Not much info, but why not run your compressed air through coiled copper or aluminum tubing inside a chamber through which you are running your hot exhaust gases?
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Those who would give up essential Liberty, to purchase a little temporary Safety, deserve neither Liberty nor Safety. Ben Franklin
Only the proper turbulence has to be created for effective heat transfer, which is easier in case of liquids.
For gases yoy may need fins/ corrugations on both inside as well as outside surfaces of the tubes. This is just a bit easier if one uses the plate exchangers rather than tubular.
You may also study something about the heat recovery systems in the powerplant where the flue gas heat is recovered, of course not to compressed air but the Primary Air.
There you are like to have the rotary exchangers (Which of course may not be economical in small scale)
Did you really expect the heat exchanger to be designed here by us? It is not so simple and now it is no more calculators and slide rules. There are softwares for this purpos.