Hydrodynamic model bubbles

Urrs, ULTB> aLs and otTB can be found from the hydrodynamic model, and CfTP is calculated in the same manner as discussed previously for bubbly flow. 

The effect of the bulk solution temperature lies primarily in its influence on the bubble content before collapse. With increasing temperature, in general, sonochemical reaction rates are slower. This reflects the dramatic influence which solvent vapor pressure has on the cavitation event the greater the solvent vapor pressure found within a bubble prior to collapse, the less effective the collapse. In fact, one can quantitate this relationship rather well (89). From simple hydrodynamic models of the cavitation process, Neppiras, for example, derives (26) the peak temperature generated during collapse of a gas-filled cavity as... 

A hydrodynamic model of fluidization attempts to account for several essential features of fluidization mixing and distribution of solids and fluid in a so-called emulsion region, the formation and motion of bubbles through the bed (the bubble region ), the nature of the bubbles (including their size) and how they affect particle motion/distribution, and the exchange of material between the bubbles (with little solid content) and the predominantly solid emulsion. Models fall into one of three classes (Yates, 1983, pp. 74-78) ... 

In all cases the underlying rationale for these hydrodynamic models rests on the observation that beds with identical solids and gas flow rates may develop either large bubbles or small bubbles depending on bed diameter, distributor design, baffle arrangement, etc. thus, bubble size must enter as the primary parameter in the model. A consequence of this argument is that models which do not allow for different bubble sizes at given imposed bed conditions certainly cannot be adequate. 

In the narrow tubes used by Beek and van Heuven, the bubbles assumed the shape of Dumitrescu (or Taylor) bubbles. Using the hydrodynamics of bubble rise and the penetration theory of absorption, an expression was developed for the total absorption rate from one bubble. The liquid surface velocity was assumed to be that of free fall, and the bubble surface area was approximated by a spherical section and a hyperbola of revolution. Values calculated from this model were 30% above the measured absorption rates. Further experiments indicated that velocities are reduced at the rear of the bubble, and are certainly much less than free fall velocities. A reduction in surface tension was also indicated by extreme curvature at the rear of the bubble. 

The following analysis holds for Type B fluidization and for Type A bubbling fluidization, when the region of particulate fluidization is so small that it can be ignored. In the framework of the two-phase model (see the subsection Hydrodynamic modeling of bubbling fluidization), the bed expansion in terms of the fraction of the bed occupied by bubbles is... 

Hydrodynamic modeling of bubbling fluidization (type B fluidization)... 

The expansion ratio profile of a continuously generated foam has been computed using various hydrodynamic models [87,88] but here again several significant simplifications are introduced. For example, a model of polyhedral bubbles was employed for all foam layers situated at different levels which, however, is not the real state in the lower foam layers. 

The first term in each case arises from bulk flow of gas into the floor of an isolated bubble and out the roof, as required by the hydrodynamic model of Davidson and Harrison (27). The weight of experimental evidence, from studies of cloud size (28,29), from chemical reaction studies (e.g. 30), and from interphase transfer studies (e.g. 31,32), is that this term is better described by the theory proposed by Murray (33). The latter leads to a reduction in the first term by a factor of 3. Some enhancement of the bulk flow component occurs for interacting bubbles (34,35), but this enhancement for a freely bubbling bed is only of the order of 20-30% (35), not the 300% that would be required for the bulk flow term Equations (1) and (2) to be valid. 

These developments led to a novel type of reactor models, the hydrodynamic models, in which the bed behavior was based on the characteristics of the rising bubbles. Several models of this kind were derived for the industrial relevant fine particle suspensions in which the rising bubbles are surrounded by very thin clouds of circulating gas. Various combinations of assumptions have been used to represent the phase interaction phenomena in this region [81] ... 

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