Fractionation models dispersion

A comparison of the dependence Af / As vs. (p(Af is film area, (p is volume fraction of dispersed phase) reported by Princen for emulsions, with the data obtained for a dodecahedral model [83] is given in Fig. 1.12. The figure shows that these dependences coincide within the whole range of expansion ratios studied. 

With this approach, even the dispersed phase is treated as a continuum. All phases share the domain and may interpenetrate as they move within it. This approach is more suitable for modeling dispersed multiphase systems with a significant volume fraction of dispersed phase (> 10%). Such situations may occur in many types of reactor, for example, in fluidized bed reactors, bubble column reactors and multiphase stirred reactors. It is possible to represent coupling between different phases by developing suitable interphase transport models. It is, however, difficult to handle complex phenomena at particle level (such as change in size due to reactions/evaporation etc.) with the Eulerian-Eulerian approach. 

In Eulerian-Eulerian (EE) simulations, an effective reaction source term of the form of Eq. (5.32) can be used in species conservation equations for all the participating species. The above comments related to models for local enhancement factors are applicable to the EE approach as well. It must be noted that interfacial area appearing in Eq. (5.32) will be a function of volume fraction of dispersed phase and effective particle diameter. It can be imagined that for turbulent flows, the time-averaged mass transfer source will have additional terms such as correlation of fluctuations in volume fraction of dispersed phase and fluctuations in concentration even in the absence... 

With a Eulerian-Lagrangian approach, processes occurring at the particle surface can be modeled when simulating particle trajectories (for example, the process of dissolution or evaporation can be simulated). However, as the volume fraction of dispersed phase increases, the Eulerian-Lagrangian approach becomes increasingly computation intensive. A Eulerian-Eulerian approach more efficiently simulate such dispersed multiphase flows. 

Interfacial tension MOSCED polarity parameter NRTL model parameter Volume fraction Volume fraction of dispersed phase (holdup)... 

As mentioned above, macroscale models are written in terms of transport equations for the lower-order moments of the NDF. The different types of moments will be discussed in Chapters 2 and 4. However, the lower-order moments that usually appear in macroscale models for monodisperse particles are the disperse-phase volume fraction, the disperse-phase mean velocity, and the disperse-phase granular temperature. When the particles are polydisperse, a description of the PSD requires (at a minimum) the mean and standard deviation of the particle size, or in other words the first three moments of the PSD. However, a more complete description of the PSD will require a larger set of particle-size moments. 

The dynamic behavior of polymer blends under low strain has been theoretically treated from the perspective of microrheology. Table 2.3 lists a summary of this approach. These models well describe the experimental data within the range of stresses and concentrations where neither drop-breakup nor coalescence takes place. The two latter models yield similar predictions as that of Palierne. The last entry in the Table 2.3 is an empirical modification of Palieme s model by replacement of the volume fraction of dispersed phase by its efiective quantity (Eq. (2.24)), which extends the applicability of the relation up to 0 < 0.449. However, at these high concentrations the drop-drop interactions absent in the Palierne model must complicate the flow and coalescence is expected. The practical solution to the latter problem is compatibilization, but the presence of the third component in blends has not been treated theoretically. 

Fig. 6. Fraction of interface molecules per particle and fraction of interfacial free energy in dependence on average particle size for the model dispersion.

Plots are provided in the Eee-Paul discussion of YJYp as a function of ui/ui, volume fraction of dispersed phase, and aija. The sensitivity of YJYp to changes in the ellipse morphology indicates that the ellipse provides a more flexible model than the disk model for adaptation of a model to experimental results. 

Despite the fact Chat there are no analogs of void fraction or pore size in the model, by varying the proportion of dust particles dispersed among the gas molecules it is possible to move from a situation where most momentum transfer occurs in collisions between pairs of gas molecules, Co one where the principal momentum transfer is between gas molecules and the dust. Thus one might hope to obtain at least a physically reasonable form for the flux relations, over the whole range from bulk diffusion to Knudsen streaming. 

When viscometric measurements of ECH homopolymer fractions were obtained in benzene, the nonperturbed dimensions and the steric hindrance parameter were calculated (24). Erom experimental data collected on polymer solubiUty in 39 solvents and intrinsic viscosity measurements in 19 solvents, Hansen (30) model parameters, 5 and 5 could be deterrnined (24). The notation 5 symbolizes the dispersion forces or nonpolar interactions 5 a representation of the sum of 8 (polar interactions) and 8 (hydrogen bonding interactions). The homopolymer is soluble in solvents that have solubility parameters 6 > 7.9, 6 > 5.5, and 0.2 < <5.0 (31). SolubiUty was also determined using a method (32) in which 8 represents the solubiUty parameter... 

CASRAM predicts discharge fractions, flash-entrainment quantities, and liquid pool evaporation rates used as input to the model s dispersion algorithm to estimate chemical hazard population exposure zones. The output of CASRAM is a deterministic estimate of the hazard zone (to estimate an associated population health risk value) or the probability distributions of hazard-zones (which is used to estimate an associated distribution population health risk). 

In the proposed model, the total volume of the dispersion (or of the reactor) is subdivided into many subvessels (or subreactors). In each, Gal-Or and Hoelscher considered a fraction of the total number of bubbles having a radius between a and a + da. The distribution of bubbles in the subreactors... 

Example 15.10 Use residence time theory to predict the fraction unreacted for a closed reactor governed by the axial dispersion model. 

Given k fit) for nny reactor, you automatically have an expression for the fraction unreacted for a first-order reaction with rate constant k. Alternatively, given ttoutik), you also know the Laplace transform of the differential distribution of residence time (e.g., k[f(t)] = exp(—t/t) for a PER). This fact resolves what was long a mystery in chemical engineering science. What is f i) for an open system governed by the axial dispersion model Chapter 9 shows that the conversion in an open system is identical to that of a closed system. Thus, the residence time distributions must be the same. It cannot be directly measured in an open system because time spent outside the system boundaries does not count as residence but does affect the tracer measurements. 

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