Disperse-phase volume transport

In order to derive a transport equation of the disperse-phase volume fraction ap, we will let the first internal coordinate pi be equal to the particle volume (Vp). The disperse-phase volume fraction is then defined by 

Starting from Eq. (4.39) with g = the transport equation for o-p is where the volume-average disperse-phase velocity is defined by 

For the case in which all particles have the same volume, Uv = Un. Note that, like Un, the volume-average disperse-phase velocity will not usually be in closed form. The particle-volume source terms are defined by 

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Electrical conductivity is an easily measured transport property, and percolation in electrical conductivity appears a sensitive probe for characterizing microstructural transformations. A variety of field (intensive) variables have been found to drive percolation in reverse microemulsions. Disperse phase volume fraction has been often reported as a driver of percolation in electrical conductivity in such microemulsions [17-20]. 

Lagues et al. [17] found that the percolation theory for hard spheres could be used to describe dramatic increases in electrical conductivity in reverse microemulsions as the volume fraction of water was increased. They also showed how certain scaling theoretical tools were applicable to the analysis of such percolation phenomena. Cazabat et al. [18] also examined percolation in reverse microemulsions with increasing disperse phase volume fraction. They reasoned the percolation came about as a result of formation of clusters of reverse microemulsion droplets. They envisioned increased transport as arising from a transformation of linear droplet clusters to tubular microstructures, to form wormlike reverse microemulsion tubules. 

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 transport equations appearing in macroscale models can be derived from the kinetic equation using the definition of the moment of interest. For example, if the moment of interest is the disperse-phase volume fraction, then it suffices to integrate over the mesoscale variables. (See Section 4.3 for a detailed discussion of this process.) Using the velocity-distribution function from Section 1.2.2 as an example, this process yields... 

Thus, Eq. (1.25) reduces to the continuity equation for the disperse-phase volume fraction. However, the disperse-phase mean velocity Up is unknown, and must be found by solving a separate transport equation. 

Figure 4.1(e) depicts a process called Ostwald ripening where the dispersed phase is transported through the continuous phase from smaller droplets to larger ones. Since larger droplets have a lower surface to volume ratio than their smaller counterparts, this process occurs with a net reduction in interfacial energy. However, the dispersed phase must be significantly soluble (or solubilised by surfactant) in the continuous phase. 

It is desirable to calculate new bulk phase Z values for the four primary media which include the contribution of dispersed phases within each medium as described by Mackay and Paterson (1991) and as listed earlier. The air is now treated as an air-aerosol mixture, water as water plus suspended particles and fish, soil as solids, air and water, and sediment as solids and porewater. The Z values thus differ from the Level I and Level II pure phase values. The necessity of introducing this complication arises from the fact that much of the intermedia transport of the chemicals occurs in association with the movement of chemical in these dispersed phases. To accommodate this change the same volumes of the soil solids and sediment solids are retained, but the total phase volumes are increased. These Level III volumes are also given in Table 1.5.2. The reaction and advection D values employ the generally smaller bulk phase Z values but the same residence times thus the G values are increased and the D values are generally larger. 

The rate constant k0 for orthokinetic coagulation is determined by physical parameters (velocity gradient du/dz, floe volume ratio of the dispersed phase, = sum over the product of particle number and volume), and the collision efficiency factor a0 observed under orthokinetic transport conditions ... 

The population balance simulator has been developed for three-dimensional porous media. It is based on the integrated experimental and theoretical studies of the Shell group (38,39,41,74,75). As described above, experiments have shown that dispersion mobility is dominated by droplet size and that droplet sizes in turn are sensitive to flow through porous media. Hence, the Shell model seeks to incorporate all mechanisms of formation, division, destruction, and transport of lamellae to obtain the steady-state distribution of droplet sizes for the dispersed phase when the various "forward and backward mechanisms become balanced. For incorporation in a reservoir simulator, the resulting equations are coupled to the flow equations found in a conventional simulator by means of the mobility in Darcy s Law. A simplified one-dimensional transient solution to the bubble population balance equations for capillary snap-off was presented and experimentally verified earlier. Patzek s chapter (Chapter 16) generalizes and extends this method to obtain the population balance averaged over the volume of mobile and stationary dispersions. The resulting equations are reduced by a series expansion to a simplified form for direct incorporation into reservoir simulators. 

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. 

As in all mathematical descriptions of transport phenomena, the theory of polydisperse multiphase flows introduces a set of dimensionless numbers that are pertinent in describing the behavior of the flow. Depending on the complexity of the flow (e.g. variations in physical properties due to chemical reactions, collisions, etc.), the set of dimensionless numbers can be quite large. (Details on the physical models for momentum exchange are given in Chapter 5.) As will be described in detail in Chapter 4, a kinetic equation can be derived for the number-density function (NDF) of the velocity of the disperse phase n t, X, v). Also in this example, for clarity, we will assume that the problem has only one particle velocity component v and is one-dimensional in physical space with coordinate x at time t. Furthermore, we will assume that the NDF has been normalized (by multiplying it by the volume of a particle) such that the first three velocity moments are... 

Consider again a system wherein all particles have the same volume and mass, and the disperse-phase momentum density is gp = ppOp. The transport equation for the disperse-phase momentum density for this case is ... 

The most frequent disposal of most of the polymer-based heterogeneous materials family takes place by the dispersed phase/matrix mode. So the dispersed phase components may be identified by the finite size of each of their domains, being surrounded by the continuous matrix. Both the size and the geometry of the particles featuring the dispersed phase together with their surface properties govern the transport phenomenon across the interphase between the dispersed particles and the continuous matrix. According to the interface approach defined in the previous section, it is obvious that the domain size and its distribution confine the interfacial volume available for effective transport flows between the matrix and the disperse phase. 

One of the most important cases is when there are two (or more) distinct regions within the reactor. This might be a packed bed of porous solids, two fluid phases, partially stagnant regions, or other complicated flows through a vessel that can basically be described by an axial dispersion type model. Transport balances can be made for each phase, per unit reactor volume ... 

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