Bubble velocity model

To evaluate the effect of holdup on bubble velocity, Marrucci (M3) used a spherical cell model of radius b such that... 

The model in its present form cannot be used for the design of gas-liquid contacting systems, for several reasons. The model requires a knowledge of the average bubble velocity relative to the fluid, U, a variable that is not available in most cases. This model only permits the calculation of the average rate per unit of area, and unless data are available from other sources on the total surface area available in the vessel, the model by itself does not permit the calculation of the overall absorption rate. 

Such a behavior agrees with results reported by Agostini et a. (2008). It was found that the elongated bubble velocity increased with increasing bubble length until a plateau was reached. An analytical model has been proposed that is able to predict this trend. 

Agostini B, Revellin R, Thome J (2008) Elongated bubbles in micro-channels. Part I Experimental study and modeling of elongated bubble velocity. Int. J. Multiphase Flow 34 590-601 Bankoff SG, Haute T (1957) Ebullition from solid surfaces in the absence of pre-existing gaseous phase. Trans ASME 79 735-740... 

The mechanistic model developed in the last section is applied to the data collected experimentally. Bubble diameter and bubble velocity calculations were based on the empirical equations obtained from frame-by-frame analysis of high-speed motion pictures taken under the respective operating conditions (Yang et al., 1984c). The equations used are ... 

Vaux (1978), Ulerich et al. (1980) and Vaux and Schruben (1983) proposed a mechanical model of bubble-induced attrition based on the kinetic energy of particles agitated by the bubble motion. Since the bubble velocity increases with bed height due to bubble coalescence, the collision force between particles increases with bed height as well. The authors conclude that the rate of bubble-induced attrition, Rbub, is then proportional to the product of excess gas velocity and bed mass or bed height, respectively,... 

A one-parameter model, termed the bubbling-bed model, is described by Kunii and Levenspiel (1991, pp. 144-149,156-159). The one parameter is the size of bubbles. This model endeavors to account for different bubble velocities and the different flow patterns of fluid and solid that result. Compared with the two-region model, the Kunii-Levenspiel (KL) model introduces two additional regions. The model establishes expressions for the distribution of the fluidized bed and of the solid particles in the various regions. These, together with expressions for coefficients for the exchange of gas between pairs of regions, form the hydrodynamic + mass transfer basis for a reactor model. 

The local gas holdup and bubble behavior were measured by a reflective optic fiber probe developed by Wang and co-workers [21,22]. It can be known whether the probe is im-merging in the gas. The rate of the time that probe immerg-ing in the gas and the total sample time is gas holdup. Gas velocity can be got by the time difference that one bubble touch two probes and the distance between two probes. Chord length can be obtained from one bubble velocity and the time that the probe stays in the bubble. Bubble size distribution is got from the probability density of the chord length based on some numerical method. The local liquid velocity in the riser was measured by a backward scattering LDA system (system 9100-8, model TSI). Details have been given by Lin et al. [23]. 

Vasconcelos variant of the slip velocity model based on bubble contamination kinetics, Eqs. (4)-(8), was used by Alves et al. [9] to interpret kl data in a double Rush-ton turbine-stirred tank. The application of Vasconcelos model to the interpretation of unreliable mass transfer co-... 

In the Davidson and Harrison s (1963) maximum stable bubble size model, the bubble disintegration takes place when the relative velocity between the bubble and the particles exceeds the particle terminal velocity. Considering that, for a vertical gas-solid flow system, choking occurs when the maximum stable bubble size is equal to the column size, Yang (1976) obtained the following choking criterion for fine particles fluidization ... 

The application of the two bubble class model is important for reactions which are mass transfer controlled. The scale-up of a mass transfer controlled operation based on a fixed gas phase residence time assumes incorrect and higher actual gas velocities. Thus, when the flow regime changes from... 

The two bubble class model is applied here to the absorption of CO2 in NaOH, which conforms to a fast pseudo-first order reaction under certain operating conditions (15). In the data reported by Schumpe et al. ( 7 ), COo was absorbed during cocurrent flow in NaOH solution in a 0.102 m diameter bubble column. The gas phase consisted of approximately 10 vol % of CO2 in N2. The gas velocities ranged from 0.025 to 0.15 m/s. Since the churn turbulent regime prevailed for gas velocities greater than approximately 0.07 m/s, only the data in the range 0.07 m/s to 0.15 m/s were considered. 

From the results described above, it is seen that the two bubble class model predicts conversions that fit the experimental data. The predicted conversions were obtained with the assumption that the large bubble size varies somewhat with the gas velocity. Thus it is required to have a knowledge of the large bubble diameter (as a function of gas velocity) to properly evaluate the validity of this model. 

Figure 5. Comparison between Plug Flow model and Two-Bubble-Class model outside range of experimental velocities. ( - Plug Flow model, A - Two-Bubble-Class model, varying d , H— Two-Bubble-Class model, d = 10 mm).

If Cq is known as a function of the capillary number and the surfactant properties, the functional form of the frequency and bubble volume can be approximated from the linear results. However, a model for Cq in constricted angular tubes does not exist. If one assumes that snap off occurs as soon as the thread becomes axisymmetric, then the base state thread radius is approximately the half width of the channel at the point snap off occurs. The experimental observations of Arriola and Ni along with the theoretical predictions of Ransohoff and Radke indicate that snap off takes place very near the constriction neck. Therefore, the radius of the bubbles formed should be slightly larger than the half width of the constriction neck. In fact, approximating Cq by the constriction half width, one observes from equations 14 and 15, that the snap off frequency and bubble volume are independent of the liquid flow rate once the critical liquid flow rate has been exceeded. Ni measured the dependence of snap off on the bubble velocity, the velocity of... 

Figure 2. a. Bubble volume—comparison between numerical model and experiment (yi) (%) numerical calculations b, Bubble velocity as function of equivalent spherical radius—comparison between numerical model and experiment (3%) ( )... 

So far, the influence of bubble wake on mean bubble velocity i b relative to the column wall has not been mentioned, since Eq. (5-3) has been formulated on the basis of uy,o, which already includes the effect of the wake (although it lacks a correction for wake fraction). In bubbling-bed models (FIO, F12, K24, L5, S18) an upward flow of solid carried by the bubbles and bubble wakes leads to a downflow of solid (that has been assumed uniform) in the remainder of the bed. Then the bubble velocity b relative to bed wall should be smaller than the slip velocity of the bubble Ms relative to emulsion, since the bubble phase is retarded by downflow of... 

Catalytic hydrogenation of ethylene by nickel- or copper-impregnated cracking catalyst is taken here for comparison. Figure 67 shows typical experimental A or values taken under a constant superficial gas velocity Uq by varying A ,. Curve LGG is based on the data by Lewis et al. (LI 2), GK by Gilliland and Knudsen (G7), and FKM those by Furusaki et al. (F18). The calculation ofA R will be explained later. The dotted curves are calculated by the two-phase consecutive model (TCM) and by the bubbling bed model (BBM) for Ug = 20 and 25 cm/sec, where the mean bubble size is 4.5 cm and the wake fracticm / = 1.0. 

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