Sky Lanterns

What is modelled

A paper lantern is an open-bottomed envelope full of hot air, heated by a wax burner at its mouth. The gas inside is warmer and so less dense than the night air, and the difference in weight is the lift. The page integrates the heat balance of the gas, the vertical motion with drag and added mass, the drift in a sheared and gusting wind, and the burn-out of the fuel. Lantern 1 of the group is the one shown in the readouts and graphs; the others follow the same equations with masses and burner powers varied by a few per cent.

Equations

• Standard atmosphere: Ta(z) = Tg − 0.0065 z, p = p0 (Ta/Tg)g/(R·0.0065), ρ = p/(R Ta), R = 287.05 J/(kg K).
• The envelope is open, so the air inside is at the outside pressure and its mass is ma = pV/(R Ti). Net upward force: F = g (ρ V − ma − m − mfuel); it vanishes at Ti = Ta / (1 − mtotal/(ρV)).
• Heat balance of the gas at constant pressure: C dTi/dt = Pburn − U A (Ti − Ta) with C = ma cp + mpaper cpaper. The burner power is P while fuel remains; fuel burns at P / 43 MJ/kg (wax). Steady state: Ti − Ta = P/(U A).
• Vertical motion: (m + ma + ½ ρ V) dv/dt = F − ½ ρ CD A v|v|. At constant gas temperature the climb speed tends to √(2F / (ρ CD A)).
• Horizontal motion: drag towards the wind, (m + ma + ½ ρ V) dvh/dt = ½ ρ Ch Aside |u − vh| (u − vh), with wind u(z) = u10 (z/10 m)α times repeatable gusts and a small veer.
• Because the lift scales with the air density, the lantern climbs until lift equals weight at the steady gas temperature: the float ceiling, found by bisection and shown in the readout and the altitude graph.

Checks run in the page

• Gas temperature needed to float: closed form against bisection on the assembled force function.
• Terminal climb speed of the integrated equation against its closed form.
• Energy conservation of the heat balance: burner heat equals wall losses plus enthalpy gained.
• Air density against the published standard-atmosphere table at five heights.
• Float ceiling by bisection against the mean height of a long time-stepped flight.

Honest simplifications

• The gas is a single well-mixed temperature; real lanterns have a hot plume near the burner and cooler air at the top. The wall coefficient U lumps convection, radiation and the loss of hot gas through the open mouth into one constant (6 W/m²K by default); burner power, mass, fuel, drag coefficients and the 60 % paper share are representative values, not measurements of any product.
• The gas does not cool by expansion as the lantern climbs, and the ambient pressure only enters through the density; the wind, gusts and veer are a repeatable synthetic profile, not weather data.
• The envelope keeps a fixed volume and shape, and has no flutter, rotation or pendulum sway; vertical and horizontal motion are only coupled through the mass.
• The group follows the same equations with small variations; the crowd, houses, lake and stars are scenery. A flying flame is a real fire hazard, and lanterns are banned in many places; this page is a physics model only.

• Run the calm-night preset: the gas heats until lift beats weight, the group is released and climbs while the wind shear carries each lantern downwind; the dashed line is the float ceiling.
• Pick the heavy lantern or the weak burner: the gas never reaches the temperature needed to float (first graph) and nothing leaves the ground.
• Try the cold night and the windy night: colder air is denser so the same burner lifts more; wind changes only the drift, not the climb.