Maximum Packing Functions

For more concentrated suspensions (q> >0.2), the sedimentation velocity becomes a complex function of At > 0.4, a hindered settling regime is usually entered whereby all of the particles sediment at the same rate (independent of size). A schematic representation for the variation of v with is shown in Figure 9.12, which also shows the variation of relative viscosity with rp. It can be seen from these data that v decreases exponentially with increase in approaches zero when cp approaches a critical value (the maximum packing fraction). The relative viscosity shows a gradual increase with increase in cp such that, when cp = the relative viscosity approaches infinity. 

In Eq 7.25 ( ) is the maximum packing volume fraction. Thus, the magnitude of the shielding function My) depends on the reduced volume fraction, = ( ) / ( ). At low concentration, —> 0, the shielding factor vanishes and Einstein s relation is recovered. However, at high concentration, —> 1, the shielding function and relative viscosity both... 

Figure 7.7. Relative viscosity of hard-sphere suspension in Newtonian fluid as a function of the volume fraction. Thomas curve represents the generalized behavior of suspensions as measured in 19 laboratories. The remaining curves were computed from Simha s, Mooney s and Krieger-Dougherty s relations assuming Einstein value for intrinsic viscosity of hard spheres, [T ] = 2.5, but different values for the maximum packing volume fraction, ([) = 0.78, 0.91, and 0.62 respectively.

The relationships between 17 and ( ) have been derived for suspensions of monodispersed hard spheres in Newtonian liquids. However, most real systems are polydispersed in size, and do not necessarily consist of spherical particles. It has been found that here also Simha s Eq 7.24, Mooney s Eq 7.28, or Krieger-Dougherty s Eq 7.8 are useful, provided that the intrinsic viscosity and the maximum packing volume fraction are defined as functions of particle shape and size polydispersity. For example, by allowing ( ) to vary with composition, it was possible to describe the vs. ( ) variation for bimodal suspensions [Chang and Powell, 1994]. Similarly, after values... 

There are numerous theories based on structural models of suspensions [Mikami, 1980]. Wildemuth and Williams [1984] considered that the maximum packing volume fraction, ( ), is a function of normalized shear stress, CTij = cTjj / M, where M is a numerical parameter. The authors derived the relation ... 

In Eq. 7.25, 4>ni is the maximum packing volume fraction. Thus, the magnitude of the shielding function X(y) depends on the reduced volume fraction,

[Pg.743]

The effect of particles and matrix properties on the shear viscosity of LDPE/ GTR blend was also studied by developing a theoretical model to predict the viscosity of the composites as a function of the rheological properties of the matrix, solid concentration, particle size distribution, particle shape, and deformability (Bhattacharya and Sbarski 1998). The real viscosity measurements were found in good agreement with the values predicted below the maximum packing fraction. 

The rate of creaming or sedimentation becomes a complex function of as is illustrated in Figure 6.25, which also shows the change of relative viscosity 7, with As seen in figure, v decreases with increasing and ultimately approaches zero when (j> exceeds a critical value (. ), which is the so-called maximum pack-... 

Figure 6.19 Variation of maximum packing as a function of the bimodal size distribution of the 15 and 78pm glass spheres. (Reprinted from Ref. 72 with kind permission from Society of Rheology, USA.)...

Figure 3 The maximum packing fraction of a powder as a function of its polydispersity (shown here as the inverse of geometric standard deviation) for a log-normal distribution.

Eq. (64), together with Eq. (58), indicates that does not depend on the size but instead the size distribution breadth expressed by Og. Experimental results show that the maximum packing fraction calculated from Eq. (58) is relatively higher than that determined with the rheological measurement [31]. The discrepancy is believed to be a result of the constant value of a, which should be a function of the particle size and particle packing... 

At higher volume fractions ((jj > 0.2), q, is a complex function of c ) and the iff-cj) curve is schematically shown in Fig. 5.20. This curve is characterized by two asymptotes [q] the intrinsic viscosity euid c )p the maximum packing fraction. 

In eqn [7], r is the particle radius, the maximum packing factor of the monodisperse partides, and

volume fraction of the partides. Figure 13 plots the interpartide distance (eqn [7]) as a function of the volume fraction of partides (solids content) for unimodal latexes. The straight line in the plot... 

For such ideal systems as suspensions of spheres of equal diameter, many equations, either theoretical or empirical have been proposed for the relative viscosity as a function of the filler volume fraction. Such a subject is obviously of tremendous importance in many fields. A thorough discussion of suspensions of rigid particles in Newtonian fluids was made by Jeffrey and Acrivos and some models available up to 1985 were discussed in detail by Metzner.59 We will consider below only the most referred equations that explicitly consider the maximum packing fraction. One of the oldest proposal was likely made by Filers in order to model the behavior of highly viscous suspensions, i.e. ... 

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