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| Fig. 1: Hot-Air Balloons in Cappadocia. (Source: Wikimedia Commons) |
Balloons have a rich history. The very first hot-air balloon was invented by Joseph and Etienne Montgolfier, which attracted "crowds gathered to watch them". [1] After their debut, hot-air balloons have become an important transportation method. Although such an importance gradually decreased as other more efficient transportation memthods emerge (e.g., airplanes and trains), hot-air balloons are still an integral part to the the economy of many regions that depend on tourism. An example of such is shown in Fig 1, with a hot-air balloon flying on top of mountains in Cappadocia. This short article will explore some of the most basic physical principles behind hot-air balloons through simple derivations and calculations.
Hot-air balloons operate under a relatively simple mechanism. In short, in order to create buoyancy, "ambient air is burned with propane and the hot exhaust gas creates buoyancy". [2] When the net buoyancy exceeds the total weight of the balloon, the balloon will rise and vice versa. To keep the balloon in stable flight (i.e., maintaing a relatively constant height), neutral buoyancy needs to be achieved, in which the net buoyancy force matches approximately with the weight of the balloon.
Traditionally, the force of buoyancy is given by
In this formula, F represents the net buoyancy force (i.e., lift), ρ represents air density, g is the acceleration because of earth's gravity, and V represents volume.
As Abe et al. points out in their book, "effective buoyant force" is "a function of the difference in the densities" between "the density of the external air and that of the internal buoyant gas". [3] In other words, the balloon's force of going upwards comes from "a buoyant gas of volume V" being "injected into a balloon". [3] With such knowledge, we can continue our calculation by calculating the effective buoyant force
In addition, the ideal gas law is given by
In this equation, p represents the absolute pressure, n is the number of moles, R is the ideal gas law constant, and T represents Kelvin temperature. The molar form of the law can be derived as
If we switch the terms around, we will get
With air outside and inside being similar in both their contents and absolute pressure, we can derive two equations: first, ρin = p ⁄ (R × Tin); second, ρout = p ⁄ (R × Tout). Combining the two equations, we obtain
If we plug this back into the effective buoyant force equation, we get
To simplify calculations, we can rearrange the equation
Let's create a hypothetical ground situation by adding a couple variables: the air temperature outside of the bag is rougly 300 Kelvin (26.85 °C), the air temperature in the bag is around 315 Kelvin (41.85 °C), the air density outside the bag is roughly 1.2 kilograms per cubic meter, the bag of the balloon is roughly 2,000 cubic meters, and the acceleration is roughly 9.8 meter per second squared. Let's also clarify on why some of the specific numbers are chosen. For temperatures both in and outside the balloons' bags, they are chosen as reasonble guesses in typical ground conditions. With the choice of air density, 1.2 kilograms per cubic meter was used in a work performed by Mandal et al. as the "reference air density". [4]
From here, we can calculate the net buoyancy force obtained. The equation above then becomes
| F | = | 2000 m3 × 1.2 kg m-3 × 9.8 m sec-2 × (1 - | 300°K 315°K |
) | = | 1120 Newtons |
When the balloon reaches steady flight, the mass such force can support a mass of
| Mass | = | F g |
= | 1120 Newtons 9.8m sec-2 |
= 114.3 kg |
Thus, in the example, if the balloon carries a total mass of roughly 114kg, it will achieve steady flight when the inside air temperature is roughly 15 degree Celcius higher than the outside air (i.e., when neutral buoyancy is achieved). If the temperature difference is greater, the balloon will climb upwards and vice versa. Adjusting the temperature of the air temperature inside the bag controls the behavior of the balloon. To keep the balloon in consistent flight, the pilot adjusts the flame to make sure the temperature difference is within the intended range.
We have explored how to calculate the net buoyancy force (in Newtons) a typical hot-air balloon provides during flight and the mass (in kg) such a lift supports when neutral buoyancy is achieved. Through a cascade of derivations, we obtained a formula that focuses on the temperature difference between air inside and outside the balloon's bag. The pilot can keep the balloon in flight through controlling the flame to adjust such temperature difference.
© Victor Wu. The author warrants that the work is the author's own and that Stanford University provided no input other than typesetting and referencing guidelines. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
[1] J. M. G. London, Fly Now!: A Colorful Story of Flight from Hot Air Balloon to the 777 'Worldliner' : the Poster Collection of the Smithsonian National Air and Space Museum (National Geographic, 2007).
[2] J. A. Jones, "Innovative Balloon Buoyancy Techniques for Atmospheric Exploration," IEEE 879305, 2000 IEEE Aerospace Conference, 25 Mar 00.
[3] T. Abe et al., Scientific Ballooning (Springer, 2009), pp 15-75.
[4] G. Mandal et al., "Comparative Analysis of Different Air Density Equations," Mapan - J. Metrol. Soc. India 28, 51 (2013).
