Particle volume fractions, effect

Hindered Settling When particle concentration increases, particle settling velocities decrease oecause of hydrodynamic interaction between particles and the upward motion of displaced liquid. The suspension viscosity increases. Hindered setthng is normally encountered in sedimentation and transport of concentrated slurries. Below 0.1 percent volumetric particle concentration, there is less than a 1 percent reduction in settling velocity. Several expressions have been given to estimate the effect of particle volume fraction on settling velocity. Maude and Whitmore Br. J. Appl. Fhys., 9, 477—482 [1958]) give, for uniformly sized spheres,... 

The next problem is to find an expression for Asg. This entropy difference is a function of the particle volume fractions in the dispersion ( ) and in the floe (<(> ). As a first approximation, we assume that Ass is independent of the concentration and chain length of free polymer. This assumption is not necessarily true the floe structure, and thus < >f, may depend on the latter parameters because also the solvent chemical potential in the solution (affected by the presence of polymer) should be the same as that in the floe phase (determined by the high particle concentration). However, we assume that these effects will be small, and we take as a constant. 

Strictly speaking, one should use 2D bins involving particle volume fraction and a Reynolds number based on slip velocity to classify the filtered drag coefficient however in these simulations, the Reynolds number effect was found to be weak and hence the data were collapsed to just volume fraction bins. 

In suspensions of particles with an aspect ratio (length to diameter) greater than 1, particle rotation during flow results in a large effective hydrodynamic volume, and Kh > 2.5 (see Figure 4.7). At particle volume fractions above about 5-10%, interaction between particles during flow causes the viscosity relationship to deviate from the Einstein equation. In such instances, the reduced viscosity is better described by the following relationship ... 

The mean-field theory has a number of shortcomings, including the approximations of a mean concentration around all particles and the establishment of spherically symmetric diffusion fields around every particle, similar to those that would exist around a single particle in a large medium. The larger the particles total volume fraction and the more closely they are crowded, the less realistic these approximations are. No account is taken in the classical model of such volume-fraction effects. Ratke and Voorhees provide a review of this topic and discuss extensions to the classical coarsening theory [8]. 

Volume-fraction effects on particle coarsening rates have been observed experimentally. For comparisons between theory and experiment, data from liquid+solid systems are far superior to those from solid+solid systems, as the latter are potentially strongly influenced by coherency stresses. Hardy and Voorhees studied Sn-rich and Pb-rich solid phases in Pb-Sn eutectic liquid over the range

presented data in support of the volume-fraction effect, as shown in Fig. 15.9 [7],... 

Figure 5.4. Determination of minimum radius of control volume for two different control volume centers (a) Effect of control volume on calculated particle volume fraction (b) Effect of control volume on relative deviation of calculated particle volume fraction.

Figure 6.14. Effect of particle volume fraction on the equilibrium velocity behind a normal shock in a gas-solid suspension (from Rudinger, 1965).

The overall solids holdup is defined as the particle volume fraction over the entire riser. Figure 10.14 shows the typical effect of the solids circulation rate on the overall... 

Consider a dilute gas-solid flow in a pipe in which the solid particles carry significant electrostatic charges. It is assumed that (a) the flow is fully developed (b) the gravitational effect is negligible and (c) the flow and the electrostatic field are axisymmetric. Derive an expression to describe the radial volume fraction distribution of the particles and identify the radial locations where the particle volume fractions are maximum and minimum in the distribution. Also, if the electrostatic charge effects are negligible, derive an expression to describe the radial volume fraction distribution of the particles. 

When the superparamagnetic theory is applied for interpretation of any measured susceptibility line, it means that some model function x(V,a>,T) depending on several material parameters, is processed through averaging like such as Eq. (4.121) or (1.122). In our case the basic set of the material parameters comprises magnetization /, anisotropy energy density K, relaxation time To, and the particle volume fraction tp. Obviously, for nanosize-dispersed systems the effective values of /, K, and t0 do not coincide with those for a bulk material. The size/volume averaging itself introduces two independent... 

The effects of the number of charges Z per particle and of the volume fraction / of the particles on the effective interaction potential between identical charged particles were examined. The number of charges per particle was varied between 150 and 1200 and the particle volume fraction / was varied between 0.005 and 0.12. The electrolyte concentration was assumed to be 10 5 or 10 4 M. 

Fig. 3. Effective interaction potential between identical charged colloidal particles versus the distance between the centers of particles (d is the diameter of the particles) for various charges per particle (Z), particle volume fraction of 0.015, and electrolyte concentration of 10 5 M. (1) Z = 300 (2) Z = 600 (3) Z = 1200.

When the particle volume fraction / was increased to 0.015, the oscillations of the effective interaction between identical charged particles became larger than those for / = 0.005 at an electrolyte concentration of 10 5 M (Fig. 3). The effective interaction between identical charged particles versus the distance between particles is plotted in Fig. 3 for various values of Z (Z = 300, Z = 600, Z = 1200). As shown in Fig. 4, the colloidal dispersion has a disordered liquidlike structure for Z = 600, but a more ordered structure for Z = 1200. When the electrolyte concentration was increased to 10-4 M, the interaction between identical charged particles became completely screened. As shown in Fig. 5, no oscillations of the effective interaction potential were present for Z = 300,600, and 1200. 

Using Monte Carlo simulations and a Debye-Htickel pah-repulsive potential, effective interaction potentials between identical charged particles were calculated and the phase structures were obtained for various charges per particle and various particle volume fractions. The simulation re-... 

The coefficient Q(Aa, a/b) in Eq. (27.45) expresses the effects of the presence of the uncharged polymer layer upon As A —> 00 or a b, soft particles tend to hard particles and Q(Aa, alb) 1 so that Eq. (27.45) tends to Eq. (27.1). For 1 0, on the other hand, the polymer layer vanishes so that a suspension of soft particles of outer radius b becomes a suspension of hard particles of radius a and the particle volume fraction changes from cj) to 4>, which is given by... 

According to general experience the Influence of sol concentration Is absent when the volume fraction remains of the order of a few percent. This may be concluded from older experiments and from recent electroacoustic studies, discussed in sec. 4.5d. Experiments involving mlcro-electrophoresls are not suited to stud3dng the volume fraction effect because the required extreme dilution may readily lead to spurious adsorption on the particles. Reed and Morrison studied theoretically the hydrodynamic interactions between pairs of different particles in electrophoresis they corrected the Helmholtz-Smoluchowski equation for various distances between the (spherical) particles and values of the electroklnetlc potentials of the particles, and... 

For adsorption onto colloidal particles two approaches are possible. The first is to determine vlscometrlcally the increase in the effective particle volume fraction upon adsorption, and the second is to measure the decrease in the particle diffusion coefficient. 

The effective volume fraction can be obtained from the intrinsic viscosity (rj) of the dispersion. According to Einstein s viscosity relation, the product of (ijl and the mass concentration c is proportional to the particle volume fraction, which is bigger for covered particles than for bare ones. The ratio (t 1 / n. where (rjlg is the intrinsic viscosity of the dispersion of bare particles, is therefore equal to the ratio between the volume of a covered pcutlcle and that of a bare one, l.e., it equals (l + d /a) . This method has been used by several authorsespecially in older work. 

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