Funny seeing this still alive....
HAI was explained to me simply as "hot air helps atomize fuel better, reducing the amount the injectors need to spray" so I understood that by adding hot air to the intake air, the fuel would burn more efficiently, and I'd use less gas pedal to move forward, but as mentioned, at the expense of horsepower taking a small drop.
Now that could be completely off, but I do know that depending on your car whether it's MAP equipped or MAF equipped, using a HAI works or it doesn't. MAF cars sense the incoming air temperature and the computer recalculates fueling to maintain optimal levels, MAP cars (like the S-Series Saturns) cannot do this, so the computer cannot alter fueling based on intake air temperatures... so fabbing a HAI and feeding the engine 160*F air effectively increases efficiencies beyond what the computer can adjust for, and you increase FE.
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John
'09 Saturn Aura 2.4L
'94 Chevy Camaro Z28 (5.7L 6sp)
'96 Chevy C1500 (5.0L 5sp)
'08 Kawasaki Vulcan 900 Custom
'01 KTM Duke 2