A lab fermentor has volume 1 L, diameter 20 cm, agitator speed 600 rpm and impeller/fermentor diameter ratio 0.3. It is scaled to 8000 L at constant impeller-tip speed, using the cube root of volume ratio as length-scale factor. Find large-reactor agitator speed in rpm.
The length-scale factor is $(8000/1)^{1/3}=20$, and geometric similarity makes impeller diameter grow by the same factor. Tip speed is πND, so constant tip speed requires $N_{large}=N_{small}/20=600/20=30$ rpm. The supplied 20 cm diameter is not needed once the specified scaling ratio is used.
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