I finally have done some testing using bottles actually filled with...
... water. Didn't even cap them.
Note the time scale on the bottom: hours. With all the "thermal mass" of the water, it takes a lot more energy to cause a temperature shift in the chamber. With air the temperature would swing out of control band in minutes; with water it takes hours and really doesn't ever venture outside the target range. You can still see the temperature shift when the relay turns off, the temperature moving up a few tenths of a degree.
(Incidentally, from a conversation I had with a guy who used to design appliances, this is why burners on ranges don't have temperature settings on their control knobs, just generic levels (like "1" to "10" or "Low" to "High"). Without knowing the type of pan being used (aluminum, cast iron, stainless steel...) and what is being cooked (grilled cheese, bit pot of water for boiling pasta, steak...) it is virtually impossible to design a control system that will keep the item being cooked at the required temperature without taking forever to reach that temperature or wildly over overshooting the temperature (that is, burn the food).)
I think we're ready to give this a shot. I'll check my supplies but I hope to have a batch of root beer going by the weekend. Expect pictures, though they will probably not be very exciting.
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