Best cylinder water container shape to prevent heat loss There's a workshop in my town and I want them to build me a boiling pot for homebrewing, my question is
What should be the proportions (radius to height) of the most efficient pot (best heat conservation)?
The pot must be made of stainless steel; I'm not sure how relevant is this but here it is
 A: In principle, heat transfer out of the pot must occur through the surface area of that pot.  The shape that holds a given volume but has the least surface area is a sphere, so the spherical shape will have the least area available for heat transfer to the environment.  However, I doubt that you want a spherical pot for boiling a liquid.  A more convenient shape is a cylinder, and the cylindrical shape that holds a given volume and also has the smallest surface area is a cylinder whose diameter equals its height.  
Such a shape will lose temperature more slowly than other shapes, but there is another way to look at this problem.  Due to the small surface area of the above cylinder, there is also less surface area available for heat transfer when you are boiling the contents of the cylinder.  Thus, it easily seen that you may require more fuel and a longer "flame time" to boil the cylinder's contents.  The question then becomes, are you interested in minimizing the amount of fuel that it takes to boil the pot's contents, or are you interested in having the pot cool down as slowly as possible?
A: there are two mechanisms of loosing heat here:
1. heat transfer from liquid (natural convection) through solid (conduction) and to the ambient air (natural convection)


*Phase change - liquid layer on free surface will evaporate and absorb heat from the liquid.


You can do the following to reduce 1:


*

*minimize the surface area where heat transfer takes place, the smallest surface area per volume is in a sphere

*make the walls thicker to increase resistance in conduction therm (you can't really do anything about the convection - in principle the convective heat transfer on vertical walls should be higher than for horizontal walls due to difference in turbulence - natural convective cells go up-down because the gravity works in this direction, but the difference will be small imho)


You can do the following to reduce 2:


*

*make tha area of interface smaller ( amount of liquid that evaporates is proportional to it)

*ensure the path from liquid-air interface to ambient air is long, so the vapour must diffuse (which is very slow) before it gets swiped by the ambient air. (it's a still-air insulation)


You will get best results by ensuring thick walls and a cap for the pot. 
edit:
you can also create cylinder with double walls - the air between solid will lower the heat loss through the walls significantly. That would be the most effective way, but probably difficult to make. 
