Designing a battery-powered hair dryer
If you design the hair dryer so that air is pulled past the batteries first, then blown through the heating wire, you will be able to extract all the heat from the process of discharging the batteries, including what would otherwise be considered waste heat.
The relevant spec will then be Watt-hours (Wh). Doing the math from this is easy:

Your 4.8 Ah 3.7 V cells have 4.8 * 3.7 = 17.76 Wh.
Four cells is then 17.76 * 4 = 71.04 Wh.
At one kilowatt of heating, this will give you a runtime of 17.04 Wh / 1000 W = 0.017 hours.
0.017 hours * 60 minutes per hour = a total runtime of 1.02 minutes.

Of course this calculation doesn't say anything about the ratio between heat dissipated in the heating wire, and &quot;waste&quot; heat from the internal resistance of the batteries.
Also, the power draw of the fan is not included, but the &quot;waste&quot; heat from the motor would also contribute to heating the air, so the end result would be roughly the same.
Other specs, like maximum continuous discharge current, maximum operating temperature, and factors like battery degradation over time, safety factor, etc, would also need to be taken into consideration.