How many plants are needed to keep someone alive?

[Brother Oni]

While industrial scaling does tend to lose efficiency (which I'm sure you know given your profession) due to concepts like "well mixed" having a tendency to get hilarious as simplifications this particular case is probably less of an issue. A glorified CSTR at low flow rate operating at the high end of viable concentration is probably going to work fine, though keeping weight low is potentially fun.

Also fun is dealing with the accumulated dead algae in the outlet stream, trying not to waste live algae by having them in the outlet stream where you suddenly care much more about that due to mass restrictions, and what the inevitable recycle process there means for the inlet stream. Granted, unseen complications and complications worse than they seem can always crop up, but there's not a lot of reason to think that this particular design would be too bad. It's fairly similar to a lot of the e. coli bioreactors used for production of organics, except that you don't have to separate out the oxygen from the water the way you do with most products.

I'm more than happy to concede expertise in the industrial scale engineering side of things to those who know more (lab scale and scale up are my experience) and working with live critters is completely out of my field of expertise.

I'd agree with your proposal that a CSTR with the relevant filters (I can hear the hysterical laughter now) pumping the live algae out into the panels with a CO2 rich water stream (and have regular CO2 feeds in/O2 filters out to maximise speed of yield return), then filtration of O2 before collection back into the CSTR could be viable as a handwaving CO2 recycling / O2 generation system. Just don't approach NASA with the idea unless you want to be laughed at.