This topic has been bounced around for a while, and I believe I have found the most economical approach to getting the job done. The short answer is a roof de-icing cable ($70-150 usd) connected to a temperature controller ($40-$60 usd).
The key to heating mud is consistent, radiant heat. The best way to get this done is electric heating cable, the same ones that heat up the bathroom floors in fancy hotels. These give off 5 watts (17 Btus) of heating per foot, and the cheapest form of these cables is roof de-icing cables, which go from 60-200 ft in length. I use a 100 ft heating cable plus 200 feet of lined heating cable for an 800 lb mud pit. As for why there's free run cable, thick mud does not heat evenly so pockets of the pit will get hot and need stirring every few hours. Multiple cables also mean I can connect to two separate breakers and not overload a circuit.
Now, for the math: 1 btu heats 1 lb of water 1 degree F. Room temp is 68F, but if you have an in-ground pit (ideally insulated), this could be much lower. The target temp on your controller should be 100F, same as safe hot tub water. In my example, this means 800 x (100 - 68) = 25,200 BTUs to get to set point. The output for 300 ft of heating cable is (300 x 17) = 5,100 BTU/h, which puts heating time to around 5 hours if mostly water. Thick mud has less water and will heat faster. A few good stirs with a paddle to even out the heat at the 2 hour mark and before use keeps the temperature consistent. This does not account for heat loss, so it could take considerably longer for a poorly insulated in-ground pit.
For consistent heating in thin mud, it's best to build a frame and evenly space your heating cable. The pit I am heating is rather small with thick mud, making it simply easier to yank out with gloves and pull off most of the clay.
Some safety precautions as well:
Mud is a poor conductor of heat. Beware of hot spots and mix thoroughly with a paddle or a wooden board. Use your temperature probe to check spot temperatures for safety. Overheated mud can cause burns, so check before you sink in.
Also, never hop in the pit when the heating element is on. GCFI outlets are recommended to avoid shocks, and small punctures in the cable or water getting into the receptacle can quickly ruin your session. The mud will retain heat for hours, so don't rush yourself. 100 F is the generally accepted safe limit for hot immersion - don't forget to hop out and hydrate after 15-20 minutes.
Finally, note the max amperage of your outlets and controllers and ensure your plug loads do not exceed them. The amp draw is listed on the product box.
Do pumps works? Do they get ruined with mud? Would the pumps handle thickness of the mud? I am wondering if a 12V Portable Macerator Pump (used for RVs) would work, or 1500W Macerator Pump like this one: https://www.amazon.com/Macerator-Function-Utility-Macerating-Basement/dp/B0GTBB8YH7 . Do you have any updates to that old thread on pumping mud?
Pumps will only work if the viscosity is close to water. If it's the viscosity of sour cream or thicker, even the bigger pumps will lock up because the displaced mud will leave a cavity and trip out on low suction (or damage the impeller if the pump is cheap).
I have yet to test with a dual diaphragm pump, which is designed for high viscosity, but it requires a compressed air feed. All that equipment generates too much noise.
A simpler option might just be a remote control servo motor that pushes down on a plastic lid in a suspended container with a ball valve and hose at the bottom. Kinda like dispensing a large toothpaste tube... of pure mess.
Can confirm. Heating cable works great and relatively economical. Removes all issues with pumps and moving fluids, etc.
As you say, GFCI is ideal and definitely unplug / power down before you ever get close to the mud!
Seemingly not mentioned, but insulation is also a big key. Ideally you could have a few inches of rigid foam on the sides and bottom of the pit, and 4+ on the top to really seal in heat. But if that isn't practical, I've found even throwing down a piece of plastic on top to stop evaporation, and an old quilted bed comforter can help keep heat in.