Wayne as a physics teacher I have to take off points because you didn't show your work in that equation.
Just kidding but I really didn't follow the math. Here is my math for my tank and pot:
40 gallons of water and milk (24 gallons of milk; 16 of water) = 151 kg liquid with an average specific heat capacity of around 4032 Joules/kg (milk is 3930 and water is 4186) (this doesn't take into consideration the metal in the pot)
raising temp from 31C to 39C (8 degrees Celsius) in 45 min or 2700 seconds
Energy needed = 4032 Joules/kg x 151 kg x 8 degrees Celsius = 4,870,656 Joules
Power needed = 4,870,656 Joules / 2700 seconds
Power = 1804 Joules/ seconds or 1804 Watts
I guess with the steel pot added and the heat lost through convection, radiation, and conduction my setup works out about right because I can turn it on and I don't have to turn it off on a full batch. It takes exactly 45 min to heat it 8 degrees. If I use less milk though I really have to watch it so I really want a thermostat. My problem is that when I have to cool it down I am in trouble.
I figured your setup (wow that's a lot of water!) and got a different answer (but close) than you so I may be figuring wrong.
100 gallons of water and milk = 378 kg liquid with an average specific heat capacity of around 4125 Joules/kg (milk is 3930 and water is 4186) (this doesn't take into consideration the metal in the pot)
raising temp from 38C to 51C (13 degrees Celsius) in 45 min or 2700 seconds
Energy needed = 4125 Joules/kg x 378 kg x 13 degrees Celsius = 20,270,250 Joules
Power needed = 20,270,250 Joules / 2700 seconds
Power = 7508 Joules/ seconds or 7508 Watts
If this is right, you could take out 1 of the elements and probably let it run wide open and have a perfect cook. I still hate thermodynamics.
I am considering putting my heaters through the walls of my water bath. What fittings, o-rings, sealants etc. did you all use to penetrate the walls of your water bath vessels?
My solution was pretty simple: I cut a hole and welded a nut the size of my element on the pot and just used Teflon tape to seal it.