Probability density distribution bubble

Figure 5.4 Data of bubble size probability density distribution in an aerated stirred vessel and fitted PSD curve based on new PSD function.

Data of bubble size probability density distribution in an aerated stirred... 

Anderson (A2) has derived a formula relating the bubble-radius probability density function (B3) to the contact-time density function on the assumption that the bubble-rise velocity is independent of position. Bankoff (B3) has developed bubble-radius distribution functions that relate the contacttime density function to the radial and axial positions of bubbles as obtained from resistivity-probe measurements. Soo (S10) has recently considered a particle-size distribution function for solid particles in a free stream ... 

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]. 

Both the RBC distribution (8) and the geometric distribution (11) are defined only for specific integer bubble sizes, and derivatives of their distribution functions do not exist. For subsequent developments we need an equivalent continuous distribution. Fortunately, for N and k large with respect to m, both discrete distributions can be closely approximated by the exponential distribution if its mean is set to the RBC mean volume given by (10). The exponential probability density is... 

Bubble equivalent diameter, m E6 Eotvos number (gD po ) f(m k) Probability density of volume m after coalescence event k F Cumulative distribution function F Exceedance function, complement of F g Gravitational acceleration, m/s k Number of coalescences Drag coefficient between bubbles and liquid, s m Bubble volume, integer multiples of initial volume M Random bubble volume corresponding to m... 

For a more general case, where the bubble size is distributed (Tsutsui and Miyauchi, 1979), let us assume a distribution density function for bubble diameter, 4>(2)b)- In this case, (Db, 4) introduced above in Eq. (48) becomes a conditional probability to get a pierced length 4 from a bubble of size Db-What we need is the probabihty density distribution... 

The liquid flow envelops the bubble surface, and the particles are entrained to a greater or a lesser extent by the liquid. The smaller the particles and the less different their density relative to that of the medium, the weaker are the inertia forces acting upon them and the more closely the particle trajectory coincides with the liquid streamlines. Thus, at the same target distance fairly large particles move almost linearly (Fig. 10.1, line 1), while fairly small particles move essentially along the corresponding liquid flow line (line 2). The trajectories of particles of intermediate size are distributed within lines 1 and 2 as the size of particles decreases, the trajectories shift from line 1 to line 2 and the probability of collision decreases. 

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