## How does a hot air balloon work: Buoyancy in gases

Buoyancy forces act not only in liquids, but also in gases. This is illustrated below using the example of a hot air balloon.

### Buoyancy

In the article Buoyancy, the physical cause and the creation of the buoyancy force has already been discussed in detail. For the sake of clarity, liquids in which the objects were immersed were always considered. It was shown that for the amount of buoyancy force on the one hand the volume of liquid \ (\ Delta V \) displaced by the immersed object and on the other hand the density of the liquid \ (\ rho_F \) is relevant (with \ (g \) as gravitational acceleration) :

\ begin {align}
& \ boxed {F_A = \ Delta V \ cdot \ rho_F \ cdot g} ~~~~~ \ text {buoyancy force} \ [5px]
\ end {align}

The Archimedes' principle was derived from this equation, which states that the buoyancy of a body is precisely the weight of the displaced liquid. For this purpose, the product of displaced liquid volume and liquid density can be interpreted as displaced liquid mass \ (\ Delta m \) in the equation above. Then the product of mass and gravitational acceleration is shown as the weight of the displaced liquid \ (F_ {G, ver} \):

\ begin {align}
& F_A = \ underbrace {\ Delta V \ cdot \ rho_F} _ {\ Delta m} \ cdot g \ [5px]
& F_A = \ underbrace {\ Delta m \ cdot g} _ {F_ {G, ver}} \ [5px]
& \ boxed {F_A = F_ {G, ver}} ~~~~~ \ text {Archimedean principle} \ [5px]
\ end {align}

### From liquids to gases

Basically, one can now also imagine a liquid whose density is kept decreasing and decreasing in your mind. At some point the density of gases will eventually be approached. So there is no reason why a buoyancy force should not also come about in gases. In fact, practice shows that buoyant forces also act in gases. These are calculated using the same principles that apply to liquids. The density \ (\ rho_F \) therefore generally stands for the density of the surrounding fluid (regardless of whether it is liquid or gas).

Compared to liquids, the calculation of the buoyancy force in gases is usually easier to the extent that a body is usually always completely immersed in the gas immersed. The displaced gas volume thus corresponds to the body volume. In the case of liquids, however, it must be noted that these can only partially be immersed in the liquid. The displaced volume of liquid then only corresponds to the volume of the body actually immersed.

Since gases have relatively low densities compared to liquids, the displaced gas mass into which a solid body is immersed is often negligibly small compared to its own mass. In such a case, the buoyancy force can usually be neglected compared to the weight of the body. For example, a person displaces around 80 liters of air through their own body volume. With an air density of approx. 1.25 g per liter, the displaced air mass is 100 g. A person weighing 80 kg is apparently 100 g lighter due to the buoyancy of the surrounding air (this corresponds to only around 0.1% of the body weight).

However, in cases where the weight of a body is relatively small compared to its volume, the buoyancy force also plays a major role in gases. This will especially be the case when considering two gaseous substances. Helium-filled balloons in the air are a typical everyday example of this, in which the buoyancy force that acts causes the balloon to rise.

In this case, the helium is lighter than the displaced air mass, so that the buoyancy force that arises according to Archimedes' principle is greater than the weight of the helium. This buoyancy force is so great that it not only lifts the helium but also the mass of the balloon into the air (plus any weight of the fastening cord). Over time, however, the helium will escape from the balloon and it will shrink, so that at some point the buoyancy force will only be sufficient to keep the balloon floating. In this case the buoyancy force corresponds to the weight of helium plus that of the balloon. If more helium escapes, the volume of the balloon will eventually be so small that significantly less air mass will be displaced. The buoyancy force sinks according to the Archimedean principle. Finally, the buoyancy force can no longer compensate for the weight of the helium and that of the balloon and the balloon sinks to the ground.

### Hot air balloon

Hot air balloons use the same principle of buoyancy. Instead of helium, simply heated air is used. A hot air balloon consists of an airtight one Balloon envelopewhich is composed of several segments sewn together. At the bottom of the balloon is that basket attached with suspension ropes. They are on a frame burner attached, which are fed by gas cylinders.

Typical data of a hot air balloon are given in the figure below, which clearly shows how it works. The envelope of the balloon has a volume of approximately 4000 m³. This means that the balloon displaces a volume of 4000 m³ of cold ambient air. At an ambient temperature of around 24 ° C, the air has a density of around 1.17 kg / m³. With a volume of 4000 m³, the balloon displaces a cold air mass of around 4700 kg. According to the Archimedes' principle, this leads to a lift force of 47 kN. If the air inside the balloon were the same as outside, then logically there would be an air mass of 4700 kg in the balloon envelope. The buoyancy force of the air would then correspond to the weight force and the air would practically float in the balloon, but would not generate any effective, upward force.

Therefore, the air inside the balloon is heated to over 100 ° C with a gas burner. As a result, the air density drops and with it the air mass in the balloon. The decrease in air mass in the balloon can clearly be explained by the fact that the heated air expands and partially flows out of the balloon. Unlike a normal air balloon or helium balloon, a hot air balloon is not a closed system, but open at the bottom, where the gas burner heats the air (and also open at the top, more on this later). Note that the volume of the balloon practically does not change during the heating process, so that nothing changes in the displaced air mass or the associated buoyancy force.

At an internal air temperature of 104 ° C, for example, the density has dropped to around 0.92 kg / m³, so that in the 4000 m³ balloon volume there is only an air mass of approx. 3700 kg. The buoyancy force of 47 kN is now only countered by a weight force of 37 kN. Thus, the heated air creates an upward force of 10 kN. This is sufficient to lift a total mass of 1000 kg! The balloon envelope with around 150 kg and the burner and gas bottles with a total of 250 kg must be taken into account. After deducting a basket weight of 100 kg, there are still 500 kg left to transport the people.

Since the buoyancy force of a hot air balloon is determined by the displaced air mass (i.e. by the balloon volume) and can practically not be changed during the journey, the air mass in the balloon must be increased again in order to lower the balloon in a targeted manner. This is achieved through a hole in the top of the balloon, which is closed when ascending and can be opened with a rope to lower it. The hot, light air can escape upwards and cold, heavy air can flow in from below. The air mass in the balloon increases again and the weight becomes greater than the buoyancy force, so that the hot air balloon now sinks downwards.

### outlook

In the article Buoyancy, the creation of buoyancy in liquids was explained by the different hydrostatic pressures on the underside and on the top of the immersed object. The fact that buoyancy forces now also act in gases therefore suggests that there must be something like “hydrostatic” pressures in gases as well. In the same way as the pressure in liquids increases with increasing depth, the pressure in gases should also increase with increasing depth or decrease with increasing height. And in fact, this is exactly the reason why the atmospheric air pressure decreases with increasing altitude - for example on mountains. More information on this in the article Barometric Height Formula.