Umm. Where shall we start.
Assume the firing of a shot imparts energy to the barrel, and each shot imparts roughly the same energy (over simplified I know but a reasonable approximation). The steel has a certain heat capacity and we'll also assume that it same material is used in both a skinny hunting barrel and a heavy target barrel. The mass of the target barrel will be greater so the average temperature rise in the heavier barrel will be less per shot. Also the heat will travel reasonably quickly through the metal (being a good conductor) and the dissipation of heat will therefore be more dependent on the interface with the atmosphere, and the rate will normally be expressed as an amount of energy (ie speed) per given AREA per measure of temperature difference. Since the heavy barrel is wider it will have a greater surface area and hence dissipate the heat quicker, certainly to start with the hotter skinny barrel will loose energy faster, but the heavier barrel will always be a lower temperature.
Yes doing engineering I lived and breathed thermodynamics for months.
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