Reynolds number bubble

The results of Massimilla et al., 0stergaard, and Adlington and Thompson are in substantial agreement on the fact that gas-liquid fluidized beds are characterized by higher rates of bubble coalescence and, as a consequence, lower gas-liquid interfacial areas than those observed in equivalent gas-liquid systems with no solid particles present. This supports the observations of gas absorption rate by Massimilla et al. It may be assumed that the absorption rate depends upon the interfacial area, the gas residence-time, and a mass-transfer coefficient. The last of these factors is probably higher in a gas-liquid fluidized bed because the bubble Reynolds number is higher, but the interfacial area is lower and the gas residence-time is also lower, as will be further discussed in Section V,E,3. 

The results reported for beds of small particles (1 mm diameter and less) are in substantial agreement on the fact that the presence of solid particles tends to decrease the gas holdup and, as a consequence, the gas residencetime. This fact may also support the observations of gas absorption rate by Massimilla et al. (Section V,E,1) if it is assumed that a decrease of absorption rate caused by a decrease of residence time outweighs the increase of absorption rate caused by increase of mass-transfer coefficient arising from the increase in bubble Reynolds number. These results on gas holdup are in... 

The boiling Reynolds number or bubble Reynolds number (Re,) is defined as the ratio of the bubble inertial force to the liquid viscous force, which indicates the intensity of liquid agitation induced by the bubble motion ... 

The rise velocity, Uo, in general depends on the bubble size, or the bubble Reynolds number but as bubble size increases, as in two-phase upflow, Ua approaches an asymptotic value that is independent of Reynolds number. The following expressions have been accepted for a single bubble rising in an infinite medium, and for one rising in a swarm of surrounding bubbles, respectively (Duckler and Taitel, 1991b) ... 

Fd was evaluated from the drag coefficient at bubble Reynolds number and the projected area of the bubble. As the Reynolds number varied from 2 to 700, the drag coefficient CD was evaluated by the Schiller and Naumann (S4) equation ... 

Fig. 8.1 Wake angle 0 for spherical-cap bubbles as a function of bubble Reynolds number.

The rise velocity of a single spherical cap bubble in an infinite liquid medium can be described by the Davies and Taylor equation [Davies and Taylor, 1950] (Problem 9.6). Experimental results indicate that the Davies and Taylor equation is valid for large bubbles (4oo > 0.02 m, in general) with bubble Reynolds numbers greater than 40, while for bubbles in fluidized beds, the bubble Reynolds numbers are typically on the order of 10 or less [Clift, 1986]. By analogy, the rise velocity of an isolated single spherical cap bubble in an infinite gas-solid medium can be expressed in terms of the volume bubble diameter by [Davidson and Harrison, 1963]... 

If special precautions are taken to avoid contamination of the bubble surface, particle deposition by diffusion to the water surface is enhtinced by internal circulation. The internal flow can be calculated for very low bubble Reynolds numbers in the creeping flow approximation (Lamb, 1953). A solution ha.s been obtained to the equation of convective... 

Grashof number D -iPAPZ ) dimensionless number of sparger orifices or sites power number P JpO jN ), dimensionless bubble Reynolds number D fjiVdimensionless impeller Reynolds number (Dj Npjp, dimensionless Schmidt number (Pc/Pc ab) dimensionless Sherwood number (kiDy /DAB), dimensionless total pressure, N/m ... 

The gas-bubble Reynolds number must be calculated with the slip velocity Rec =... 

The spherical-cap bubbles observed in fluidized beds can best be characterized by the included angle as shown in Fig. 9. The angle was found to be dependent only on the bubble Reynolds number, as shown in Eq. (71)... 

Re = bubble Reynolds number based on = particle volume... 

Small bubbles in polar liquids can be treated as rigid spheres because of the effects of surfactants. Provided that the bubble Reynolds number is 0(1) or smaller, one can use the Maxey-Riley equation, which is given in Eq. (14). However, this may be a very restrictive assumption. For example, based on its rise velocity, a 120-/im bubble in water has a Reynolds number roughly equal to unity. It reasonable to assume that bubbles as large as 1 mm are spherical in water. Using Ryskin and Leal s [94,95,96] finite difference methods, McLaughlin [68] found that the axis ratio of a freely rising 1-mm bubble in pure water (i.e., a mobile interface) was 1.12. However, the axis ratio of a fully contaminated 1 -mm bubble was 1.01. In the latter case, the Reynolds number based on the rise velocity and the equivalent spherical diameter was 110. This... 

Hetsroni [37] reported that turbulence is generated in the wake of bubbles when the bubble Reynolds number Rcb is higher than 400. According to Kawakami et al. [24], the mean bubble diameter is 1.5 times as large as Lb, provided that the bubbles are spherical in shape, and thus the bubble Reynolds number Re is defined as... 

The discrepancy between the measurement and (6.42) for Vs = lOcSt can be explained as follows According to Hetsroni [18], the wake of a bubble becomes turbulent when the bubble Reynolds number R, defined in terms of the relative velocity between bubbles and molten slag, mean bubble diameter and the... 

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