This may be calculated. (This formula is only for balloons using hotair. Not other gases)
When making the model for the Hot-Air balloon simulation (http://www.powersim.no/WebSims/Hotair), I got the formulas from a hot air balloon pilot (Ole Anton Haugland) at the Norwegian University of science in Tromsø. He teaches physics as well, and he gave me the following formula for the lift of a hot air balloon: F = V * g * P * 1.29 * 273.15 * (1/To - 1/Ti) Where: F = The lift force of the balloon in Newton V = Volume of balloon in cubic meters (m^3) g = accelleration of gravity = 9.81 m/s^3 (In Norway, and further north it is 9.82, because of the rotation of the planet
P = Air pressure in bar (usually 1 at sea level) 1.29 is a constant for how much the air expands when the temperature is increased 273.15 is the absolute point of zero temperature To and Ti is the temperature outside and inside of the balloon in deg KELVIN! (You add 273.15 to the temperature in deg Celsius)
If you want your answer in kg you have to divide by g. (or just remove g from your formula) You may try this against some of your tables for lift, to verify it's correct.
Back to your question: What is your maximum possible height? As you see, the lift force is dependant on both the air pressure and the difference in temperature. When climbing, you get advantage of the decreased temperature, but disadvantage of the lost air pressure. Near ground level the advantage is more significant than the lost pressure, but when getting higher you will need more and more temperature to keep the lift force.
What you really want to know, is how high may the balloon go with a constant high temperature inside (your maximum allowed temperature). Let's say you keep a constant temperature of 103 deg Celsius, in a balloon of 1590 m^3, and your total weight is 320 kg. We also assume the outside temperature to be constant -40 deg Celsius. This could be the temperature at a very high altitude. To = -40 + 273.15 = 233.15 Ti = 103 + 273.15 = 376.15 This gives the lift force: (in kg. ...I ignore the g) F = V * g * P * 1.29 * 273.15 * (1/To - 1/Ti) Kg_lift = 1590 * P * 1.29 * 273.15 *(1/233.15 - 1/376.15) Kg_lift = 1590 * P * 0.57455 Kg_lift = (P * 913.5) kg*bar
This says that if the pressure is 1 bar, the lift of balloon will be 913.5 kg You want to know how low the pressure may be to lift your balloon; 320 kg. Than we have to turn the formula: Kg_lift = (P * 913.5) kg*bar P = Kg_lift / 913.5 P = 320 / 913.5 P = 0.35029 bar
That pressure should be at an altitude higher than Mount Everest, where the pressure is about 0.4 bar. If we want to do this calculation more advanced we would combine it with a variable value for the outside temperature, and so on. In this calculation the temperature is -40 deg regardeless of the height. I guess the temperature would be even lower at an altitude of 8000-9000 meters. (and give even better lift force)
QUESTION: Do anyone have a table showing altitudes for a given air pressure? This shouldn't be too hard to find....
The complete calculation would be: F = V * g * P * 1.29 * 273.15 * (1/To - 1/Ti) Kg_lift = V * P * 1.29 * 273.15 * (1/To - 1/Ti) P = Kg_lift / (V * 1.29 * 273.15 * (1/To - 1/Ti))
Just to verify: P = 320 / (1590 * 1.29 * 273.15 * (1/233.15 - 1/376.15)) P = 320 / (1590 * 1.29 * 273.15 * 0.00163057) P = 320 / (1590 * 1.29 * 0.44539) P = 320 / (1590 * 0.57455) P = 320 / 913.54 P = 0.35 qed.
This became more than I planned, Sorry for using much of your time. If you have any comments to these calculations, or the simulator, please give response.
Kind regards, Steinar Moen
