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I was wondering about a bubble moving with constant velocity in a liquid, and how the motion affects the mass transfer. Since the viscosity of the gas is significantly smaller than the viscosity of the liquid, the condition ## \tau_{gas} = \tau_{liquid} ## tells me that

$$ \mu_{gas} ( \frac {dv_{gas}} {dy} )_{boundary} = \mu_{gas} ( \frac {dv_{liquid}} {dy} )_{boundary} → (\frac {dv_{gas}} {dy} )_{boundary} >> ( \frac {dv_{liquid}} {dy} )_{boundary} $$

So we can assume the system is static (because there is no significant variation from the velocity of the bubble and the velocity of the fluid in the penetration layer). Now my problem arise:

my professor and all the books I read, introduce the concept of "residence time" ##tc = x/v##, but what exactly is ##x##? I figured out it has to be the diameter of the bubble, but why?? Can someone please explain me this, I spent 2 hours thinking on it and still I don't get it.... I feel like I'm close to the solution, but I'm not 100% sure

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# I Higbie's penetration theory in a bubble

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