Hot air balloons float in the air because of the difference in density between c

Question

Hot air balloons float in the air because of the difference in density between cold and hot air. In this problem, you will estimate the minimum temperature the gas inside the balloon needs to be, for it to take off. To do this, use the following variables and make these assumptions:

The combined weight of the pilot basket together with that of the balloon fabric and other equipment is W.

The volume of the hot air inside the balloon when it is inflated is V.

The absolute temperature of the hot air at the bottom of the balloon is Th (where Th>Tc).

The absolute temperature of the cold air outside the balloon is Tc and its density is ?c.

The balloon is open at the bottom, so that the pressure inside and outside the balloon is the same.

As always, treat air as an ideal gas.

Use g for the magnitude of the acceleration due to gravity.

Question>>>

For the balloon to float, what is the minimum temperature Tmin of the hot air inside it?

For the balloon to float, what is the minimum temperature  of the hot air inside it?

WcWTc

WWcTc

WcWc+WTc

WcWc?WTc

Explanation / Answer

(Wc/Wc-W) * Tc

"For a given W (sometimes known as the payload), the larger the balloon volume (i.e. larger Wc), the lower the temperature required for the balloon to float.

Similarly, for a given balloon volume, the lower the payload to be lifted, the lower the required temperature.

Sports balloons have a volume of about 3000m3?90,000cubicfeet. This means that Wc?30,000N. The payload may be about 500kg or 5000N. So Tmin?60?C=140F.

Here is a surprisingly different approach to thinking about the balloon. You may have learnt that the pressure p in a medium decreases with the height h as

p=pbottom??gh.

where ? is the density of the medium. Here, since ?h<?c, the pressure outside the balloon decreases with height at a much faster rate than the pressure inside the balloon. This pressure difference causes the balloon to rise. This approach also suggests a method to improve on this model. In general the density will not be constant throughout the balloon, however, calculating the density leads us to the study of convection, which quickly becomes quite complicated."

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