Random coalescence-dispersion

Models reported in the literature have tended to focus on one of two parts of the problem. Simple reaction models (e.g reacting slabs, random coalescence-dispersion, or multienvironment models) are designed to fit overall reaction data. More complex theoretical approaches require a model of the turbulence (see Section 2-5). The problem here is the adequacy of the turbulence model over a wide range of flow conditions. There still is no theory that takes into account structural aspects of turbulence with or without superimposed chemical reaction, although steady progress is being made in this direction [some cnrrent approaches are discussed by Fox (1998)]. 

Here, (jf n) indicates (p value on the nth particle. In the coalescence/dispersion model proposed by Curl (1%3), the state of/at f + Zif is calculated from the collision frequency co. The compositions of a pair of particles, which are selected at random (denoted by wi and W2), change as ... 

Equation (17) indicates that the entire distribution may be determined if one parameter, av, is known as a function of the physical properties of the system and the operating variables. It is constant for a particular system under constant operating conditions. This equation has been checked in a batch system of hydrosols coagulating in Brownian motion, where a changes with time due to coalescence and breakup of particles, and in a liquid-liquid dispersion, in which av is not a function of time (B4, G5). The agreement in both cases is good. The deviation in Fig. 2 probably results from the distortion of the bubbles from spherical shape and a departure from random collisions, coalescence, and breakup of bubbles. 

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. 

The amorphous phase is not usually a desirable state for the API because the formation process is more random and difficult to control than a crystallization. A second dispersed liquid phase is usually formed just prior to freezing and may coalesce or disperse under the influence of hydrodynamic forces in the crystallizer, making the process sensitive to micro-mixing effects on scale up. Amorphous solids also have significantly lower thermodynamic stability than related crystalline material and may subsequently crystallize during formulation and storage. Because of the non-uniformity of the amorphous solid it can more easily incorporate molecules other than the API, making purification less effective. 

A simplified homogeneous dispersed-phase mixing model was proposed by Curl (C16). Uniform drops are assumed, coalescence occurs at random and redispersion occurs immediately to yield equal-size drops of the same concentration, and the dispersion is assumed to be homogeneous. Irreversible reaction of general order s was assumed to occur in the drops. The population balance equations of total number over species concentration in the drop were derived for the discrete and continuous cases for a continuous-fiow well-mixed vessel. The population balance equation could be obtained from Eq. (102) by taking the internal coordinate to be drop concentration and writing the population balance equation in terms of number to yield... 

Incorporation of Copolymers (Nonreactive Compatibilization). Block or graft copolymers with segments that are miscible with their respective polymer components show a tendency to be localized at the interface between immiscible blend phases. The copolymers anchor their segments in the relevant polymer, reducing interfacial tension and stabilizing dispersion against coalescence (24-52). Random copolymers, sometimes also used as compatibilizers, reduce interfacial tension, but their ability to stabilize the phase structure is limited... 

We have explored the possibility that deactivation of polydimethylsiloxane-hydrophobed silica antifoams by disproportionation is a consequence of the random distribution of particles across the drops formed when the antifoam is dispersed to achieve a steady state by processes of drop splitting and coalescence. Simple mass balance considerations permit the estimation of the composition of deactivated antifoam dispersions if we assume they are made up of particle-free drops together with particle-rich drops and agglomerates of known silica content From that composition, it is possible to calculate the ratio of the number of particles to the number of drops, NM, in a deactivated antifoam if the distribution of particles across drops is assumed to be random. This analysis reveals that for monodisperse particles and drops where NIM < 1 this necessarily implies a proportion of particle-free drops irrespective of the nature of any distribution If, on the other hand, NIM > 1, then the probability of finding particle-free drops is always vanishingly small. 

Viewed in a simplified way, microemulsions form when the amphiphile adsorbs at the oil-water interface to lower the interfacial tension between two immiscible liquids (e.g., oil and water) to ultralow values (often <0.001 dynes/cm) allowing random thermal motions to spontaneously disperse the two immiscible liquids into each other (23). Unlike conventional emulsions, microemulsion domains may fluctuate in size and shape and undergo spontaneous coalescence and breakup (24, 25). Microemulsions typically contain discrete domain microstmcture that can be described as a continuum of possible topologies ranging from watef continuous, to bicontinuous, and oil continuous (recognizing that the water and oil subphases may contain small amounts of additional dissolved components). These subphase domains typically range in size from 100 to 1000 A and are illustrated qualitatively in Fig. 16.1. 

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