Friends and I have made several portable hot tubs over the years. Anything from a low volume two person to a 20 seater. Here is what I can share from our experiance...
1. It takes a great deal of heat to bring water to tub temp. You need massive heat output and a long time. Our best versions required at least 6 hours to heat up. If you want to use it in less time go with a sauna. A sauna is MUCH less work, uses about 1/50 as much fuel and heats up in as little as 30 minutes.
2. The lower the volume the less water to heat, reduce the height of the water to a minimum and be thinking a tight 4 person tub not the 20 seater. (We built a 20 seater once, it took over 14 hours and about 1 1/2 cords of wood! Once we got it going it would coast along with just a little wood now and then.)
3. Almost any water tight structure can be made into a hot tub. Kiddie pools, 6 mil. plastic tarp and a ring of hay bales, foam sleeping pads wrapped into a ring and lined with a light plastic painters tarp. Continous liners work well, joints of any kind did not.
4. We had the best results from submersible type heaters. We found local wood or 20 lbs propane cans to be the best fuel sources. External heat exchangers were not as efficient and required a pump for circulation. A submersable wood heater can be as simple as a 35 gallon drum weighted down with rocks. An easy propane heater can be made from a U shaped piece of iron pipe and a propane weed eater like this..
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Submerge the pipe with both ends above water and attach the burner nozzle to one end.
Note: things below the water line do not get too hot for short skin contact, however, things above the water line can get red hot. Don't let the heater have direct contact with the liner, used some kind of standoff for at least a 3/4" seperation.
5. Smaller is better, go for the lowest seating capacity and the minimum water volume to reduce fuel cost and heating time. Insulating the bottom by placing foam sleeping pads below the liner worked well, insulating the top (other than a wind cover) or sides did not have a big effect.. You can make up for poor insulation with more heat, fuel might be more portable than insulation. Increase your heater efficiancy by designing it to have a maximim wet surface. The larger the under water surface area the better, avoid large above water surfaces that just waste heat.
Wood fired heaters benifit from a flue and a fresh air vent, adding small fan to the fresh air vent can significantly increase heat output. Think of the difference between a cozy camp fire and a forge, HUGE amounts of heat can be created with a forced air wood fire. On the other hand, the propane heaters are super simple, just a nozzle and a pipe. As a result they are the most portable and require the least attention.
6. Hot water gets really skanky after a day, use some bromide tablets to go for a few days.