Bubble formation models

Part 1. Presentation of the model. Int J Heat Mass Transfer 47 3375-3385 Tiselj I, Hetsroni G, Mavko B, Mosyak A, Pogrebnyak E, Segal Z (2004) Effect of axial conduction on the heat transfer in micro-channels Int J Heat Mass Transfer 47 2551-2565 Triplett KA, Ghiaasiaan SM, Abdel-Khalik SI, Sadowski DL (1999) Gas-liquid two-phase flow in microchannels. Part I. Two-phase flow patterns. Int J Multiphase Flow 25 377-394 Tsai J-H, Lin L (2002) Transient thermal bubble formation on polysihcon micro-resisters. J Heat Transfer 124 375-382... 

The physical situation in a fluidized bed reactor is obviously too complicated to be modeled by an ideal plug flow reactor or an ideal stirred tank reactor although, under certain conditions, either of these ideal models may provide a fair representation of the behavior of a fluidized bed reactor. In other cases, the behavior of the system can be characterized as plug flow modified by longitudinal dispersion, and the unidimensional pseudo homogeneous model (Section 12.7.2.1) can be employed to describe the fluidized bed reactor. As an alternative, a cascade of CSTR s (Section 11.1.3.2) may be used to model the fluidized bed reactor. Unfortunately, none of these models provides an adequate representation of reaction behavior in fluidized beds, particularly when there is appreciable bubble formation within the bed. This situation arises mainly because a knowledge of the residence time distribution of the gas in the bed is insuf-... 

Bubble Dynamics. To adequately describe the jet, the bubble size generated by the jet needs to be studied. A substantial amount of gas leaks from the bubble, to the emulsion phase during bubble formation stage, particularly when the bed is less than minimally fluidized. A model developed on the basis of this mechanism predicted the experimental bubble diameter well when the experimental bubble frequency was used as an input. The experimentally observed bubble frequency is smaller by a factor of 3 to 5 than that calculated from the Davidson and Harrison model (1963), which assumed no net gas interchange between the bubble and the emulsion phase. This discrepancy is due primarily to the extensive bubble coalescence above the jet nozzle and the assumption that no gas leaks from the bubble phase. 

The computation performed in this study is based on the model equations developed in this study as presented in Sections II.A, III.A, III.B, and III.C These equations are incorporated into a 3-D hydrodynamic solver, CFDLIB, developed by the Los Alamos National Laboratory (Kashiwa et al., 1994). In what follows, simple cases including a single air bubble rising in water, and bubble formation from a single nozzle in bubble columns are first simulated. To verify the accuracy of the model, experiments are also conducted for these cases and the experimental results are compared with the simulation results. Simulations are performed to account for the bubble-rise phenomena in liquid solid suspensions with single nozzles. Finally, the interactive behavior between bubbles and solid particles is examined. The bubble formation and rise from multiple nozzles is simulated, and the limitation of the applicability of the models is discussed. 

All the above conclusions and the data reported will be discussed in detail later, when a general model is proposed for bubble formation under constant flow conditions. It will then be shown that each of the above observations is correct although the conclusions drawn from them are applicable only in limited range. 

Fig. 8. Idealized sequence of bubble formation in a liquid—Davidson and Schuler s model (D8).

Fig. 10. Idealized sequence of bubble formation—Kumar and Kuloor s model (K18).

Fig. 11. Comparison of Kumar and Kuloor s model (K18) with the data and model of Davidson and Schuler (D9) for bubble formation in inviscid liquids.

Fig. 13. Comparison of the model (K19) with the experimental data for bubble formation in viscous liquids without surface-tension effect.

There exists the maximum amount of contradiction regarding the role of viscosity in bubble formation. The present model indicates that for both the stages of expansion and of detachment—... 

Fio. 14. Comparison of the general model (Rl) with the data collected for bubble formation in viscous liquids. 

The study of bubble formation in non-Newtonian fluids has not been reported in literature in spite of the great industrial uses of these fluids. Recently, Subramaniyan and Kumar (S16) have studied bubble formation under constant flow conditions in fluids following the Ostwald-de-Waele rheological model. The model of Kumar and Kuloor (K16, K18, K19) has been extended to take into consideration the drag variation caused by the complexity of the rheological equation. 

This model is a modification of the model developed by Kumar and Kuloor (K18) for bubble formation in inviscid fluids in the absence of surface-tension effects. The need for modification arises because the bubble forming nozzles actually used to collect data on bubble formation in fluidized beds differ from the orifice plates in that they do not have a flat base. Under such conditions the bubble must be assumed to be moving in an infinite medium and the value of 1/2 is more justified than the value 11/16. 

Coughlin and Canevari (1%9) have published experimental data on two systems at a variety of operating conditions the extraction of xylene from polypropylene and the extraction of methanol from polypropylene. These studies were conducted in a single screw extruder at low pressures and w was assumed to be small in comparison with w. Coughlin and Canevari developed a model which they used in conjunction with their experimental data to obtain a value for the diffusion coefficient. The values that they computed were of the order of 10 mVsec, which obviously means that the model is incorrect. Coughlin and Canevari also computed values for the mass transfer coefficient and found it to be independent of screw speed. This observation is particularly noteworthy since they saw no evidence of bubble formation. 

Theoretical Models for Bubble Formation at a Submerged Orifice... 

In view of the complexity of the bubble formation process, it is not surprising that the models are successful only under restricted conditions. The simplest models, and the only ones to give simple analytic expressions for the volume of the bubble produced, apply for constant flow formation. All the models have inherent limitations ... 

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