Rigid-sphere dispersion

Following Einstein s work (Einstein, 1906, 1911) (Equation 2.24) on dilute rigid sphere dispersions, models for estimating viscosity of concentrated non-food dispersions of solids are based on volume fraction (p) of the suspended granules and the relative viscosity of the dispersion, % = (vIVs), where t] is the viscosity of the dispersion rjs is the viscosity of the continuous phase (Jinescu, 1974 Metzner, 1985). 

The important role of volume fraction on the structure of rigid sphere dispersions has been uncovered recently as the volume fraction of hard spheres is increased, the equilibrium phase changes from a disordered fluid to coexistence with a crystalline phase (0.494<<0.545), then to fully crystalline ( = 0.545), and finally to a glass (0 = 0.58) (Pham et al. 2002). 

Variables in the Rheology-Structure Relation for Rigid-Sphere Dispersions... 

The theoretical foundations of these rules are, however, rather weak the first one is supposed to result from a formula derived by London for dispersion forces between unlike molecules, the validity of which is actually restricted to distances much larger than r the second one would only be true for molecules acting as rigid spheres. Many authors tried to check the validity of the combination rules by measuring the second virial coefficients of mixtures. It seems that within the experimental accuracy (unfortunately not very high) both rules are roughly verified.24... 

The viscosity of a polymer solution is one of its most distinctive properties. The spatial extension of the molecules is great enough so that the solute particles cut across velocity gradients and increase the viscosity in the manner suggested by Figure 4.8. In this regard they are no different from the rigid spheres of the Einstein model. What is different for these molecules is the internal structure of the dispersed units, which are flexible and swollen with solvent. The viscosity of a polymer solution depends, therefore, on the polymer-solvent interactions, as well as on the properties of the polymer itself. 

For dispersions of non-rigid spheres (e.g. emulsions) the flow lines may be partially transmitted through the suspended particles, making k in Einstein s equation less than 2.5. 

Changing the shape of the dispersed species while flowing also has an impact. Since emulsion droplets and foam bubbles are not rigid spheres, they may deform in shear flow. In the cases of electrostatically interacting species, or those with surfactant or polymeric stabilizing agents at the interface, the species will not be noninteracting, as is assumed in the theory. Thus, Stokes law will not strictly apply and may underestimate or even overestimate the real terminal velocity. 

The classical thermodynamic and kinetic model is that of a rigid sphere impenetrable by water. A spherical geometry has been observed in many polysaccharide systems, notably hyaluronic acid-protein complexes (Ogston and Stainer, 1951), dispersed gum arabic (Whistler, 1993), and spray-dried ungelatinized starch granules (Zhao and Whistler, 1994). Spherulites of short-chain amylose were obtained by precipitation with 30% water-ethanol (Ring et al., 1987), and spherulites of synthetic polymers were obtained... 

Wildemuth and Williams (1984) modeled the jr of rigid sphere suspensions with a shear-dependent maximum volume fraction (pm)- The apphcability of a shear dependent food dispersions has not been tested. 

The mass-transfer coefficient in each film is expected to depend upon molecular diffusivity, and this behavior often is represented by a power-law function k . For two-film theory, n = 1 as discussed above [(Eq. (15-62)]. Subsequent theories introduced by Higbie [Trans. AIChE, 31, p. 365 (1935)] and by Dankwerts [Ind. Eng. Chem., 43, pp. 1460-1467 (1951)] allow for surface renewal or penetration of the stagnant film. These theories indicate a 0.5 power-law relationship. Numerous models have been developed since then where 0.5 < n < 1.0 the results depend upon such things as whether the dispersed drop is treated as a rigid sphere, as a sphere with internal circulation, or as oscillating drops. These theories are discussed by Skelland [ Tnterphase Mass Transfer, Chap. 2 in Science and Practice of Liquid-Liquid Extraction, vol. 1, Thornton, ed. (Oxford, 1992)]. 

As is known [24,28 ] in most cases disperse silicas have globular structure. For rigid spheres of a b radius with an uniformly distributed charge the following equation is proposed ... 

Calderbank and Moo-Young also confirmed the classical correlation for mass and heat transfer through natural convection in dispersions of fine droplets or fine gas bubbles for so-called rigid spheres" ... 

For dispersed multiphase flows a Lagrangian description of the dispersed phase are advantageous in many practical situations. In this concept the individual particles are treated as rigid spheres (i.e., neglecting particle deformation and internal flows) being so small that they can be considered as point centers of mass in space. The translational motion of the particle is governed by the Lagrangian form of Newton s second law [103, 148, 120, 38] ... 

The maximum internal phase ratio that can be attained without deforming the drop spherical shape depends on the drop size distribution. For monodispersed rigid spheres, the most dense tessellation is the hexagonal packing at about 74% of internal phase. As an example, for randomly settled monodispersed spheres it could be 65%. For very poly dispersed emulsions it might be higher than 90% (16). 

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