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Yes but then you're erasing some of your fuel efficiency gains on take-off by running a significantly less efficient APU to charge it back up. Also I think the APU is a built in unit designed entirely for in-flight loads (like an alternator on a car) and not intended to provide the kind of energy you'd need to even moderately recharge this pack after takeoff/ascent.
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I wonder if a supercapacitor could be integrated into this setup
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