Dilute suspension model

Compared to other models (e.g., Voigt-Reuss, Halpin-Tsai, modified mixture law, and Cox), the dilute suspension of clusters model promulgated by Villoria and Miravete [255] could estimate the influence of the dispersion of nanofillers in nanocomposite Young s modulus with much improved theoretical-experimental correlation. 

Charge transfer occurs when particles collide with each other or with a solid wall. For monodispersed dilute suspensions of gas-solid flows, Cheng and Soo (1970) presented a simple model for the charge transfer in a single scattering collision between two elastic particles. They developed an electrostatic theory based on this mechanism, to illustrate the interrelationship between the charging current on a ball probe and the particle mass flux in a dilute gas-solid suspension. This electrostatic ball probe theory was modified to account for the multiple scattering effect in a dense particle suspension [Zhu and Soo, 1992]. 

A regime map of Fo versus the solid volume fraction, ap, for various gas-solid flows was presented by Hunt (1989), as shown in Fig. 4.3. Hunt (1989) suggested that except when Fo > 1 and ap > 0.1, use of the pseudocontinuum model is inappropriate. Thus, from Fig. 4.3, it can be seen that the pseudocontinuum model is applicable to packed beds, incipient fluidized beds, and granular flows, whereas it is not applicable to pneumatic transport flows, dilute suspensions, bubbling beds, and slugging fluidized beds [Glicksman and Decker, 1982 Hunt, 1989]. 

In dilute suspensions clays tend to form gels. The classical model is the house of cards structure of kaolinite in which the face-to-edge association leads to an open 3-D structure (van Olphen, 1965). In the case of smectite-water systems it now seems more likely that the microstructure is mainly controlled by the face-to-face interactions (Van Damme et al., 1985). 

The simplest model to predict the viscosity of liquids containing solid particles is that derived for a dilute suspension of uniform, monodisperse, noninteracting hard spheres in a solvent of viscosity rj. ... 

In this chapter, we extend the electrokinetic theory of soft particles (Chapter 21), which is applicable for dilute suspensions, to cover the case of concentrated suspensions [1-3] on the basis of Kuwabara s cell model [4], which has been applied to theoretical studies of various electrokinetic phenomena in concentrated suspensions of hard colloidal particles [5-23]. 

Electrokinetic equations describing the electrical conductivity of a suspension of colloidal particles are the same as those for the electrophoretic mobility of colloidal particles and thus conductivity measurements can provide us with essentially the same information as that from electrophoretic mobihty measurements. Several theoretical studies have been made on dilute suspensions of hard particles [1-3], mercury drops [4], and spherical polyelectrolytes (charged porous spheres) [5], and on concentrated suspensions of hard spherical particles [6] and mercury drops [7] on the basis of Kuwabara s cell model [8], which was originally applied to electrophoresis problem [9,10]. In this chapter, we develop a theory of conductivity of a concentrated suspension of soft particles [11]. The results cover those for the dilute case in the limit of very low particle volume fractions. We confine ourselves to the case where the overlapping of the electrical double layers of adjacent particles is negligible. 

As indicated above, such an agreement is perhaps expected. On the other hand, it is remarkable that a rather complex phenomenological theory postulated for an LC continuum can be reconciled with an even more complex molecular theory built on the concept of intermolecular potential. Perhaps the only other such happy instance is the agreement between the continuum Oldroyd-B model for viscoelastic liquids and the molecular model based on a dilute suspension of linear Hookean dumbbells in a Newtonian solvent. ... 

Auton [7], Thomas et al [152] and Auton et al [8] determined a lift force due to inviscid flow around a sphere. In an Eulerian model formulation this lift force parameterization is usually approximated for dilute suspensions, giving ... 

In one of the proposed modeling approaches the production of granular temperature represented by the gas-particle velocity covariance term is interpreted as a mechanism that breaks a homogeneous fluidized bed with no shearing motion into a non-homogeneous distribution. Koch [74] proposed a closure for these gas-particle interactions for dilute suspensions. Koch and... 

The interaction in a two-body collision in a dilute suspension has been expanded to provide a useful and quantitative understanding of the aggregation and sedimentation of particulate matter in a lake. In this view, Brownian diffusion, fluid shear, and differential sedimentation provide contact opportunities that can change sedimentation processes in a lake, particularly when solution conditions are such that the particles attach readily as they do in Lake Zurich [high cc(i,j)exp]. Coagulation provides a conceptual framework that connects model predictions with field observations of particle concentrations and size distributions in lake waters and sediment traps, laboratory determinations of attachment probabilities, and measurements of the composition and fluxes of sedimenting materials (Weilenmann et al., 1989). 

We give here a rather complete summary of the results for the kinetic theory of a dilute suspension of dumbbells (i.e., for N = 2) and we relate these results to experimental data or to continuum mechanics and rheology wherever possible. It might at first glance seem questionable to treat dumbbell models exhaustively. We feel, however, that a thorough summary is timely and useful for these reasons ... 

A model for the bulk effective resistivity of a dilute suspension (disperse phase) of noninteracting conducting spheres (not necessarily mono-dispersed) of material resistivity 9id and void fraction ad suspended in a continuous medium of material resistivity 9ic was derived by Maxwell (1954). His result is... 

To deseribe wave propagation in marine sediments mathematieally, various simple to eomplex models have been developed which approximate the sediment by a dilute suspension (Wood 1946) or an elastic, water-saturated frame (Gassmann 1951 Biot 1956a, b). The most common model whieh considers the microstructure of the sediment and simulates frequency-dependent wave propagation is based on Biot s theory (Biot 1956a, b). It includes Wood s suspension and Gassmann s elastic frame model as low-frequency approximations and combines acoustic and elastic parameters - P- and... 

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