Mathematical Influences on the Design of Hot Air Balloons

The Basics

Let's look at the main components of the modern day hot air balloon. (drawing is courtesy of www.howstuffworks.com; see http://travel.howstuffworks.com/hot-air-balloon.htm)

From the design aspect, the key elements are:

Envelope - This consists of the panels, the gores and the parachute valve.

The gores provide the majority of the envelope and are made of nylon with sewn-in webbing. Nylon provides strength, is lightweight and has a relatively high melting temperature...always a consideration, when you'll be heating the air within to more than 100 degrees above the ambient air temperature.

The panels are what actually make up the gore, reaching from the skirt of the balloon to the parachute valve.

The parachute valve is controlled via a cord that will run to the basket area. The valve is used to both control the rate of ascent, as well as allowing the balloon to sink.

The "lightbulb" design of the balloon envelope came via trials with using a sphere as the initial shape. When the sphere was inflated, is was seen the there was no easy way to "fill out" the bottom of the sphere, since the realtively warmer air would stay at the top. The designers then removed the excess fabric that would not "inflate" and this resulted in the familiar "lightbub" shape. Other envelope designs have resulted that have taken note of this inflation issues, such that some uniques "cubicle" shapes and combination shapes have resulted.

A mathematical problem would be to determine how much fabric is saved via the change in shape from a sphere to the "lightbulb" or a "filled ice cream cone". In this case, we would need to calculate:

1. The volume of a cone that is , say 3X's the radius in height. i.e. 1/3 * pi * r * r * (3r) --> pi * r * r * r

2. Add in half of a sphere at the top of the cone. i.e. 2/3 * pi * r * r * r

3. Then subtract off the volume that is removed from the bottom of the cone up to the skirt. Take this height to be at r/2. Note, from similar triangles, we can determine that the radius at this point in 1/6 the radius at the top of the cone. (How??? because the height of this triangle is only 1/6 that of the total...3r compared to r/2) i.e. 1/3 * pi * (r/6)*(r/6) * (r/2)

Design of the envelope centers around the amount of wieght that must be lifted. It is known that at sea level, the density of air is such that 1 cubic foot of air weighs about 28 grams. By increasing the air temperature by 100 degrees F, we can achieve a density of 21 grams per cubic foot. This means, that a 1 cubic foot balloon, heated to 100 degree F will lift about 7 grams. Since there are about 455 grams per pound, that means, we need to size our balloon by 65 cubic feet to lift 1 pound.

Most balloons are sized to lift about 1,000 lbs. So that would mean an envelope that would enclose 65,000 cubic feet. If we could construct a spherical envelope that enclosed this volume, then the radius of our sphere would be:

cube root of 15,525 or about 25 feet.

If our balloon were spherical and 30 feet in diameter, we would be able to lift about 1,740 lbs.

The reader can experiment with the volumes of other shaped balloons. Another typical shape is that of a cylinder (or the "Soup Can"). Again, this type of balloon may be somewhat modified to include the skirt and appropriate tapering to get to the skirt needs to be determined.

Wicker Baskets - Wicker was chosen due to the fact that it is lightweight and it can withstand the shock of landing/running into obstacles. A more rigid basket would relay the shock of landing to the passengers, whereas with the wicker, much of the shock is absorbed and not transferred.

Burners - Includes the controls. The burners can be operated to allow the liquid propane to be burned as liquid or as gas. The burning of the propane gas is more efficient than the liquid, however it is noisier. Burning of liquid propane is preferred in quieter suburban areas and also around livestock in rural areas so as not to disturb the livestock.

Burner design takes into account this functionality of either preheating the liquild propane, to allow it to turn into a gas, or to turn off this preheating, thus allowing the liquid propane to be burned directly. Safety and control also come into play with the burners.

Burner, looking up into the balloon.

Propane Tanks - Number and size relative to amount of time and the size of the balloon. These are kept within the basket area, so that transfers from tank to tank can be easily handled by the pilot and crew.

The size and number of the propane tanks is relative to the amount of time the pilot cares to have th balloon loft and the size of the balloon. The typical air time is about 2 hours. This is done while the sun is no more than 30 degrees above the horizon. These 2 hours are taken typically just after sunrise and then again just before sunset. Since the size of tanks can vary from mfg. to mfg. as well as the burner control design, there is no general way to calculate this value. It will vary from balloon to balloon. Typically 3 to 4 tanks are used, with a reserve capacity of 25% of the total always on hand.

Skirt - That area around the base of the envelope that is made of nylon and coated with fire-resistant material. This keeps the balloon from being ignited by the flame.