Wood ash has been used for centuries (as Early as 500 A.D!) in all kinds of processes such as creating fertilizer in the form of potash and producing lye which among other things can be used as an alternative for lime in the leather working process. Perhaps down the line it could also be used for pottery glazes or soap making.
Since you do need a high heat fire like a large fire pit to get the right kind of ash, maybe you could get it by creating something similar to a charcoal pit but without the top being covered?
Once you have the ash you could use the already existing barrel mechanics, adding wood ash to water and waiting for the water to dissolve the soluble materials to produce lye water.
If you wanted to go for a more realistic approach by modifying a barrel to act as filter by placing loose gravel and straw in the bottom of it and drilling a few small holes into the bottom. After this is set up you simply place the wood ash in the barrel and run water though it several times until it leaches enough lye from with wood ash.
In order to get the dried version you simply boil the water off in a clay pot (which is actually where the name 'pot ash' comes from)
