7.3 Bubble
7.3.1 Equivalent diameter
8 9 10
0.1 1
L- UC
α
Bubble Slug Eq. 7.5
Figure 7.3:Normalized unit cell length ¯LUCas a function of the mean void fraction α. Symbols: experimental data, line: prediction given by Eq.7.5.
where C0and ¯LLhave been taken as 1.22 and 1.39, respectively, according to results in previous Chapters. Therefore, ¯LUCreaches a minimum possible value of 1.7 when the mean void fraction tends to zero.
Fig. 7.3shows the measured values of ¯LUC as a function of the mean void fraction. According to the assumption of LLS =LL, only experimental data cor-responding to the saturation regime are plotted in the figure. ¯LUC behavior is well predicted by Eq. 7.5. The data scattering can be justified on the basis of experimental errors associated to the experimental apparatus and the measure-ment of the lengths.
7.3 Bubble
7.3.1 Equivalent diameter
When bubbles grow reaching a size close to the capillary diameter, they deform longitudinally as a consequence of the increasing interaction with the capillary wall, acquiring an elongated shape within the tube. When the bubble length achieves a size several times the diameter of the capillary, bubbles acquire a cylindrical main body with a bullet-shaped nose. These bubbles are frequently
76 7 Characteristic Lengths
referred to as Taylor or Dimitrescu bubbles. In the case that the new boundary conditions allow it, when bubbles leave the capillary, they try to readjust to a spherical shape as a result of the surface tension action. Such a situation occurs when a train of bubbles generated in a T-junction is injected into a large enough cavity filled with quiescent fluid. Other additional phenomena can also arise.
For example, given that pressure is generally lower at the capillary exit than in-side the capillary, gas also tends to expand and increase in volume.
In any case, taking into account the previous assumptions of high regularity and consequently small size dispersion during the bubble generation, an equiv-alent bubble diameter φB can be defined for each bubble. This diameter can be estimated by means of the diameter of an equivalent sphere with the same volume:
7.3.1.1 Dependence with the gas and liquid superficial velocities
Scaling the equivalent diameter with the capillary diameter and turning QG to USG, a normalized expression of φBin terms of the gas superficial velocity and the generation frequency can be obtained from Eq.7.6:
φB= 6 φ¯Bis plotted in Fig.7.4as a function of the superficial gas velocity. Eq.6.4 has been used in combination with Eq.7.7as a prediction of the experimental data shown in this figure. The fitting of USG0(see Eq.6.8) has also been super-imposed in this figure in order to distinguish the separation between the linear and saturation regions. It is shown in this figure that bubble size increases as a function of USG1/3, as expected. It can also be noted that bubble size reduces when increasing USL. It can be explained since increasing USL also increases drag, making easier the bubble detachment. Dimensionless sizes close to 1 are obtained, which proves that it is possible to control the bubble size accurately, getting bubbles of the order of the capillary diameter (see Section 3.3). Bub-ble and slug flow regimes can then be distinguished in this figure by the value φ¯B=1.
Moreover, one would expect a constant bubble size for small USGand given USL. For USGtending to zero and using Eq.6.1, Eq.7.7becomes:
7.3 Bubble 77
Figure 7.4: Normalized equivalent diameter ¯φB as a function of the gas superficial velocity USGfor given USL. Symbols: experimental data, lines:
prediction of Eq.7.7by using Eq.6.4, as well as USG0from fitting of Eq.6.8.
φ¯B= 3 2 ¯a
1/3
(7.8) where a is normalized with φc. The initial slope being constant for a given USL, as discussed in Subsection6.1.3, indicates that ¯φB should also be constant.
However, this tendency is not clearly observed in Fig.7.4due to experimental uncertainties, which are amplified in estimating ¯φBwhen both USG and f tend to zero.
In order to express ¯φBin terms of gas and liquid superficial velocities, Eq.7.7 can be rewritten by using Eq.6.2as:
φ¯B= 3 · ¯LLS As long as most of the experimental points are inside the saturation regime as shown in Fig.7.4, and in order to simplify what is essentially a more complex analysis, we assume the following approximation LLS ≈ LL = 1.39 · 10−3m, which is equivalent to imposing f ≈ fsat. Under this assumption, Eq.7.7can be rewritten as:
78 7 Characteristic Lengths Fig.7.5shows the normalized equivalent diameter calculated with Eq.7.7, by using the actual f measured from the experiments, as a function of the ratio of superficial velocities USG/USL. As expected, slight dispersion appears over the straight line (Eq.7.10), corresponding these scattered data to the linear region, in which f differs from fsat. Since f is lower than its corresponding fsatfor these points, their equivalent diameters are expected to be greater than the estimation provided by 7.10.
We show in Fig.7.6the normalized equivalent diameter obtained using fsat
(Eq.6.6) instead of the actual f . Experimental data fit well to Eq.7.10in this second situation. Although Eq. 7.10describes the behavior of ¯φB only in the saturation region, the number of experimental data in the linear region is lower than in the saturation region. In addition, the quantitative variations with data of the linear region are small. Therefore, Eq.7.10turn out to be a simple enough prediction for most of the flows studied in this work and is used as a general prediction hereafter for these flows.
7.3.1.2 Dependence with the Weber number
In order to check the theoretical prediction available for the slow gas flow rate regime (see Section3.3), the normalized equivalent bubble diameter has been plotted in Fig.7.7as a function of the square root of the nominal Weber number of the liquid cross-flow based on the superficial velocity. It can be observed that bubble size depends linearly on WeSL1/2, as expected. Points corresponding to the smaller gas flow rates (roughly up to 0.106 m/s) tend to superpose in the fig-ure, which is consistent with the theoretical prediction that in this regime sizes are independent of USGand depend on WeSL1/2just through USL.
From the fitting of these small gas flow rate data with Eq.3.39we can obtain a value for the parameters of the theory, namely γ and Wec. Nevertheless, calcu-lation of bubble sizes for small USGare again subject to larger uncertainties, and the superposition of the points is not perfect. For this reason parameter values range as follows: γ=0.86 ±0.06 and Wec=8.64 ±2.2. Line in Fig.7.7, which has been plotted as a guide to the eye, corresponds to γ=0.85 and Wec=10. Again, it is shown in Fig.7.7that sizes generated are values close to 1.
7.3 Bubble 79
0.1 1 10
0.01 0.1 1 10
φ- B
USG / USL
Figure 7.5: Normalized equivalent diameter ¯φB as a function of the ratio of the superficial gas/liquid velocities. Symbols: experimental data, line:
prediction of Eq.7.10.
0.1 1 10
0.01 0.1 1 10
φ- B
USG / USL
Figure 7.6: Normalized equivalent diameter ¯φB as a function of the ratio of the superficial gas/liquid velocities. Symbols: experimental data, line:
prediction of Eq.7.10.
80 7 Characteristic Lengths
Figure 7.7:Normalized equivalent diameter ¯φBas a function of the square root of the Weber number. Symbols: experimental data for different USG, line: prediction of Eq.3.39, with γ=0.85 and Wec=10.