I didn't find out how much a given volume of air masses, but I believe it is 1.25 kg/m^3. I can live with that. I estimate that a car, with 5 m^3 of air and 5 m^2 of surface area, heats up 15 degrees C rather quickly; easily 5% Kelvin above the ambient temperature (288 K). I guesstimate that hot air in such a car masses ~5% less than in a car with open windows, and provides ~60 g/m^3 of 'lift' (I'm mixing the wrong units- sue me!). If I had a 1 m^3 balloon with a 1 m^2 cross-section and massing <60 g, it would rise.
Let us make a plastic hot air balloon 10 meters across, with a 78 m^2 cross-section and a volume of ~500 m^3. The sun heats the air enough to raise ~97 kg of air 15 degrees C, or ~630 kg of air ~2.4 C, or 2.4 Kelvin, .9% above ambient, something like 5.5 kg of 'lift'. That works out to 17 g/m^2, or ~1/2 ounce per square yard. Possible, but iffy. On the other hand, if the sun heats the thing all day, we could see 6 ounces of lift per square yard of material. Although, the sun is heating up the ambient air, too...
I am making the following assumptions-
• Passive solar heating provides energy to heat air equal to 60g/m^3 times 150 K above ambient temperature (I really wish I could put that in Joules, Newtons and Calories or BTUs...).
• At lower pressures, the air masses less and is heated up more.
• The material is thermally opaque, retaining heat fairly well (whatever that means 8-). 24/7, the internal volume is warmer and lighter than the external air.
A 100 meter Hot Air Solar Power Bubble (TM 8-) is about 7,800 m^2 of cross-section, 31,500 m^2 of surface area, and half a million m^3 or 650,000 kg of air, which we can probably heat up 60*150*7,800/650,000 = 9,000*7,800/650,000 = 9*7.8/.65 = 9*12 = 108 K. 396/288 or rather, 288/396 = 72/99 = 24/33 = 8/11 is 3 pounds of lift for every 11 pounds of air, ~180,000 kg. 650,000 kg of air weighs 715 tons, or 195 tons of lift. The surface area of the hot air balloon is ~35,000 square yards, at 2 pounds/SY, is 35 tons. There's 160 tons left for an overage of 5-fold, 10 lbs/SY, and the power generation part of this scheme.
A 100 meter bubble has about 10 MW/s of sunlight falling on it during the day and in direct sunlight; at 1% efficiency, that's 100 KW. Nothing to sneeze about! A 1km bubble should be good for 10 MW, and heat up 1000 times as much air 1/10 as much, 10.8 K, 288/298.8, or ~8/8.3. But that still works out to a hundred times as much lift and a hundred times as much weight in material.
Now, on to the 'Cheap Access To Space'! If I wanted this bubble up at 22 km for my own purposes, we need to figure out how much lift less dense air, heated more, could potentially generate. That's four halvings, or ½ to the fourth, 1/2^4, 1/16 times, about 80 g/m^3 to 173 K. the ambient is less at that altitude, but we'll go with 258 K, 15 C below freezing. We're talking about 173/431 times 80 g/m^3, ~40%, 32 g/m^3, a little over one ounce of lift. Fortunately, that's ~16 billion grams, 16 million kg, >7,000 tons of lift. The thing weighs in at 3,500 tons, half the lift. Even a 10 MW of wind turbine, at 1kg/100W, is <100 tons (or 25 MW and 250 tonnes). The 4 million kg per km of maglev launcher (based on SpaeTram) I want to hang under it is a problem, but we can always scale the sucker back a little, 100 tonne orbiter, 35 tonnes of cargo or 100 passengers.
(Whew! Take a breath and say 'paper rocketry', Vinnie!)