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A packs OFF t/o does require the APU to be operating until the packs are returned to their required position for the flight which does increase the fuel burn along with APU maintenance....i guess it all depends on your taxi time
Not true. No requirement to have the APU running in order to take off with packs off.
Maybe you're thinking of running the packs from the APU to increase Go around gradient?
Without APU,it would technically be an unpressurized takeoff rather than a bleeds off takeoff
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Some airlines allow flex AND packs OFF ,some dont.
I don't get this: why would an airline not allow packs off and flex?
One chief pilot may prefer to save the engines as much as possible and another would prefer the odds to have the bleeds ON as long as perf is ok.