Fluid systems dispersion model

Dispersion modeling equations for water systems take the same form as those presented later in this chapter for the atmosphere. Analytical solutions tire not nearly as complicated or difficult, since the bulk motion of the fluid (in this case, wtiicr) is a weak vtiriablc with respect to m.ignitude, direction, lime, and position as it is when the fluid is air. 

The CD model was first proposed by Curl (1963) to describe coalescence and breakage of a dispersed two-fluid system. In each mixing event, two fluid particles with distinct compositions first coalesce and then disperse with identical compositions.75 Written in terms of the two compositions (f>A and [Pg.292]

Equation (I-l) is the general representation of the dispersion model. The dispersion coefficient is a function of both the fluid properties and the flow situation the former have a major effect at low flow rates, but almost none at high rates. In this general representation, the dispersion coefficient and the fluid velocity are all functions of position. The dispersion coefficient, D, is also in general nonisotropic. In other words, it has different values in different directions. Thus, the coefficient may be represented by a second-order tensor, and if the principal axes are taken to correspond with the coordinate system, the tensor will consist of only diagonal elements. 

The two-phase theory of fluidization has been extensively used to describe fluidization (e.g., see Kunii and Levenspiel, Fluidization Engineering, 2d ed., Wiley, 1990). The fluidized bed is assumed to contain a bubble and an emulsion phase. The bubble phase may be modeled by a plug flow (or dispersion) model, and the emulsion phase is assumed to be well mixed and may be modeled as a CSTR. Correlations for the size of the bubbles and the heat and mass transport from the bubbles to the emulsion phase are available in Sec. 17 of this Handbook and in textbooks on the subject. Davidson and Harrison (Fluidization, 2d ed., Academic Press, 1985), Geldart (Gas Fluidization Technology, Wiley, 1986), Kunii and Levenspiel (Fluidization Engineering, Wiley, 1969), and Zenz (Fluidization and Fluid-Particle Systems, Pemm-Corp Publications, 1989) are good reference books. 

The dimensionless term (9/u0 L, where 9 is the axial dispersion coefficient, u0 is the superficial fluid velocity, and L is the expanded-bed height) is the column-vessel dispersion number, Tc, and is the inverse of the Peclet number of the system. Two limiting cases can be identified from the axial dispersion model. First, when 9/u0L - 0, no axial dispersion occurs, while when 9/u0 L - 00 an infinite diffusivity is obtained and a stirred tank performance is achieved. The dimensionless term Fc, can thus be utilized as an important indicator of the flow characteristics within a fluidized-bed system.446... 

So far, only the axial dispersion model has been used for scaleup purposes. Very little knowledge on the effects of reactor configuration and flow conditions on the parameters of more complex macromixing models (e.g., the two-parameters model, etc.) is available. Since these complex models are more realistic, more information on the relation between their parameters and the system conditions, such as packing size, fluid properties, and flow rates, needs to be obtained. At present, complex models are not very useful for scaleup purposes. 

The mesoscale models for momentum transfer between phases differ quite substantially depending on the multiphase system under investigation, and different semi-empirical relationships have been developed for different systems. Since the nature of the disperse phase is particularly important, the available mesoscale models are generally divided into those valid for fluid-fluid and those valid for fluid-solid systems. The main difference is that in fluid-fluid systems the elements of the disperse phase are deformable particles (i.e. bubbles or droplets), whereas in fluid-solid systems the disperse phase is constituted by particles of constant shape. Typical fluid-fluid systems for which the mesoscale models reported below apply are gas-liquid, liquid-liquid, and liquid-gas systems. The mesoscale models reported for fluid-solid systems are valid both for gas-solid and for liquid-solid systems. As a general rule, the mesoscale model for Afp should be derived starting from a single-particle momentum balance ... 

S.D. Kolev, E. Pungor, Description of an axially-dispersed plug flow model for the flow pattern in elements of fluid systems, Anal. Chim. Acta 185 (1986) 315. 

FIGURE 26.31 The particle system, which models the dispersion of a cubic colloidal slab positioned at the interface of counter flow sketched in (a). The flows are accelerated in both directions. The colloidal slab is made of colloidal particles (approximately 10 particles) and fluid consists of 10 DPD particles (invisible). The shade of gray of colloidal particle represents particle velocity. The dark gray and black colors indicate the largest velocity of particles. Figures (b) and (c) represent the projection of the colloidal particles after 2000 timesteps on x-y and x-z plains, respectively. In figure (d) the break-up instant (after 3000 timesteps) is displayed. 

Performing the review as intended under item 3 is a more problematic task. Competence to judge scientific quality requires a sound expert knowledge of dense gas dispersion as a specific scientific domain linked to fluid dynamics, thermodynamics, and meteorology. Generally accepted scientific principles are based on verification and falsification of theories and assumptions. However, nowadays dense gas dispersion models are complex systems that cannot be simply verified or falsified. Scientific assessment requires that the following aspects be considered carefully ... 

Repulsive interactions are important when molecules are close to each other. They result from the overlap of electrons when atoms approach one another. As molecules move very close to each other the potential energy rises steeply, due partly to repulsive interactions between electrons, but also due to forces with a quantum mechanical origin in the Pauli exclusion principle. Repulsive interactions effectively correspond to steric or excluded volume interactions. Because a molecule cannot come into contact with other molecules, it effectively excludes volume to these other molecules. The simplest model for an excluded volume interaction is the hard sphere model. The hard sphere model has direct application to one class of soft materials, namely sterically stabilized colloidal dispersions. These are described in Section 3.6. It is also used as a reference system for modelling the behaviour of simple fluids. The hard sphere potential, V(r), has a particularly simple form ... 

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