Presets
Playback
Balloon and air
Burner and vent
Wind and altitude hold
Layers
Model check
The science
What is modelled
A hot-air balloon is an open-bottomed bag, so the air inside is at the same pressure as the air outside but hotter and therefore less dense. The ideal-gas law gives the hot-air density rho_in = p / (R T_in), and the net upward force is L = (rho_air - rho_in) V g - m g = V rho_air (1 - T_air/T_in) g - m g, where m is the mass of envelope, basket, burner, fuel and crew without the hot air. The balloon floats when L = 0, at the temperature T_float = T_air / (1 - m / (V rho_air)).
Heating and cooling
The envelope air obeys m_in c_p dT_in/dt = Q_burner - UA (T_in - T_air) - (vent term). UA is an effective loss conductance (fabric, porosity and the open mouth lumped together, h_loss x envelope area). With constant burner power the temperature excess settles at Q/UA, with time constant m_in c_p / UA, a few minutes: this is why a balloon answers a burn slowly and why pilots fire short blasts. The parachute vent releases hot air, modelled as a first-order pull towards the outside temperature. Fuel (propane, 46 MJ/kg, 90 % burner efficiency) is consumed and the balloon gets lighter.
Vertical motion
(m + m_in + C_a rho V) dv/dt = L - 1/2 rho C_d A v|v| with a sphere-like envelope, C_d = 0.45 and added-mass coefficient C_a = 0.5. The terminal speed is sqrt(2 L / (rho C_d A)). Air properties come from the ISA troposphere: temperature falls 6.5 K per km and pressure follows (T/T0)^5.256, so lift per cubic metre falls with height and sets a ceiling where lift vanishes at the 120 C fabric limit.
Simplifications and checks
Lumped envelope temperature (no stratification), no solar heating, no humidity, constant volume (no envelope breathing), a straight wind profile u = u0 + shear h carrying the balloon, and a drag coefficient taken from a sphere. In-page checks: (1) the closed form L = V rho_air (1 - T_air/T_in) g matches buoyancy minus hot-air weight, and ISA sea-level density is 1.225 kg/m3; (2) the float temperature found by bisection matches the formula; (3) a constant-force ascent integrated numerically matches the exact tanh solution, and a heat balance at a chosen burner power settles at Q/UA above the air.