosity, rj, is increased relative to that of the medium, jjj, by... 

To examine the developed model, 3D numerical simulations of silica slurry dryiug iu the adopted spray chamber have beeu carried out. The slurry consists of amorphous silica spherical particles dispersed homogeneously in water with initial average moisture content of 1.35 kg H20/kg dry solid. The size of silica particles is 272 mn and the density is 1950 kg/m [40]. The critical moisture content calculated using Equation 10.6 is equal to 0.342 kg HjO/kg solid, and the final moisture content of dried particles is assumed to be 0.05 kg H20/kg solid. 

Gravity separations dqtend essentially on die density differences of the gas, solid, or liquids present in the mix. The particle size of the dispersed phase and the properties of the continuous phase are alw factors with the separation motivated by die acceleration of gravity. The simplest representation of this involves the assumption of a rigid spherical particle dispersed in a fluid with the terminal or free-settling velocity represented by... 

More quantitative information depends on the use of models. The Takayanagi models were already mentioned in connection with Figure 6.29. More analytical models have been evolved by Kerner/ Hashin and Shtrikman/ and Davies.Briefly, the first two theories assume spherical particles dispersed in an isotropic matrix. From the modulus of each phase, the composite modulus is calculated. An upper or lower bound modulus is arrived at by assuming the higher or lower modulus phase to be the matrix, respectively. The theory is reviewed elsewhere. 

Hashin and Shtrikman ako provided an upper bound model for the thermal conductivity of spherical particles dispersed randomly in a continuous matrix... 

The TEM micrographs presented in Figure 4.9 show typical results obtained by means of the freeze fracturing technique applied to overbased calcium alkyl benzene sulfonate (OCABS) dispersion in dodecane. The OCABS aggregates, clearly identified by X-ray analysis on the extractive replica, appear as approximately spherical particles dispersed in the apolar medium [4, 35]. 

The classical theory of Einstein describes a relationship between the solid volume fraction and the viscosity for monosized spherical particles dispersed randomly in a fluid [46] ... 

Let us first consider a dilute system of rigid spherical particles dispersed at a volume fraction m Newtonian liquid of shear viscosity rjo. According to Einstein,the Newtonian viscosity of the whole system, rj, is increased by the presence of the particles according to the equation... 

Borstnik, A., H. Stark, and S. Zumer. 1999. Interaction of spherical particles dispersed in a liquid crystal above the nematic-isotropic phase transition. Phys. Rev. E 60 4210-4218. 

Viscosity—Concentration Relationship for Dilute Dispersions. The viscosities of dilute dispersions have received considerable theoretical and experimental treatment, partly because of the similarity between polymer solutions and small particle dispersions at low concentration. Nondeformable spherical particles are usually assumed in the cases of molecules and particles. The key viscosity quantity for dispersions is the relative viscosity or viscosity ratio,... 

The relative viscosity of a dilute dispersion of rigid spherical particles is given by = 1 + ft0, where a is equal to [Tj], the limiting viscosity number (intrinsic viscosity) in terms of volume concentration, and ( ) is the volume fraction. Einstein has shown that, provided that the particle concentration is low enough and certain other conditions are met, [77] = 2.5, and the viscosity equation is then = 1 + 2.50. This expression is usually called the Einstein equation. 

FIG. 16-27 Constant pattern solutions for R = 0.5. Ordinant is cfor nfexcept for axial dispersion for which individual curves are labeled a, axial dispersion h, external mass transfer c, pore diffusion (spherical particles) d, surface diffusion (spherical particles) e, linear driving force approximation f, reaction kinetics. [from LeVan in Rodrigues et al. (eds.), Adsorption Science and Technology, Kluwer Academic Publishers, Dor drecht, The Nether lands, 1989 r eprinted with permission.]... 

Information on the morphology of the nanohybrid sorbents also was revealed with SEM analysis. Dispersed spherical polymer-silica particles with a diameter of 0.3-5 pm were observed. Every particle, in one s turn, is a porous material with size of pores to 200 nm and spherical particles from 100 nm to 500 nm. Therefore, the obtained samples were demonstrated to form a nanometer - scale porous structure. 

Fig. 8. Calculations performed considering metallic spherical particles (i.e., N = 1/3) with intrinsic Crude parameters fttOp = I eV, fiV = 0.01 eV, dispersed in an insulating matrix with parameters fttOp i = 2 eV, ftP] = I eV and fttO] = 5 eV, and filling factor /between 0.2 and 1.

How does yield stress depend on the size of particles We have mentioned above that increasing the specific surface, i.e. decreasing an average size of particles of one type, causes an increase in yield stress. This fact was observed in many works (for example [14-16]). Clear model experiments the purpose of which was to reveal the role of a particle s size were carried out in work [8], By an example of suspensions of spherical particles in polystyrene melt it was shown that yield stress of equiconcentrated dispersions may change by a hundred of times according to the diameter d of non-... 

It follows from general considerations that the role of the shape of the filler particles during net-formation must be very significant. Thus, it is well-known that the transition from spherical particles to rod-like ones in homogeneous systems results in such radical structural effect as the formation of liquid-crystal phase. Something like that must be observed in disperse systems. 

The value of the first coefficient b, for the dispersion of spherical particles is well known and generally accepted. This is Einstein coefficient b, = 2.5, taking into account the viscosity variation of the dispersion medium upon introducing noninteracting solid particles of spherical form into it. Thus, for tp [Pg.83]

Third, a complicated question on the role of the dispersion of particles dimensions of particles dimensions is of independent value it is known that the viscosity of equi-concentrated dispersions of even spherical particles depends on the fact if spheres of one dimension or mixtures of different fractions were used in the experiments and here in all the cases the transition from monodisperse particles to wide distributions leads to a considerable decrease in viscosity [21] (which, certainly, is of theoretical and enormous practical interest as well). 

Even if the peculiarities of net-formation of nonspherical particles are not taken into account, at least two fundamentally new effects arise during the flow of dispersion. First, this is the possibility to be oriented in the flow, as a consequence of which the medium becomes anisotropic. And second, this is the possibility to rotate the spherical particles in the flow (spherical particles can, of course, rotato too, but their rotation does not affect the structure of the system as a whole). 

This is obvious for the simplest case of nondeformable anisotropic particles. Even if such particles do not change the form, i.e. they are rigid, a new in principle effect in comparison to spherical particles, is their turn upon the flow of dispersion. For suspensions of anisodiametrical particles we can introduce a new characteristic time parameter Dr-1, equal to an inverse value of the coefficient of rotational diffusion and, correspondingly, a dimensionless parameter C = yDr 1. The value of Dr is expressed via the ratio of semiaxes of ellipsoid to the viscosity of a dispersion medium. 

Most of the electrochemical promotion studies surveyed in this book have been carried out with active catalyst films deposited on solid electrolytes. These films, typically 1 to 10 pm in thickness, consist of catalyst grains (crystallites) typically 0.1 to 1 pm in diameter. Even a diameter of 0.1 pm corresponds to many (-300) atom diameters, assuming an atomic diameter of 3-10 10 m. This means that the active phase dispersion, Dc, as already discussed in Chapter 11, which expresses the fraction of the active phase atoms which are on the surface, and which for spherical particles can be approximated by ... 

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