The argument in the linked wikipedia entry is that combustion temperatures are lower in this cycle because the better fuel/air mixing means it can burn much leaner and thus at a lower temperature. But a leaner mixture means lower power for a given displacement too, which means lower efficiency than a comparable traditional diesel (which are already hard-pressed to see gains like the 40% claimed against well-tuned gas engines).
Honestly the whole thing sounds very snake oily to me. I don't deny that it's possible such a thing could be tuned to operate as well as a traditional engine, but... it sounds awfully fiddly. I'd want to see numbers from a production engine in a real car before placing any bets. Electric continues to look like a much better bet to my eyes.
Throwing f/a into the mix, the HCCI mode is apparently only active during low-power cruise, so the lean mixture is fine. The Mazda innovation is controlling the changeover from HCCI to SI when more power is demanded.
My understanding is that because gasoline is refined to a higher degree it burns a fair bit cleaner in similar environments(and also makes it a PITA to store for more than a few months).
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We've got a pair of tractors around the house here, '81 diesel(pre-emissions), and '47 gasser.
I can tell you which one I prefer to run from an exhaust perspective, the diesel may have torque and is a dead-simple engine but you don't want to be downwind of it when cutting heavy brush.