Bubble Formation under Constant Flow Conditions

Bubble Formation under Constant Flow Conditions 

A general model applicable to the formation of bubbles under all kinds of operating conditions is not yet available. The complex nature of the problem may very well delay the emergence of such a theory. Even under constant 

The Influence of Liquid Properties on Bubble Volume as Reported by Various Investigators0 

Davidson and Schuler (D8) Positive (large) None— constant flow Positive— constant pressure Negative 

Siemes and Kaufmann (SI 1) Positive (large) None None 

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IV. Bubble Formation under Constant Flow Conditions. 277... 

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. 

These authors have extended the concepts developed by Kumar and Kuloor (K16, K18, K19) for bubble formation under constant flow conditions to the situations of constant pressure conditions. 

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. 

Fig. 20. Effect of orientation of orifice on bubble formation under constant flow conditions.

The drop formation is considered to proceed exactly in the same fashion as the bubble formation under constant flow conditions, viz. the two step (the expansion and detachment) mechanism. The tensile force does not arise in the expansion stage because there is no neck formation. 

Equations (156) and (158) are applicable to bubble formation under constant flow conditions because all the pertinent equations can be obtained from Eqs. (156) and (158). 

Fig. 12.1 Variation of dimensionless bubble volume, V = VgAp/d a, with dimensionless gas flow rate Q = for bubble formation under constant flow conditions predictions of...

Ramakrishnan S, Kumar R, Kuloor NR (1969) Studies in bubble formation -I Bubble formation under constant flow conditions. Chem Eng Sci 24(4) 731-747. Ranade VV (2002) Computational Flow Modeling for Chemical Reactor Engineering, Academic Press San Diego... 

Fio. 3. Equipment for bubble formation under constant flow and constant pressure conditions. 

Equation (54) is quite general in nature, and can be used for the evaluation of Vfb in any situation when the bubble formation is taking place under constant flow conditions. Thus, if the last term on the right-hand side of Eq. (54) is omitted, the equation reduces to that of the viscous case without surface-tension effects. Similarly, if the second term on the right-hand side is dropped, the resulting equation is the one for the inviscid case with surface tension. Further, if both the second and the third terms on the right-hand side are deleted, the equation reduces to that of the inviscid case without surface tension. 

Fig. 17. Comparison of the model (S16) with the collected data for bubble formation in non-Newtonian liquids under constant flow conditions.

From the above discussion, it is seen that the bed acts as an inviscid fluid without surface tension, and that the bubble formation takes place under constant flow conditions. Thus, the theories which are applicable to the above conditions should also be applicable to bubble formation in fluidized beds. 

The first quantitative attempt (K12) in this direction was made with vertical orifices, under constant flow conditions. Here, the bubble formation is considered to be occurring in two distinct steps. In the first stage, the bubble is assumed to expand at the tip, moving vertically at the same time. As the bubble is formed at an angle to the vertical, a vertical component of the surface tension force will be operative during this stage. The first stage is... 

For many purposes, approximate predictions suffice, and may be obtained from the results for constant flow formation using some simple guidelines. Bubbles obtained under constant pressure tend to be larger than under constant flow conditions at the same time-mean flow rate, Q, because most of the flow with variable Q occurs during the latter stages of formation. It is convenient to define a ratio of bubble volumes formed under constant pressure and constant flow conditions as... 

The effect of pressure on the initial bubble size under various bubble formation conditions is shown in Fig. 3. The solid lines in the figures represent the model predictions. Under constant flow conditions (Yc < 1), the pressure effect on the initial bubble size is not significant however, under variable flow conditions Nc > 1), pressure has a significant effect on the... 

A common dimensionless number used to characterize the bubble formation from orifices through a gas chamber is the capacitance number defined as Nc — 4VcgpilnDoPs. For the bubble-formation system with inlet gas provided by nozzle tubes connected to an air compressor, the volume of the gas chamber is negligible, and thus, the dimensionless capacitance number is close to zero. The gas-flow rate through the nozzle would be near constant. For bubble formation under the constant flow rate condition, an increasing flow rate significantly increases the frequency of bubble formation. The initial bubble size also increases with an increase in the flow rate. Experimental results are shown in Fig. 6. Three different nozzle-inlet velocities are used in the air-water experiments. It is clearly seen that at all velocities used for nozzle air injection, bubbles rise in a zigzag path and a spiral motion of the bubbles prevails in air-water experiments. The simulation results on bubble formation and rise behavior conducted in this study closely resemble the experimental results. 

The resistance offered by the disk is so large that the bubble formation takes place under essentially constant flow conditions. 

For bubble formation from a single orifice without a gas chamber, the motion equation of the rising bubble itself is sufficient to predict the initial bubble size. For the case in which the orifice is connected to a gas chamber, the gas flow rate through the orifice is not constant and depends on pressure fluctuations in both the chamber and the bubble. In order to simulate bubble formation under such conditions, the pressure fluctuations in the gas chamber and in the bubble must be considered to account for the time-variant orifice gas flow rate as illustrated below. 

The bed depth has no influence on the size of the bubble produced. This indicates that the bubbles are foxmed under either constant flow or constant pressure conditions. In the intermediate region, Padmavathy, Kumar, and Kuloor (PI) have shown that the bubble volume in an air-water system is highly sensitive to the variation in the depth of the liquid column above the bubble forming nozzle. As the bed has no surface tension, no variation of flow is expected during bubble formation, and the conditions of constant flow are approximated. This explanation is due to present authors. 

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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