Why not just burn coal and air in an oven and capture the CO2
Because only part of the air gets converted to CO2. Most of the air is nitrogen, and only ~21% is oxygen. Even if you have complete conversion of the oxygen to CO2 (not going to happen), you'd end up with exhaust gas that's mostly nitrogen with some carbon dioxide mixed in. This nitrogen/carbon dioxide mix is difficult to deal with. To do anything with the CO2 you'd have to separate it from the nitrogen and residual oxygen, which gets complicated and expensive.
The hard part is surely the CO2 capture, not the burning.
Exactly. This new method attempts solve that by separating the CO2 generation stage from the air-using stage. If you could effectively separate them, you'll get a pure CO2 stream in one half of the reactor (which if you can keep closed you can pump off into storage tanks) and you'll keep the nitrogen/depleted-oxygen mix in the other half of the reactor, away from your pure CO2.
The way it works is to use iron oxide as an oxygen shuttle. The iron oxide pellets grab oxygen from the air half of the reactor, and are then transferred as a relatively gas-free solid to the coal half of the reactor, where they give up their oxygen to produce a relatively pure stream of CO2. The pellets are then separated from the coal ash and transferred as a relatively gas-free solid back to the air half of the reactor, where they are recharged with oxygen. If you engineer it right, you could conceivably make it a continuous feed operation, where you shuttle the iron oxide beads back and forth through airlocks, keeping most of the CO2 in the sealed reactor where it can be pumped off as a comparatively pure gas.
Source: http://rss.slashdot.org/~r/Slashdot/slashdotScience/~3/feibLZvQ4AQ/story01.htm
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