Bubble terminal rise velocity

It has been assumed that the terminal rise velocity of a single bubble is known so the characteristic velocity can be calculated. In general the bubble s velocity depends on its size, which in turn depends on the design of the distributor. An excellent survey of bubble and droplet formation has been given by Kumar and Kuloor (1970). 

An accepted empirical correlation for the terminal rise velocity of single isolated bubbles is that of Peebles and Garber (1953), who identified four regions as shown in Table 7.1. 

As Wallis (1969) points out, the upper limit of region 4 is with very large bubbles when their rise is dominated by inertial forces. Under these conditions, the terminal rise velocity is readily calculated from potential flow theory and is given by... 

Where u, is the terminal rising velocity of the bubbles given later. 

In the presence of a surfactant (Terpineol), Rice et al. (2) obtained a very good fit between theory and experiment using ( )( ) = (l- ) which is the Wallis model with n = 2. In the work cited, average bubble size was around 1 mm diameter. This particular structure shows, according to equation (3), that the slip velocity approaches terminal rise velocity as voidage becomes small, as one expects. 

In bubble columns, the estimation of parameters is more difficult than in the case of either gas-solid or solid-liquid fluidized beds. Major uncertainties in the case of bubble columns are due to the essential differences between solid particles and gas bubbles. The solid particles are rigid, and hence the solid-hquid (or gas-solid) interface is nondeformable, whereas the bubbles cannot be considered as rigid and the gas-liquid interface is deformable. Further, the effect of surface active agents is much more pronounced in the case of gas-liquid interfaces. This leads to uncertainties in the prediction of all the major parameters such as terminal bubble rise velocity, the relation between bubble diameter and terminal bubble rise velocity, and the relation between hindered rise velocity and terminal rise velocity. The estimation procedure for these parameters is reviewed next. 

In the same Fig. 5, two more curves based on experimental data of Gaudin (1977) are given for air-water systems. From this figure, one of the key problems in estimation of terminal bubble rise velocity becomes apparent. For a widely used system such as air-water, if a bubble diameter of 2 mm is taken, the rise velocity can have any value between 150 mm/s and 300 mm/s. Conversely, if we take the terminal rise velocity of bubbles to be 200 mm/s, bubble diameter can be anywhere between 1 and 10 mm, depending upon the degree of contamination, which is difficult to quantify and can change over a period of time. In the present work, we have used correlations of Clift et aL (1978) for the predictions. We also have used the... 

Fig. 5. Various relations between bubble diameter and terminal rise velocity air-water system.

Fig. 24. Effect of bubble diameter-terminal rise velocity on transitions bubble columns [oi = 0.5, Cv = 1.0, m = 1.9].

For an air-water system, terminal rise velocity of bubbles is not very sensitive to bubble diameter. Therefore, for bubbles with diameters in the range 3 to 8 mm, the ratio of drag coefficient to the bubble diameter (Cdb/ b) can be considered as approximately constant. Ranade (1997) carried out simulations by setting the ratio of drag coefficient to bubble diameter (Cdb/ b) equal to 290 m . 

In the first alternative, if the terminal rise velocity of gas bubbles is known (or can be estimated with confidence), the top surface of the dispersion may be defined as an inlet . Normal liquid velocity may be set to zero while normal gas velocity may be set to terminal rise velocity. The implicit assumption here is that gas bubbles escape the dispersion with terminal rise velocity. It should be noted that even after defining the top surface as an inlet, gas volume fraction at the top surface is a free variable. There is no implicit forcing of gas volume fraction distribution. Alternatively, the top surface of the dispersion can be modeled as a no shear wall. This will automatically set normal liquid velocity to zero. It will also set normal gas velocity to zero. In order to represent escaping gas bubbles, an appropriate sink may be defined for all the computational cells attached to the top surface (Figure 11.7) ... 

In view of the fact that chains of bubbles are formed, this regime is sometimes referred to as chain bubbling regime. The terminal rise velocity of a small discrete bubble is relatively very low. However, the drag on an individual bubble is affected by the presence of bubbles in its vicinity. The resulting hindrance increases with inaeasing number of bubbles or gas holdup. The drag and consequent turbulence is... 

This correlation reqnires information on u, which can be estimated using Equation 10.7. This latter equation requires data on as a function of superficial gas velocity to evaluate the terminal rise velocity of the bubble, These data can be obtained throngh simple gas holdup measurements. The drift flux model of Zuber and Findlay (1965) can be used to obtain as per Equation 10.10 ... 

Bubble size is expressed in terms of volume rather than diameter. Though the behavior of a single bubble (e.g., terminal rise velocity, drag, shape) is correlated by equivalent diameter, coalescence effects are best treated with volume. Volumes are added in coalescence, regardless of bubble shape, which is seldom spherical in parameter ranges of interest. If necessary, of course, equivalent diameter can be computed from the volume whenever required, but volume remains the primary measure. 

Re Reynolds number (pU Djr ) Velocity of the liquid phase, m/s Bubble terminal rise velocity Relative velocity of bubbles to the surrounding liquid, m/s V Bubble volume, m ... 

The behavior of PAA solutions in response to agitation and aeration was different from the behavior in CMC and XTN solutions. In PAA solutions, small spherical, as well as inverted tear drop bubbles, were observed. This latter shape is the result of the interaction of elastic and surface tension forces, and has been reported to occur in stagnant [26,27] as well as in mildly stirred solutions [28]. It is known that in free climb motion, PAA bubbles have lower terminal rise velocities than... 

Determination of gas hold-up from Equation 26 requires a knowledge of the superficial liquid circulation rate, given by Equation 9 and the single bubble terminal rise velocity Most researchers have used = 0.25 msThe gas holdup and liquid circulation data in 250 L pilot-scale internal-loop airlift bioreactor for Saccharopolyspora erythmea (n = 0.55) were satisfactorily correlated by this model. 

The method of dynamic gas disengagement (Sriram and Mann, 1977 Patel et al., 1989) to obtain an estimate of bubble size distribution is worthy of mention since it is convenient to use, sometimes even in real systems, especially for lower gas fractions. The impeller is stopped and the trend in the level measured against time. This trend indicates the bubble size distribution if terminal rise velocities are known and if coalescence is negligible. 

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