How many plants are needed to keep someone alive?

[Brother Oni]

I think it's pretty reasonable to assume that it is not a large engineering burden to create greater efficiencies than a completely passive pond. Placing that same cubic meter of water into 1cm thick clear panels, for example, is going to massively increase effective surface area, and since you're going to need to be circulating the liquid anyway for gas exchange, presumably it's also easy to optimize nutrient flow for our little green buddies. I seem to recall that algae can reproduce at absurd rates under the proper conditions.

The problem is that the link which mentions the use of algae ponds for CO2 capture, doesn't mention their depth or the species, only the dry cell weight concentration.

Say we pick an algae earmarked for biofuel potential, Chlorella vulgaris. From this link, the optimal concentration of chlorella vulgaris is 20g dry weight per litre without light intensity shenanigans.

1 L over 1 square metre means you only need a panel 0.1 cm thick to hit that target of 439.9 m2 per person per day.

From the paper, their largest reaction vessel was no more than 4cm diameter, so presumably your panels can't get more than 2cm thick without the yield efficiency dropping off, making (439.9 m² x 0.1cm / 2 cm) 22.0 m² per person per day the smallest you could go. That's still about double the 11m² bamboo footprint, but with a significantly reduced vertical footprint.

Algae panels aren't as nice to look at though.

Edit: With regard to the algae concentration disparity in the lab and the algae ponds, I suspect that 20g dry weight per litre is probably not feasible outside of small scale lab conditions with tender loving care and you'd probably ideally want a lower concentration so that your algae has space to grow into as it captures the CO2.