Alas, none of your questions has anything to do with programming. Perhaps there are Forums or hot-air balloon enthusiasts.
The amount of lift depends on the difference between the mass of hot air inside the balloon and the mass of the same volume of air at ambient temperature. Because the heat causes the air to expand the mass of a volume of hot air at atmospheric pressure will be lower the mass of the same volume of cold air at the same pressure. PV = MRT where P is the pressure, V is the volume, M is the Mass, R is the universal gas constant and T is the absolute temperature. Details in any physics text book.
Alas, none of your questions has anything to do with programming. Perhaps there are Forums or hot-air balloon enthusiasts.
Also, I am sorry, but I thought that questions involving a propane burner controlled by an arduino, and its power source, would fall under the category Mechanics and Power.
You only get about 1/5th an ounce of lift per cubic foot of balloon. Getting enough lift just to carry a propane tank, much less a pneumatic solenoid, would need a big balloon. I'd suggest you consider a burner of lower complexity like a camp fuel stove. That would also allow a simpler adjustment of the flame by using a simple RC servo to twist a knob.
nehalda:
I thought that questions involving a propane burner controlled by an arduino, and its power source, would fall under the category Mechanics and Power.
Your questions are about selecting a burner - not about controlling it. When you have selected something suitable for your project we should be able to assist if you need advice about the control program or the interface between an Arduino and your chosen burner.
Not many people here have experience of proportional valves used with propane, try specialist suppliers
of propane equipement?
As for lift, that is a function of the balloon, not the burner, although obviously for each size of balloon there
must be a minimum size of burner it needs (probably a maximum too).
There will be regulations covering balloon aviation, have you done your research?