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The science
Buoyancy of hot air
Archimedes: the envelope of volume V displaces cold air of density ρa, while the hot air inside weighs less. Treating both as ideal gases at the same pressure, the net buoyant force is L = ρa V g (1 - Ta/Ti), with temperatures in kelvin. Lift-off needs L > m g, so the neutral temperature is Ti = Ta / (1 - m/(ρa V)).
Motion, heat and atmosphere
Newton: (m + ρiV + ½ρaV) dv/dt = L - m g - ½ ρa Cd A v|v|, where the extra terms are the carried hot air and the added mass of the sphere, Cd = 0.5 and A = πr². The envelope air obeys ρi V cp dTi/dt = 20 % of the burner power (10 MW at 100 % duty) - loss · (Ti - Ta). Air temperature falls 6.5 K per km and pressure follows the standard-atmosphere law p = p0 (T/T0)5.256, so the same temperature gives less lift higher up.
Honest simplifications: a spherical envelope, constant volume, no wind or turbulence, a fixed burner efficiency and heat-loss coefficient, no vent. Numbers are typical, not those of a particular balloon.
What to try
- Press Morning lift-off: the balloon stays on the ground until the air passes the neutral temperature, then climbs.
- Raise the total mass and watch the neutral temperature rise; the heavy preset barely lifts.
- Use Hold 1000 m: the pilot modulates the burner and the altitude settles.
- Set the burner to 0 % in flight: heat loss cools the air and the balloon descends.