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Let's take this image as our fireball in question:

enter image description here

And let's further assume that it travels about 30 meters before burning out (twice the range of a WWII flamethrower) and travels at the speed of 20 m/s (which, apparently, is about how fast a 10-year-old can pitch a baseball). The flame's temperature is in the region of 1,300 celsius, in line with actual flamethrowers.

What how much energy, measured by kilocalories, would be required to fuel such?

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    $\begingroup$ Depends on the process of creating the ''fireball'' if you throw a molotov it consumes maybe 4-7 kilocalories , if you want to ionize air by consuming calories instead of mana or other things then it might cost some hundred of kilocalories but then you'd obtain more a giant explosion instead of a fireball... I don't really know how magic works. $\endgroup$ – Charon Aug 22 '16 at 15:18
  • $\begingroup$ @渡し守シャロン This isn't the energy of the person, but the actual energy consumed by a flame of that description. So, borrowing your molotov example, would use a lot more kilocalories than 4-7, since I'm talking about the flame and not the arm throw. $\endgroup$ – Ranger Aug 22 '16 at 15:19
  • $\begingroup$ @渡し守シャロン To further clarify, the mechanics of how the magic works should be immaterial to this question; this is about the implied energy consumption of the fireball described being that it is actual fire. $\endgroup$ – Ranger Aug 22 '16 at 15:20
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    $\begingroup$ It can not be immaterial as the energy given off by the flame is dependant on what is burning. Fire isn't per se an element that has intrinsic energy it is the manifestation of a violent release of energy. So just changing what is burning changes the energy of the flame even though the travel distance (which is the travel distance of the propelled elements being burned) and speed is the same $\endgroup$ – Riff Aug 22 '16 at 15:26
  • $\begingroup$ @Nicolas If you're referring to the burning temperature, I did forget to include that, and will do so now. But the unit requested is a calorie, a simple unit of measure. The same temperature of the same volume (heat energy) for the same duration will produce the same caloric measurement of energy expended no matter the material. $\endgroup$ – Ranger Aug 22 '16 at 15:31
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Like all great physics problems I'm going to make some assumptions.

The fireball is going to be approximated as a sphere of superheated air. Nothing is actually burning to provide the energy.

The specific heat of air is around 1.0 kJ/kg K and air density is about 1.225 kg/m^3 (at sea level) so to raise a ~1 cubic meter of air to 1300 C from around 20 C is going to take:

E = 1 m^3 * 1.225 kg/m^3 * 1280 K * 1.0 kJ/kg K = 1568 kJ = 396.3 kcals

This would be a low estimate to just create a "fireball" that instantly dissipates to the environment. A more impressive moving fireball would deal with the air expanding under heating and things around the fireball absorbing heat, things like moving the "fireball" would also raise the calorie requirement. In addition the magic heating process may not be 100% efficient requiring even more energy.

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  • $\begingroup$ Awesome, thanks for the answer with the clear, easy to follow explanation. Mind if I ask what the requirements would be to stop the fireball from dissipating? How often would the kJ requirement need to be renewed to maintain it? $\endgroup$ – Ranger Aug 22 '16 at 15:59
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    $\begingroup$ Normally heating a point to high temperatures like this would cause the air to expand and move away, so to maintain it a magic force-field could be used to keep the air from expanding, otherwise it get really complex in term of gas expansion and airflow. Bonus when you drop the force field it explodes/ $\endgroup$ – Josh King Aug 22 '16 at 16:02
  • $\begingroup$ Could you maybe phrase kcal in form of example meals? ;) $\endgroup$ – Mołot Aug 22 '16 at 16:15
  • $\begingroup$ >what the requirements would be to stop the fireball from dissipating Cast it into a small room that can contain the pressure from the sudden application of heat. A magically-sudden 1300 C fireball filling a room that had previously been 15 C and 1 standard atmosphere of pressure would see a pressure increase of about 5 1/2 times. Sturdy walls could probably take it, windows maybe not. It's particularly easy to figure because it's not actually a combustion reaction, it's just dumping in energy. True combustion changes the number of gaseous particles in play and complicates things. $\endgroup$ – SudoSedWinifred Aug 22 '16 at 16:18
  • $\begingroup$ Of course, unless you can cast through walls, it's inadvisable to cast a fireball "into" an enclosed space because it's going to blow back into your face instantly. $\endgroup$ – SudoSedWinifred Aug 22 '16 at 16:23

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