Fraction of the Dispersed Phase

With an increase in dispersed-phase volume fraction, the viscosity of an emulsion increases. This increase in viscosity is linear at a low droplet concentration (McClements, 1999) the viscosity of an emulsion of milk fat globules in milk plasma increases linearly with fat content up to 30% (Bakshi and Smith, 1984 Kyazze and Starov, 2004), whereas the viscosity of low-fat milk ( 2.0% fat) increases in a near linear fashion with fat content (Phillips et al., 1995). However, above a certain volume fraction of the dispersed phase, the droplets in emulsions are packed so closely that flow is impaired, giving the emulsion a gel-like character (McClements, 1999). For instance, the viscosity of cream increases rapidly with increasing fat content when the fat content is 50% (Prentice, 1968 Mulder Walstra, 1974). 

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Holdup and Flooding. The volume fraction of the dispersed phase, commonly known as the holdup can be adjusted in a batch extractor by means of the relative volumes of each Hquid phase added. In a continuously operated weU-mixed tank, the holdup is also in proportion to the volume flow rates because the phases become intimately dispersed as soon as they enter the tank. 

Example. The viscosity of the continuous phase liquid is 20. The viscosity of the dispersed phase liquid is 30. The volume fraction of the dispersed phase liquid is 0.3. The nomograph shows the emulsion viscosity to be 36.2. 

Optical systems can be used in multiphase flows at a very low volume fraction of the dispersed phase. Through a refractory index matching of hquid-liquid or liquid-solid systems, it is also possible to measure at high void fractions. However, it is not possible to obtain complete refractory index matching since the molecules at the phase boundary have different optical properties than the molecules in the bulk. Consequently, it is possible to measure at a higher fraction of the dispersed phase with larger drops and particles because of the lower surface area per volume fluid. 

This response time should be compared to the turbulent eddy lifetime to estimate whether the drops will follow the turbulent flow. The timescale for the large turbulent eddies can be estimated from the turbulent kinetic energy k and the rate of dissipation e, Xc = 30-50 ms, for most chemical reactors. The Stokes number is an estimation of the effect of external flow on the particle movement, St = r /tc. If the Stokes number is above 1, the particles will have some random movement that increases the probability for coalescence. If St 1, the drops move with the turbulent eddies, and the rates of collisions and coalescence are very small. Coalescence will mainly be seen in shear layers at a high volume fraction of the dispersed phase. 

As a matter of fact, in comparison with the Euler-Lagrangian approach, the complete Eulerian (or Euler-Euler) approach may better comply with denser two-phase flows, i.e., with higher volume fractions of the dispersed phase, when tracking individual particles is no longer doable in view of the computational times involved and the computer memory required, and when the physical interactions become too dominating to be ignored. Under these circumstances, the motion of individual particles may be overlooked and it is wiser to opt for a more superficial strategy that, however, still has to take the proper physics into account. 

In contrast to the highly interconnected pores mentioned previously, closed pores can also be obtained by microemulsion polymerization if the initial volume fraction of the dispersed phase is kept lower than 30%. Recently two systems have been reported where the polymerization of the continuous phase and the subsequent removal of the Hquid dispersed phase resulted in the formation... 

In a subsequent theoretical analysis, Princen [26] initially used a model of infinitely long cylindrical drops to relate the geometric and thermodynamic properties of monodisperse HIPEs to the volume fraction of the dispersed phase. Thus the analysis could be restricted to a two-dimensional cross-section of the emulsion. Two principle emulsion parameters were considered the film thickness between adjacent drops (h) and the contact angle (0) [27-29]. The effects of these variables on the volume fraction, , both in the presence and absence of a compressive force on the emulsion, were considered. The results indicated that if both h and 0 are kept at zero, the maximum volume fraction () of the uncompressed emulsion is 0.9069, which is equivalent to = 0.7405 in real emulsions with spherical droplets (cf. Lissant s work). If 0 is zero (or constant) and h is increased, the maximum value of decreases on the other hand, increasing 0 with zero or constant h causes to increase above the value 0.9069, again at zero compression. This implies that, in the presence of an appreciable contact angle, without any applied compressive force, values of <(> in excess of the maximum value for undeformed droplets can occur. Thus, the dispersed phase... 

An important quantity, which characterizes a macroemulsion, is the volume fraction of the disperse phase 4>a (inner phase volume fraction). Intuitively one would assume that the volume fraction should be significantly below 50%. In reality much higher volume fractions are reached. If the inner phase consists of spherical drops all of the same size, then the maximal volume fraction is that of closed packed spheres (fa = 0.74). It is possible to prepare macroemulsions with even higher volume fractions volume fractions of more than 99% have been achieved. Such emulsions are also called high internal phase emulsions (HIPE). Two effects can occur. First, the droplet size distribution is usually inhomogeneous, so that small drops fill the free volume between large drops (see Fig. 12.9). Second, the drops can deform, so that in the end only a thin film of the continuous phase remains between neighboring droplets. 

Figure 12.9 An oil-in-water emulsion and a water-in-oil emulsion with low volume fractions of the dispersed phase d. In addition, a water-in-oil emulsion with high volume fraction is shown. Volume fractions above 0.74 can occur due to the polydispersity of the drops. Small drops can fill the spaces between large drops.

A mechanically strong and elastic interfacial film This is particularly important when the volume fraction of the dispersed phase is high. 

If the emulsion consisted of an assembly of closely packed uniform spherical droplets, the dispersed phase would occupy 0.74 of the total volume. Stable emulsions can, however, be prepared in which the volume fraction of the dispersed phase exceeds 0.74, because (a) the droplets are not of uniform size and can, therefore, be packed more densely, and (b) the droplets may be deformed into polyhedra, the interfacia] film preventing coalescence. 

Surface of separation. Where volume fraction of the dispersed phase is high so that the area of contact between contiguous particles of the second phase is unnegllgible the extent of the surface of separation between the second phase and the matrix has to be considered in measuring the degree of dispersion. For a two-phase composite the degree of separation dp is defined as ... 

Single phase microemulsions are treated in the next section. Two general thermodynamic equations are derived from the condition that the free energy of the system should be a minimum with respect to both the radius r of the globules as well as the volume fraction of the dispersed phase. The first equation can be employed to calculate the radius while the second, a generalized Laplace equation, can be used to explain the instability of the spherical shape of the globules. The two and three phase systems are examined in Sections III and IV of the paper. 

Here y is a generalized interfacial tension, Ci and C2 are bending stresses associated with the curvatures ci and eg, respectively A is the internal interfacial area per unit volume of microemulsion ui and ni are the chemical potentials per molecule and the number of molecules of species i, respectively 0 is the volume fraction of the dispersed phase and P2 and pi are the pressures inside the globules and in the continuous phase in the space between the globules. Here the actual physical surface of the globule (to the extent to which it can be defined) of radius r is selected as the Gibbs dividing surface. 

Colloidal interactions between emulsion droplets play a primary role in determining emulsion rheology. If attractions predominate over repulsive forces, flocculation can occur, which leads to an increase in the effective volume fraction of the dispersed phase and thus increases viscosity (McCle-ments, 1999). Clustering of milk fat globules due to cold agglutination increases the effective volume fraction of the milk fat globules, thereby increasing viscosity (Prentice, 1992). 

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