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Average hot air balloons have a diameter of about 18 m. The largest out there is roughly 32 m.
My question is, are there any reasons other than fuel concerns that would prevent having a larger balloon, say around 50-60 m as a lifting body? I.e. problems with air mixing, pressure differentials, etc in such a large volume?
You need to figure out how to land a large balloon in conditions with no wind, without the passengers ending up underneath the collapsed balloon.
In windy conditions, you also need a larger landing area to avoid damaging the collapsed balloon, and at some size point you will need machinery to pack it for transportation because of its own weight.
The largest passenger carrying balloon (as of 2017) was around 40m tall, which gives some idea how much ground area is required to operate it, remembering that balloons are not steerable so you don't have the option of landing at a purpose-built site.
Also, the only real purpose of having a bigger envelope is to carry more load, and eventually you hit design problems trying to make a completely flexible structure like a balloon strong enough.
As the volume of a hot air balloon is increased, so increases its surface area- through which the burner heat is lost. Since the volume increases faster than the area for a spherical balloon, more volume means more lift and lower heat losses for that amount of lift. However, more volume also means more fuel burned to fill the balloon to stay aloft, and beyond a certain balloon size the burners have to be running continuously to maintain lift. That determines the practical maximum size of a hot air balloon for a standard set of burners.
This also means that in general, a balloon is designed for a certain payload specification. This sets the balloon volume and hence the fuel burn rate, and from that comes the burner capacity in BTU's.
What you need to do is look closely at the physics of the inflated envelope and how the shape is determined. Let's limit the variables and hold the temperature differential constant as we make the balloon bigger. As the envelope height dimension grows, the pressure differential between the inside and outside grows. So that if the envelope tension is the same, the Gaussian curvature will increase. That's not what you want, you want the curvature to decrease in order to make the balloon bigger. So now you have to decrease increase the envelope tension in proportion to the square of the size change in order to maintain dimensional similitude. You want most of the envelope tension to come from the payload support tendons, and little from the hoop tension.
So you have to solve a fairly complicated but reasonably well conditioned set of equations that account for -
You can definitely set this up in Excel and use the add-in solver to optimize the envelope parameters.
As sanity check, if we start with a balloon with 20 tendons at 25# tension each, and double the linear scale, we now need 40 tendons to maintain panel size, and need to tension them at 100# to keep similarity with the original. Thus lift (total tension) goes with volume, as we would expect for similar shapes.
