Bubble diameter calculations

In hen of careful independent checks of predictive accuracy, the results of the comprehensive theoretical work will not be presented here. Simpler, more easily understood predictive methods, for certain important limiting cases, will be presented. As a check on the accuracy of these simpler methods, it will perhaps be prudent to calculate the bubble diameter from the graphical representation by Mersmann (loc. cit.) of the resiJts of Kumar et al. (loc. cit.). 

Gal-Or and Resnick (Gl) have developed a simplified theoretical model for the calculation of mass-transfer rates for a sparingly soluble gas in an agtitated gas-liquid contactor. The model is based on the average gas residencetime, and its use requires, among other things, knowledge of bubble diameter. In a related study (G2) a photographic technique for the determination of bubble flow patterns and of the relative velocity between bubbles and liquid is described. 

Yoshida and Miura (Y3) reported empirical correlations for average bubble diameter, interfacial area, gas holdup, and mass-transfer coefficients. The bubble diameter was calculated as... 

Particularly high stress occurs when bubbles burst on the surface of the liquid, whereby droplets are eruptive torn out of the surface [32-36]. According to theoretical calculations, maximum energy densities occur in the region of the boundary surface shortly before the droplets separate [36]. The results calculated by Boulton-Stone and Blake [34] show that these are exponentially dependent on bubble diameter dg. Whereas these authors found values of e = lO mVs with dg = 0.5 mm, these are only e 1 m /s with dg = 5 mm. The situation may be different regarding the droplet volume separated from the surface by the gas throughput and thus the number of particles which are exposed to high stress. The maximum for this value occurs with a bubble diameter of dg = 4 mm (see [34]), and it is therefore feasible that there could be an optimal bubble size. 

The highly interactive nature of the balance and equilibria equations for the distillation period are depicted in Fig. 3.66. An implicit, iterative algebraic loop is involved in the calculation of the boiling point temperature at each time interval. This involves guessing the temperature and calculating the sum of the partial pressures, or mole fractions. The condition required is that Zyi + yw = 1. The iterative loop for the bubble point calculation is represented by the five interconnected blocks in the lower right hand corner of Fig. 3.66. The model of Prenosil (1976) also included an efficiency term E for the steam heating, dependent on liquid depth L and bubble diameter D. 

To calculate the total ebullition cycle time, the length of the bubble growth period, 0rf, is also needed, which can be estimated by first estimating the bubble diameter at the time of departure and then estimating the time for the bubble to grow to that size, as will be discussed in later sections. 

Bubble Dynamics. To adequately describe the jet, the bubble size generated by the jet needs to be studied. A substantial amount of gas leaks from the bubble, to the emulsion phase during bubble formation stage, particularly when the bed is less than minimally fluidized. A model developed on the basis of this mechanism predicted the experimental bubble diameter well when the experimental bubble frequency was used as an input. The experimentally observed bubble frequency is smaller by a factor of 3 to 5 than that calculated from the Davidson and Harrison model (1963), which assumed no net gas interchange between the bubble and the emulsion phase. This discrepancy is due primarily to the extensive bubble coalescence above the jet nozzle and the assumption that no gas leaks from the bubble phase. 

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

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 value of Am strongly depends on the size distribution of the bubble diameter in the system and the volume fraction of the bubbles. These factors are strongly dependent not only on the liquid properties, but also on the gas/liquid flow ration and the energy dissipation. Opposite to the situation with respect to kiiq it is not possible to calculate Am with theoretical equations. This is the reason that in practise almost always the product of kuq and Am is considered. This product can easily be determined experimentally. 

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). 

The gas-liquid interfacial area per unit volume of gas-liquid mixture a (L 1. or L ), calculated by Equation 7.26 from the measured values of the fractional gas holdup and the volume-surface mean bubble diameter d, were correlated... 

The equations given before require iterative solution. Because, in order to calculate the interfacial area, the gas hold-up and the bubble diameter are required, and this depends on the terminal velocity of the bubbles, which in turn depends again on their diameter. The procedure for doing this has been given [11], The iteration is started with u, = 21 cm/s, which is close to its final value, and then one should proceed using the following algorithm ... 

Inga [1] suggested a modified expression of the Wilkinson equation (cited in ref. 1) to calculate the mean bubble diameter in the presence of catalyst particles and at different pressures and gas velocity ... 

In order to substitute numerical values, the interfacial area per unit volume of dispersion a needs to be calculated from the mean bubble diameter volume fraction to = (1 - eL) = 0.34. Let there be nt bubbles per unit volume of the dispersion, all of the same size. 

Note that re-arranging the above relationship to give d/, = 6ea/a shows how a mean bubble size might be calculated from measurements of and a. A mean bubble diameter defined in this way (Volume 2, Chapter 1) is called a Sauter mean (i.e. a surface volume mean see also Section 4.3.4). 

In our model the bubble diameter is assumed as constant throughout the bed and it is calculated from the equation... 

The k a results were combined with the interfacial area values to yield kL values. The values of bubble diameter were calculated using the methods outlined by Sridhar and Potter (27). The results are shown in Figure 3 as a plot of k against bubble diameter. It is seen that the k values are essentially constant. Calderbank and Moo-Young (19) proposed the following correlation ... 

For a given value of O and y in the log-normal distribution function, the mean and variance of the distribution function were computed and compared with the mean and variance of the measured bubble lengths. A regula falsi technique was used to minimize the difference between observed and calculated mean and variance. The values of O and y that minimized the difference between observed and calculated mean and variance were then employed in Equation (1) to describe the local bubble diameter distribution. The Sauter mean bubble diameter was evaluated from the second and third moments of Equation (4). 

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