The Gas Fraction in a Mobile Flooded Packed Bed

If the probability for a motion caused only by bubbles-liquid interactions is Pa = uAt, then the following 1 — p = 1 — uAt, gives the probability for a displacement due to the bubble-bubble interactions. When we have the last type of motion in the interval of time At, one bubble changes its velocity v to velocity v  

Relation (4.239) shows that k bubbles (bubbles having velocity v ) reach point x at time t + At because of the interaction with the other types of bubbles (the probability for this event is 1 — aAt) or because of the interaction with the composite liquid-solid medium (the probability for this event is aAt). At the same time, the bubbles that originate from the position x — v At without interaction with the nearly bubbles keep their velocity so the local distribution function of these individuals velocities is f (x, v, t). Due to the stochastic character of the described process, the transition probabilities from the state e to all k states verify the unification condition. Consequently, the probability p j will be written as 

Pjjj = 1 — Pke Slid relation (4.239) will be rewritten as follows  

The determination of the transition probabilities (p j, V k,e = 1,. ..m) is a problem that requires careful analysis. Experimental observations show that, for bubbles moving in a liquid, two interaction rules can be accepted  

The assembly resulting from a bubble-bubble interaction, which takes the velocity of the bubble having the higher velocity  

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