Here is the situation: We have 100w heatload coming from little die which has only 1cm^2 of surface area. We cool it with Ln2 as cold as possible (in normal atmospheric pressure) by using simple round container which is made of 99.99% copper. Lets assume that system is perfectly isolated so all the energy for Ln2 boiling comes from the 100w 1cm^2 surface area. Lets easen the situation by choosing cooling elements wall thickness 5mm so that we have to optimize only the bottom thickness. The inside bottom surface area is roughly 3,25*3,25*3,14=33cm^2 and the wall surface area inside 6,5*3,14*20-x (around 390cm^2). There are many holes in the task. First of all we should know the amount of wattage 1cm^2 of copper surface area can transfer to Ln2. This also depends on the distance from the heat source so the core surface area above the core moves energy much more efficiently to Ln2 than in upper parts of the pipe due the temperature difference. Lets assume the cooling element has smooth shining inside surface (bad for boiling but it makes sense for this task since the bottom thickness will be this way something else than "thinnest possible" since the extra surface area from upper section of the cooling element is needed for most cold result). I dont know what level of knoweledge this forum has but im thinking this as an "breakfast challenge" for some physics teacher or professor. I will give more information if there is anyone who has slightest idea how to calculate this (i know this goes 3-dimensional stuff when done right). Here is picture from the system: http://koti.mbnet.fi/ilkkahy/sekalaiset/container.JPG"