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I don't really have time or inclination to correct you math, I pointed out it was wrong I would think you could have worked out your error with a little effort. Are you still at school? Area of a 12" diameter circle is 12pi. Area of a 10 inch diameter circle is 10pi. The area of the gap is...

just briefly five errors there. 1) The main error. It's not 30% more thermal mass, its 13% more, check your math. 2) Your surrounding water is open to the air so will lose/gain a lot more heat than the enclosed barrel encased in plastic. 3) your surface area to voume ratio relates to a sphere...

It will add to the thermal mass that is true. But unless you sit the carboy in a container much bigger than the carboy it won't be a lot. Easy to work out, the increase in thermal mass is the volume of the water left in the container the carboy sits in (not the volume of the container) divided...

Don't do that water is a conductor not an insulator. You would just be increasing the surface area contact of the environment with the beer. Better to wrap the carboy in a towel and sit it on some carpet or cork or polystyrene.

the book I'm reading says after fermentation is over, cool the cider and rack it off. It was written in 1917 though. I'm still thinking leave it warm for a couple of weeks before racking. Surely this is a resolved question over the 90 odd years since 1917 (the book also says wine benefits...

Right I'm sure I can describe this without pictures, might just takle a thousand words. First thing is I don't ferment in a bucket. I ferment in a plastic beer brew barrel, pretty solid, holds 5 gallons and is the same as the one I rack into to store and keep the cider. The barrels are...

about 11C. The barrels have aquarium heaters in them which keep the liquid at a constant 20C (as I said) so does it matter?

I have two 5 gallons barrels about to finish primary fermentation. They are bubbling about 5 a minute. They both have aquarium heaters in them keeping them at a constant 20C (68F). My question is after the bubbling stops completely do I leave them in the barrels with the heaters on or off...