2 updated to reflect that it violates the rules
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I have an answer, but I'm not sure ifI'm not sure if it breaks the rules or not it breaksgoes against the rules or noton two fronts - drilling holes in the wand (thanks JBH) and violates the 13th century rules (thanks Nosajimiki). SoBut, I'm going to throw it out there and see if it worksanyway as an apocryphal answer.

I have an answer, but I'm not sure if it breaks the rules or not. So, I'm going to throw it out there and see if it works.

I have an answer, but I'm not sure if it breaks the rules or not it goes against the rules on two fronts - drilling holes in the wand (thanks JBH) and violates the 13th century rules (thanks Nosajimiki). But, I'm going to throw it out there anyway as an apocryphal answer.

1
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I have an answer, but I'm not sure if it breaks the rules or not. So, I'm going to throw it out there and see if it works.

First of all, you're completely right; this is a classic heat sink problem. But, you're also wrong; a gun barrel heats up because of convection from the gases in the gun barrel; the barrel itself is metal for strength, and also because you actually want the barrel to get hot so that the next bullet doesn't discharge prematurely because of a massive heat buildup in the rifle. This was the whole point of gatling guns back in the day; give each barrel time to cool down before being used to fire another bullet.

In point of fact, the whole problem you have with your wand in terms of thermodynamics is that it isn't like a gun barrel, which would make better sense. What you really want to do is conduct the heat away quickly so that it an dissipate in the atmosphere rather than cooking your hand. This is also why gun stocks and handles are not made of metal - They're designed out of wood, pearl, composites, etc. to shield your hand from the excess heat buildup.

I'm going to assume that your wand is some form of magical wood or other element that traps the heat to insane levels without spontaneously combusting. That is about the only model which could work the way you describe, although it also means that the biggest issue with your wand isn't that it gets hot, it's that it doesn't cool down fast enough. That said, the solution is twofold; firstly you need to reinvent a 'stock', or wand handle, and then you need to borrow an ammunition concept from science fiction.

Put more simply, you need to insulate the hand AND absorb the heat out of the wand fast enough so as not to overwhelm the insulative properties of the stock which you use to grip.

Let's deal with the stock first. If it wasn't for their combustible nature, wood or paper would actually make a good option here. But, in this instance, I'm going to suggest a fibrelass handle with a styrofoam core. This would have several benefits;

. You can turn this into a simple collar or tube which your wand slides into
. The styrofoam will grip the wand quite well, meaning less chance of it slipping out in a fight
. Easily replaceable if the styrofoam starts to wear.

Secondly, we need to create something like a thermal clip for your wand. Basically, the idea of a thermal clip in scifi games is that we have all moved to energy weapons, but those weapons can't disperse their heat so they capture it in a disposable thermal clip, which basically stores the heat and can be replaced quickly in battle. In gaming the idea is to keep the idea of ammunition. In your scenario, it's actually a pretty good analogue for the problem you face.

I'm going to suggest that you drill a small hole down the centre line of your wand (hence the idea that it could be outside the rules) and put a small rod of something like Tungsten in it, with a screw thread on the back end. Why tungsten? Because it has the highest melting point but still pretty good thermal conduction properties. Then what you need is a supply of Liquid N2 bulbs that screw or clip on at the base of the wand.

The idea would be that the tungsten conducts the heat to the bulb, which absorbs the heat and converts the nitrogen back to gas. Once it has done that to all the nitrogen in the bulb, you discard and replace.

Now, I know this isn't a perfect solution; how do you keep your N2 bulbs cold in the field? How many shots can a nitrogen bulb acting as a thermal clip absorb? Seriously, you want me to insert what in my wand? How is that going to affect the magical properties?

I suggest that you get one of your friends who wants to be a 'level 6' magician to assist you with some field trials (preferably one you don't like very much) but ultimately, the engineering of this is quite simple - you need to dissipate heat quickly while insulating your hand from it until the heat has dissipated.

For that reason, gun barrels getting hot is actually a good thing; it's touching them that's bad. Your wand needs a similar dynamic. Putting a styrofoam / fibreglass grip around it will hopefully protect your hand while the 'thermal clip' sucks as much of the excess heat out as quickly as it can, reducing the load on the insulated grip.

In any case, be very careful during your testing and don't let anyone do this who will be indispensable in your future magical career.