I saw a video on the MinuteEarth channel on YouTube discussing Lord Kelvin's error in estimating the age of the earth. Their explanation left me a bit...cold. They say that his primary error was his failure to account for convection in the mantle. (They also say that radioactivity doesn't account for the discrepancy, but that's another discussion.) I'm trying to mentally compare the cooling behavior of a static body against that of a body with convection going on, but I don't have the mathematical expertise to do it. So I'm hoping that someone with the math skills can help me to visualize it well enough to get a qualitative feel for the comparison.

I can imagine a static planet being made up of infinite concentric spheres, each losing heat to its containing sphere. So the one in the very middle would lose heat more slowly than all the others, and the outermost would lose heat faster than all the others. So far, so good, I think.

If I imagine a planet with convection going on, it seems that the zone of convection would behave, more or less, as a single sphere at a more or less even temperature. It would gain heat from the inner non-convecting zone, and lose heat to the outer non-convecting zone. Still, I think, so far, so good.

But I'm stuck on one thing: it seems to me that a convecting planet would lose heat more quickly overall than a static planet. If this were the case, then it seems that Kelvin was wrong in the other direction: if he found, for example, that a deep mine shaft was at 100˚ and concluded that a (static) Earth is 20m years old, then it seems that the deep mine shaft on a convecting Earth would cool to 100˚ in less than 20m years because of the accelerated loss of heat. Someone suggested to me that the temperature of the mine shaft on the convecting earth would take longer to cool to 100˚ because the top of the convection zone is much hotter than it would be if it weren't convecting. That makes sense too, but I don't have the math to figure out which effect (accelerated cooling or transfer of extra heat from the extremely hot core) would take precedence.

Can anyone help me with this, or do I need to take two semesters of physics to figure it out?