Hot Air Balloon Shape and Pressure Gradient Within the Envelope

[Yachtsman]

I am interested in the math involved to calculate the ideal natural teardrop shape for a hot air balloon. I want to learn the details of what is involved to calculate this accurately.

I read this https://www.physicsforums.com/showthread.php?t=658802 which was a really nice start, but it unfortunately did not get into the details of how this is calculated.

Regarding the pressure gradient within the balloon, I'm curious how to calculate the pressure change from the top to the bottom, and the math involved in calculating the shape.

I appreciate the help. Thank you.

 

[CWatters]

Have you tried contacting "K^2" ?

 

[Yachtsman]

for some reason last night when I tried it did not let me email K^2, but now today it will. I must have been tired when I wrote my posted my note. What I did was to just repost what I wrote into that thread so that others can benefit from any response.

 

[256bits]

The parametric equation for a teardrop curve
http://mathworld.wolfram.com/TeardropCurve.html

 

[PJC01]

I understand your interest in the math behind calculating the ideal natural teardrop shape for a hot air balloon. It is important to note that the shape of a hot air balloon is determined by a combination of factors, including the pressure gradient within the envelope and the forces acting on the balloon.

To calculate the ideal shape, we must first understand the concept of buoyancy. Buoyancy is the upward force that is exerted on an object immersed in a fluid, in this case, the hot air inside the balloon. The amount of buoyancy is determined by the density of the fluid and the volume of the object. In the case of a hot air balloon, the volume is determined by the shape of the envelope.

The shape of the envelope plays a crucial role in determining the pressure gradient within the balloon. The pressure gradient is the change in pressure from the top to the bottom of the balloon. This gradient is directly related to the buoyancy force and the weight of the balloon. As the balloon rises, the air pressure decreases, causing the balloon to expand. This expansion is what gives the balloon its characteristic teardrop shape.

To accurately calculate the ideal shape, we must also consider the forces acting on the balloon. These forces include the weight of the envelope, the weight of the basket and passengers, and the drag force caused by air resistance. The ideal shape of the balloon will be one that minimizes the drag force while still providing enough buoyancy to lift the weight of the balloon and its contents.

The math involved in calculating the ideal shape is complex and involves equations that take into account all of the factors mentioned above. It also involves some trial and error to find the perfect balance between buoyancy and drag. The specific equations and calculations will vary depending on the size and weight of the balloon, as well as the atmospheric conditions.

In conclusion, the ideal shape of a hot air balloon is determined by a combination of factors, including the pressure gradient within the envelope and the forces acting on the balloon. The math involved in calculating this shape is complex and involves considering the buoyancy force, weight, and drag force. I hope this provides some insight into the details of calculating the ideal shape of a hot air balloon.