Volume fraction maximum solids

Initially, there is no gas flow through the reactor, and the true volume fraction of solids in the bed is about that of maximum packing. However, Lindborg et al [92] adopted the customary approach of specifying the initial conditions in correspondence with the state of a bubbling bed operating at minimum fluidization conditions. The bed height at minimum fluidization conditions was thus set to an estimated value Lmf, and the gas volume fraction was set to amf at the bed levels below Lmf and unity in the freeboard. The pressure... 

The dependence of viscosity on volume fraction sohds is shown in Fig. 8.88. At high particle concentrations, viscosity of the suspension increases more rapidly than predicted by the above equation due to interparticle interactions. Several empirical equations are available to relate viscosity to the solid concentration behavior of suspensions. As the volume fraction of solids is increased further, a stage will be reached where the particles will be interlocked and no flow will occur (i.e., viscosity approaches infinity). The volume fraction of sohds at which this occurs is called the maximum packing fraction and its... 

You must determine the maximum feed rate that a thickener can handle to concentrate a waste suspension from 5% solids by volume to 40% solids by volume. The thickener has a diameter of 40 ft. A batch flux test in the laboratory for the settled height versus time was analyzed to give the data below for the solids flux versus solids volume fraction. Determine ... 

Thus, just as for incompressible single-phase flow, the pressure p constrains the velocity fields to ensure (in the case of multiphase flows) that the sum of the phase volume fractions equals unity. In the presence of mass transfer, the right-hand side of Eq. (148) is nonzero nevertheless, the role of the pressure is still the same. Finally, we should note that in gas-solid flows the maximum volume fraction of the solid phase is less than unity due to physical constraints (i.e., when particles are close packed there is still room for the gas phase so that 0solid-pressure term ps that becomes extremely large when ag approaches its minimum value (e.g., oc — 0.4). 

We can recommend the expression proposed by Thomas [20], valid for solid concentration up to the maximum volume fraction attainable ( 60 vol. %) ... 

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. 

Figure 10.8 The density of network junctions as a function of the volume fraction of paraffinic oil in EPDM/oil vulcanisates [74], The solid line represents the result of a linear regression analysis of the dependence (intercept = 453 5 mmol/kg slope = -6.2 0.0.3 mmol/kg the correlation coefficient = 0.996). Maximum torque in the rheometer curve for the vulcanisates is shown on the right ordinate...

The concept of the free volume of disperse systems can also be correlated with the change in the structure of the composite of the type solid particles — liquid — gas during its compaction. In that case the value of the maximum packing fraction of filler (p in Eq. (80b) remains valid also for systems containing air inclusions, and instead of the value of the volume fraction of filler, characteristic for a solid particles — liquid dispersion-system solid particles — liquid — gas should be substituted. This value can be calculated as follows the ratio of concentrations Cs x g/Cs, to the first approximation can be substituted by the ratio of the densities of uncompacted and compacted composites, i.e. by parameter Kp. Then Eq. (80b) in view of Eq. (88), for uncompacted composites acquires the form ... 

A similar study was conducted by Poslinski et al. (36) on the effect of a bimodal size distribution of solids. They confirmed the findings of Chong et al. (28) in that the relative shear viscosity can exhibit a minimum for a plot of relative viscosity versus volume percent of small particles in total solids. Moreover, the primary normal stress also exhibited a minimum. Poslinski et al. showed that the relative viscosity can be predicted from the knowledge of the maximum packing volume fraction of the bimodal solids systems. 

The effective volume fraction increases with a relative increase of the dispersant layer thickness. Even at 10% volume fraction, a maximum packing (< = 0.67) is soon reached, with an adsorbed layer thickness that is comparable to the particle radius. In this case, overlap of the steric layers wiU result in significant increases in viscosity. Such considerations may help to explain why solids loading can be severely Hmited, especially with small particles. In practice, soUds loading curves can be used to characterize the system, and take the form of those illustrated in Figure 11.6... 

Fig. 8 Fracture strengths, in terms of in-plane macrostress (triangles) and maximum "equivalent noni stress" in PSZ phase (circles), plotted against PSZ volume fraction. Solid and open markings refer to the cases with and without taking account of residual stresses.

Experience has shown structured fluids to be more difficult to manufacture, due to the complexity of their rheological profiles. In addition to elasticity, dilatancy, and rheopexy, certain structured fluid compositions may exhibit solid-like properties in the quiescent state and other flow anomalies under specific flow conditions. For emulsions and solid particulate dispersions, near the maximum packing volume fraction of the dispersed phase, for example, yield stresses may be excessive, severely limiting or prohibiting flow under gravity, demanding special consideration in nearly all unit operations. Such fluids pose problems in... 

The problem of packing a maximum volume of solids into a given space is common to numerous branches of physics and technology. It suffices to note that the relative viscosity of suspensions is a function of the reduced volume fraction, ()) = (()/ ( ), to realize the importance of < >. Experimentally, it was demonstrated that < > calculated from dry packing of solid particles agrees well with the value determined for a suspension. 

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