What Damage Does A Normal Bullet Do?
As with any new weapon, we first have to ask whether this is a marked improvement over what we already have. Is it worth the bother?
Let's run the energies involved, because bullets wound by transferring energy to the target. Let's assume a standard M855 5.56 NATO rifle bullet fired from an M16 with a 20" barrel, because that's what you'll see on a battlefield, and anything smaller is almost useless.
Leaving the muzzle, the bullet has 1900 J of energy. This drops off to about 1300 J at 100 meters, and about 600 J at 300 meters. After that, you probably won't hit anything.
This is already enough energy to pierce a steel helmet and Type 1 and 2 body armor. Upon hitting flesh, the kinetic energy of the bullet, plus cavitation effects, does some very bad things indeed.
Source. Also, ow.
This is already enough to give the target a pretty bad day. Can heating the bullet significantly improve on that?
First, Some Problems
There's the problem of how we heat just the bullet to such high temperatures quickly without also heating the surrounding metal of the rifle. There's also the problem of heating, and setting off, the powder charge. Tiny lasers inside the chamber? What's the power source? This all sounds very fragile and expensive.
Second, lead melts at 327°C. Steel at 1500°C. Let's go with tungsten at 3422°C for our bullets. Very expensive tungsten.
Third, a high temperature bullet would severely damage the rifling on your rifle no matter what it's made of. You could only fire a few of these before you'd be unable to consistently hit targets and have to change barrels (and maybe other parts).
Fourth, a high temperature bullet might melt your rifle. Melting points vary by alloys, but generally aluminum melts between 500°C and 1300°C. Steel at about 1500°C. And you certainly can't use any composites. You'd need a very expensive tungsten chamber, barrel, bolt, and receiver.
IANAL, but legally you can probably shoot as many soldiers with this as you like so long as they're not near civilians. If you used it in a city you'd probably be in violation of the UN Protocol on Prohibitions or Restrictions on the use of Incendiary Weapons Article 2 Section 3 which nearly everyone has signed.
It is further prohibited to make any military objective located within a concentration of civilians the object of attack by means of incendiary weapons other than air-delivered incendiary weapons, except when such military objective is clearly separated from the concentration of civilians and all feasible precautions are taken with a view to limiting the incendiary effects to the military objective and to avoiding, and in any event to minimizing, incidental loss of civilian life, injury to civilians and damage to civilian objects.
We'll let the engineers solve those problems, and the accounts figure out how to pay for it all, and the lawyers to make it legal.
How Much Energy Is In A 1000°C Bullet?
If we heated the 4 gram tungsten bullet to 1000°C what sort of energy would that transfer to the bullet (and later to the victim's flesh)?
$$ \text{energy} = \text{specific heat of tungsten} \times \text{mass of bullet} \times \text{temperature change} $$
$$ \text{specific heat of tungsten} = 0.132\,\frac{\mathrm{J}}{\mathrm{g} \times \mathrm{K}} $$
$$ \text{mass of bullet} = 4\,\mathrm{g} $$
$$ \text{temperature change} = 1000\,\mathrm{K} $$
528 J. Not an insubstantial amount of energy, but significantly less than the bullet itself delivers. And it's also going to lose that energy, radiate it away to the air, as it travels to the target. Tungsten has a very low specific heat which means it does not retain heat well.
Let's get this out of the way, this will add nothing to its penetration capability. It won't "burn through" the armor, there just isn't enough energy. To give you an idea, 528 J is about the energy of a decent photography flash being relatively slowly and inefficiently transferred to the air and armor. Whereas a bullet transfers all of its kinetic energy very quickly by slamming into the target and rapidly decelerating.
"Ah ha, but there will be this burning hot hunk of metal inside the flesh of the target!" Well, this brings us to our next problem: water. Water absorbs heat like nothing else.
Our 4 grams of hot bullet will be cooled as it passed through the surrounding flesh. Flesh is mostly water. Unlike tungsten, water has a very high specific heat and can absorb over 30 times as much energy as tungsten by mass. And it takes even more energy to turn liquid water to steam (which would be really bad inside a body).
How much would 528 J of heat to do, say, just 10 grams of water representing the flesh in contact with the bullet?
$$ \text{temperature change} = \frac{\text{energy}}{\text{specific heat of water} \times \text{mass of water}} $$
$$ \text{specific heat of water} = 4.18\,\frac{\mathrm{J}}{\mathrm{g} \times \mathrm{K}} $$
$$ \text{mass of water} = 10\,\mathrm{g} $$
$$ \text{energy} = 528\,\mathrm{J} $$
13°C change. Lukewarm. Likely no appreciable additional effect to an already heavily traumatized area.
What about 2000°C? Double the temperate change, double the energy: 1056 J which can heat 10 g of water by 26°C.
What about 3000°C? Triple the temperate change, triple the energy: 1,584 J which can heat 10 g of water by 39°C. If it was already 37°C (body temperature) you're at 76°C which is hot enough to scald, but it's just 10 grams.
You might have even done them a favor by cauterizing the wound.
Modern Incendiary Rounds
Incendiary rounds exist primarily for two reasons: tracer and anti-material. Rather than heating the bullet, the incendiary component is provided by a chemical reaction. This is much more energetic than just heating metal.
Tracer rounds are designed to glow so you can see them in flight to assist with aiming. It's more about light than heat.
Anti-material rounds are designed to use their kinetic energy to penetrate the hard outer shell of a vehicle or building, and once through ignite. Usually the bullet contains some chemical that spontaneously combusts on contact with air after the bullet deforms on hitting a hard target. Unlike the wet flesh inside of a person, a combat vehicle contains lots of dry, highly flammable things inside to ignite and burn. Fire is a combat vehicle crew's worst nightmare.
However, your typical intermediate combat round (5.56 NATO or 5.45 Soviet) is too small to carry a significant incendiary charge. You need to scale up to full power rifle rounds like 7.62mm for the bullet volume to be large enough to make an incendiary payload worthwhile.