The Sabatier reaction is how CO2 is scrubbed to provide breathable air on the International Space Station: https://en.wikipedia.org/wiki/Sabatier_reaction#International_Space_Station_life_support
It happens when you combine CO2 with H2 at 300–400°C, 30 bar, in the presence of a nickel catalyst.
The three-way cycle
What I propose (and sanity-check me on this please), is a three-point cycle: the Sabatier reaction, electrolysis, and breathing.
- the Sabatier reaction takes in CO2 from exhaled air, and H2 from electrolysis, and outputs methane and water. CO2 + 4H2 → CH4 + 2H2O
- electrolysis takes in water, outputs H2 (to feed the Sabatier reactor) and O2 (to feed the person). 2H2O → 2H2 + O2
- breathing takes in O2 and outputs CO2 (into the Sabatier reactor), on a one-mole-per-mole basis according to http://www.madsci.org/posts/archives/2004-09/1096283374.En.r.html
This is not a perfect closed loop system, unfortunately, because the Sabatier reaction requires four hydrogen molecules.
Add hydrogen at step 4:
- Start with 2H20
- Electrolyse that into 2H2 and O2
- Breathe the O2. Now you have 2H2 and CO2
- Add bottled hydrogen. Now you have 4H2 and CO2
- That gives CH4 + 2H2O
According to Starfish Prime's answer here, our hero requires ~5mmol of oxygen per second, aka 0.3mol/min. So we need to add/consume 0.6 moles of hydrogen gas per minute at step 4, which (given H2's molecular weight of 2.01588g/mol) 1.209528 g/min or 72.57168 g/hour of hydrogen to keep the system running, scrubbing the aquanaut's CO2 and providing their O2. The system will produce 0.3mol/min of methane, 4.812738 g/min or 288.76428 g/hour, so if you take methane's energy density to be energy 50 MJ/kg, that's 14.438214 MJ/hour, or 4.010615 kilowatt hours per hour, better known as kilowatts, of methane fuel. According to Starfish Prime's answer already linked, the electrolysis required 2.38kW (and the temperature and pressure of the Sabatier reactor requires energy too). Note that you actually won't be able to use that methane until you get home, as you have no oxygen with which to combust it.
Use more water than is needed, offgas oxygen at step 3:
- Start with the 2H20 from the end of this cycle. Add 2H20 from somewhere to
- Electrolyse that into 4H2 and 2O2
- Chuck out some O2. You now have 4H2 and O2.
- Breathe the O2. Now you have 4H2 and CO2
- The Sabatier reaction turns that into CH4 + 2H2O
Disadvantages: electrolysis is energy-expensive, as we've seen. By doubling the amount of water you're electrolysing, you waste 2.38kW. If our hero is swimming in the sea, she'll need to desalinate the water before electrolysing it, or die from chlorine gas inhalation. And she has enough contraptions on her back already without adding desalination! Probably better to carry a tiny bit of distilled water: the input (to get 0.3mol of O2 per min) is 0.6g of water per minute, 36g (aka millileters) of water per hour. Still waste the energy though. Another disadvantage is that you're blowing gas bubbles, which will spook some fish (a problem with normal SCUBA too); many sea-creatures are very sensitive to sounds.
Close the loop by making it a four-way cycle by adding methane pyrolysis
Sabatier reaction is CO2 + 4H2 → CH4 + 2H2O
Electrolysis is 2H2O → 2H2 + O2
Methane pyrolysis is CH4 → C + 2H2
This completes the loop; all inputs feed back in, apart from a little charcoal as a waste product. Specifically 0.3mol of charcoal per hour, or 3.6 grams.
The methane pyrolysis (according to Musamali, R., & Isa, Y. M. (2018). Decomposition of methane to carbon and hydrogen; a catalytic perspective. Energy Technology. doi:10.1002/ente.201800593) can be done with a catalyst at 850°C and "The energy required for the production of one mole of hydrogen (45.1kJ/mol (H2) at 1073K)". As we said above "we need to add/consume 0.6 moles of hydrogen gas per minute", that's an added requirement of 1623.6 kJ/hour, or 451 Watts. This is in addition to the 2380W required for electrolysis, and the energy to heat the Sabatier reactor. Overall you might need 3-4kW, which is quite a lot for a wearable apparatus, but not unheard of, e.g. here's a 3-3.5kW backpack apparatus. This is a lot more energy-efficient than solution 2.
A bonus of this is that the heat generated could feasibly be redirected to actively warm the user in cold water; hypothermia is a danger during long dives.