Side view of a balloon climbing through a thinning atmosphere. Heat the envelope with the burner, watch lift beat weight, and see where it levels off.
Envelope air at temperature Ti and ambient air at T0 share the same pressure, so by the ideal gas law ρi = ρa T0/Ti (kelvin). Archimedes gives the net lift F = (ρa - ρi) V g - ms g, where ms is the fabric, basket, burner and payload mass. At sea level one cubic metre of 100 °C air lifts about 0.28 kg more than one of 15 °C air.
The atmosphere is isothermal here: ρa(z) = ρ0 e-z/H, H = R T0/(M g), about 8.4 km. Lift therefore falls with height and the balloon floats where ρa(z)(1 - T0/Ti) V = ms.
Motion: (ms + ρiV + ½ρaV) dv/dt = F - ½ ρa Cd A v|v| (the ½ρaV term is the added mass of the air pushed along). Heat balance: ρi V cp dTi/dt = Pburner - U S (Ti - T0), so the steady envelope temperature is T0 + P/(US) with S the envelope area.
Simplifications: no temperature lapse or wind, uniform envelope temperature, constant U, no vent or deflation, a burner treated as a continuous average power. Fabric temperature is limited to about 120 °C.