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A barrel/drum with a diameter of 60cm is rotating at 20RPM to get a good mixing of the fluid contained (type thick oil).

At what RPM should a barrel of 30cm rotate to get the same mixing efficiency with the same fluid? To get the same peripheral velocity, the RPM should be 40RPM, but will this give the same mixing of the fluid? I guess gravitational and centripetal forces play a role her...

EDIT: More information to my otherwise incomplete question :)

The barrel is rotating around the length axis and the degree of fill is 90%.

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This actually sounds like a fun question, and should be approachable by scaling/dimensional analysis arguments. Mixing is a microscopic kinetic effect, so is really something to do with vicosity. Clearly, the size of the container matters, perhaps both radius and height. Finally, there is the rotation speed. What dimensionless quantities may be made with these? Are there other factors which might be important (density? I don't have a very good intuition about fluids...) – genneth Jan 28 '11 at 11:53
The orientation of the drum relative to gravity is still missing: ie does the drum stand upright or does it lie horizontally with respect to its axis of symmetry? – Georg Jan 28 '11 at 12:47
It lies horizontally.. just like a barrel rolling down a hill. – williamx Jan 28 '11 at 14:07

2 Answers

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Knowing a bit about process engineers' attitude to science I would say you just need to have same Reynolds number. $U\approx\Omega R$, $L\approx R$, then $Re\approx\frac{\Omega R^2}{\nu}$ so 80RPM.

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Knowing process engineers, I'd call that their attitude to problem solving. (Normally they will invoke 3 dimensionless numbers at least.) If that engineer does really know this problem, he knows that "sciences" use is very limited in such cases. I know of dozens of inventions (traded or tinkered locally) in the field of mixing/heating in "small" vessels. So I know, that there is no simple/positive answer here! On top of all physics in those drums You always have to pay attention to several economic factors. – Georg Jan 28 '11 at 13:02
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If I take thick oil to mean what I think it does, this should be a very viscous fluid, so Reynolds number, which relates inertial forces to viscous forces should be so small as to not be applicable. I think in the purely linear viscous regime, that good solutions exist. My first guess would be that you need the same number of rotations to achieve the same degree of mixing, so 20RPM should probably achieve the same rate of mixing per unit of time. Of course for the smaller barrel, you may be able to run it faster and have the process take less time.

If you have a way to measure the quality of mixing, you might be able to experiment with it. Perhaps the density difference between two samples of the oil. My assumption of number of rotations comes from assuming the geometry is what matters, i.e. the free surface is roughly horizontal, and the flow pattern is the same as you change size/rotation speed. Obviously at some rotation speed, the flow will be too slow so the the free surface (air bubble) may simply be carries around the barrel, so I'm guessing there is a max rotation rate, and I'm not sure how that scales with size/viscosity.

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