Life of a heated coil using a 9V battery
You must know the current drawn by your load (your heater coil).  Ohm's Law calculates the current (Amps) = Voltage divided by resistance (ohms). 

Then you must know the capacity of your battery. This is typically stated in Amp-Hours (Ah) or mili-Amp Hours (mAh) for small batteries. Those little rectangular 9V batteries have capacities ranging from 175 mAH (NiMH) to 1200 mAh (lithium).

So then it is a rather simple matter of solving the equation for the current drawn, and the capacity of the battery.

Then you must also know what is your "cutoff" point at which the battery can no longer generate enough heat for your system.

Note that those little 9V rectangular batteries have the LOWEST capacity of all the commonly available consumer batteries. Even less than a AAA cell.


Ohm's Law Calculator:  http://www.rcrowley.com/eirp.htm
Battery capacity Cylindrical cells: https://en.wikipedia.org/wiki/List_of_battery_sizes#Cylindrical_cells
Battery capacity Rectangular cells: https://en.wikipedia.org/wiki/List_of_battery_sizes#Non-cylindrical_batteries