Bubble velocity calculations

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 ... 

The Eulerian gas velocity field required in both the mass balance and the above transport equation for nh is found by an approximate method first, the complete field of liquid velocities obtained with FLUENT is adapted downward because the power draw is smaller under gassed conditions next, in a very simple way of one-way coupling, the bubble velocity calculated from the above force balance is just added to this adapted liquid velocity field. This procedure makes a momentum balance for the bubble phase redundant this saves a lot of computational effort. 

Ratio of Minimum Bubbling Velocity to Minimum Fluidization Velocity (Umb/Umf). This ratio can be calculated as follows ... 

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. 

Fig. 6 shows the FFT spectrum for calculated bed pressure drop fluctuations at various centrifugal accelerations. The excess gas velocity, defined by (Uo-U ,, was set at 0.5 m/s. Here, 1 G means numerical result of particle fluidization behavior in a conventional fluidized bed. In Fig. 6, the power spectrum density function has typical peak in each centrifugal acceleration. However, as centrifugal acceleration increased, typical peak shifted to high frequency region. Therefore, it is considered that periods of bubble generation and eruption are shorter, and bubble velocity is faster at hi er centrifugal acceleration. 

If the bubble frequency, bubble diameter, and bubble velocity are known, the solids mixing rate can be calculated. 

Next, the bubble velocities are calculated to determine the flow regime ... 

Thus, the bubble velocity is within the intermediate-bubble regime, and equation 23.3-6b is used for fb. Parameters u b, Kbe, fb, and Lj[ are calculated as follows ... 

This equation has been deduced from studies conducted with bed diameters of 7.6-130 cm, minimum fluidization velocities of 0.5-20 cm/s, solid particle sizes of 0.006-0.045 cm, and us - wfm< 48 cm/s. To calculate an average value of the bubble velocity, an average bubble diameter should be used. This diameter can be taken to be equal to the bubble diameter at z = Hfl2. Thus, to calculate the bubble diameter and thus the bubble velocity, the fluidized bed height should be known. To solve the problem, an iteration method should be used (Figure 3.60). 

Here, it has to be noted that for calculating the Peclet number in fixed beds, the actual velocity has to be used, i.e. the interstitial velocity, which influences the degree of mixing. In slurry bubble column reactors, the real velocity of the fluid is the bubble velocity, which is much higher than the gas superficial velocity. The mean bubble rise velocity for a batch liquid is (eq (3.201))... 

To test the viability of in quantifying fluidizing characteristics, it is plotted against the ratio of incipient bubbling velocity to incipient fluidization velocity, the latter being calculated after Geldart. Figure 69 shows... 

As shown in Fig. 4, the effect of pressure is to increase the corresponding kLa. Similarly to a, kLa depends on pressure only for gas and liquid velocities above some critical velocities. The effect of pressure can be due either to an increase in kL or in a, or in both. A literature review on the pressure effect on kL in different gas-liquid contactors reveals that kL may be considered as independent of reactor pressure. Hence, kLa should vary with pressure only via the effect of the interfacial area. Following the assumption of the presence of small bubbles in the liquid films, gas-liquid mass transfer can be split into a mass transfer from the continuous gas to the liquid film, with a mass transfer coefficient equal to the one at atmospheric conditions and a mass transfer from the bubbles to the surrounding liquid, as if bubbles were suspended in a stagnant medium. Then, contribution brought about by bubbles is calculated as the product of the excess interfacial area ab and the mass transfer coefficient of a bubble in a stagnant medium (Sh = 2) ... 

The large bubble rise velocities calculated from equation 19 are shown in Table II and they are significantly higher than the rise velocities of single bubbles of the same diameter. 

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%) ( )... 

Fig. 37. Mean bubble velocity along the column axis experimental data are for FCC-catalyst beds. Full curves are calculated by + o with u, = 49.5 cm/sec andito...

The relation between proposed equations (6-10) and (6-18) merits discussion. When = 1 (no reaction) and m = (no partition to particles), kat, as calculated by Eq. (6-10) or (6-16) is smaller than that by Eq. (6-18) for fine particles, since the former takes s for bubble velocity and the latter uses Md, which is larger than u. Equations (6-20) and (6-13) are generally different because m and e r re not necessarily equal. The bubble-void resistance to mass transfer has been assumed negligible in Eq. (6-21). This equation is rendered applicable to the case of arbitrary m by utilizing Eq. (6-22). Equation (6-13) can be rewritten in a form similar to Eq. (6-22), with the observations that m = Cfe + wJs e. and Cf =... 

Thus the hydrodynamic force is larger than it would be in the absence of the thermocapillary (Marangoni) contribution to the shear stress at the bubble surface. As a consequence, the bubble moves slower. Indeed, at steady-state, the bubble velocity can be calculated from the overall force balance ... 

Figure 7-17. Streamlines for thermocapillary motion of a gas bubble for a = 0.8 (a2pg//3), where the bubble velocity is reduced to 20% of its value in the absence of thermocapillary effects. The stream-function values are calculated from Eq. (7-244) with coefficients C and D from Eqs. (7-248). Contour values are plotted in increments of 0.7681.

There are two small parameters in Eq. (10.42), (n - T ) and Up /a. The first or the second term can predominate in Eq. (10.42). The bubble buoyancy cannot be sensitive to the existence of f s.fz., not even at very small angle rt - T. In Section 8.7.3 this was proven for small Re. The coincidence of the measured bubble velocity with that calculated for a solid sphere is not the ground to calculate the collision efficiency neglecting the mobility of the bubble surface. Its influence can predominate according Eq. (10.2.23). 

Table 10.1 Measured and calculated bubble velocities at different bubble diameters...

Collision efficiency was calculated by the method proposed for the first time by Dukhin Derjaguin (1958). To calculate the integral in Eq. (10.25) it is necessary to know the distribution of the radial velocity of particles whose centre are located at a distance equal to their radius from the bubble surface. The latter is presented as superposition of the rate of particle sedimentation on a bubble surface and radial components of liquid velocity calculated for the position of particle centres. Such an approximation is possibly true for moderate Reynolds numbers until the boundary hydrodynamic layer arises. At a particle size commensurable with the hydrodynamic layer thickness, the differential of the radial liquid velocity at a distance equal to the particle diameter is a double liquid velocity which corresponds to the position of the particle centre. Such a situation radically differs from the situation at Reynolds numbers of the order of unity and less when the velocity in the hydrodynamic field of a bubble varies at a distance of the order ab ap. At a distance of the order of the particle diameter it varies by less than about 10%. Just for such conditions the identification of particle velocity and liquid local velocity was proposed and seems to be sufficiently exact. In situations of commensurability of the size of particle and hydrodynamic boundary layer thickness at strongly retarded surface such identification leads to an error and nothing is known about its magnitude. 

The determination of the particle velocity after its inelastic collision with a bubble is based on the calculation of the energy losses spent in overcoming the viscous resistance during the approach and thinning of the film formed between the bubble and the particle (cf Section 11.2). This may be estimated by taking the ratio between the particle velocity after collision v and its initial velocity (equal to bubble velocity). 

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