Solar Balloon Part 2-lets do the math
The solar balloon is by far the largest project I have ever attempted and so an element of risk comes with it. I usually don’t like doing math when I do my tinkering projects, however, both me and Andy decided that if we were going to be making a full-scale solar balloon capable of carrying one or two people then we might want to take some safety precautions. One of the first precautions we wanted to take was to know the math so that we would at least know that our paper works on paper. If you find math to be very boring then you might want to move on to the next post but if you find this interesting or you are attempting this yourself then you should stick around.
The math behind a solar balloon or any hot air balloon is very similar to a boat. the reason is a lighter than air device like a balloon literally floats on top of the air like a boat floats on water. to solve for this, we can use the buoyancy equation which says that the buoyancy force on an object is equal to the weight of the air (or water) that you displace. in math-speak that is:
or the pressure of the displaced air (P) multiplied by the pull of gravity (G) and the volume of displaced air (V). For our solar balloon that is modified to:
(Po= outside pressure and Pi= inside pressure)
or the difference in the density between the air outside of the balloon and inside the balloon (Po-Pi), multiplied by gravity (G) and the volume (V). Though this will get you the buoyant force, we wanted the only variable to be the temperature of the air and to incorporate this we use the ideal gas law which is:
or the pressure of the gas (P) multiplied by the volume of gas in question (V) is equal to the amount you have (the number of moles)(N) multiplied by a constant (the ideal gas constant)(R) and by the temperature of the air (in Kelvin )(T). When you combine this with the buoyancy equation you get:
Once we had this equation, Andy, who knows programming, but this into a program so we could easily modify the terms and find a volume that works for our purposes. The code he wrote looks like this.
we then did some research and found the lowest temperature difference recorded in a solar balloon was about 2 degrees and so decided to use that as our test temperature. we will probably get more heat than that in our balloon and our next balloon is going to be a test platform to find out stuff like this but for now, we figured that this would be a good baseline. We started to test the program at different volumes and these are the numbers we came up with.
- A spherical balloon with a 5-foot radius (or 10 feet across) with 1 degree of temperature difference will lift 0.594 kg and with 2 degrees of temperature difference will lift 1.185kg
- A spherical balloon with a 10-foot radius (or 20 feet across) with 1 degree of temperature difference will lift 4.754 kg and with 2 degrees of temperature difference will lift 9.487kg
- A spherical balloon with a 25-foot radius (or 50 feet across) with 1 degree of temperature difference will lift 74 kg and with 2 degrees of temperature difference will lift 148kg
This data shows that a balloon with a 2-degree temperature difference and a 25-foot radius will lift a 220 lb person with ease. This is fun to imagine but for now, we will be making a 5-foot radius balloon and its cargo will be a video camera (mostly because we wanted a cargo and though that pictures might be kinda fun). this balloon is probably going to be much too big for just lifting a camera but that means that we can use it as a test balloon for a long time coming.
I recognize that math can be very boring but we decided that it was worth knowing so we don’t take weeks making a balloon that doesn’t work. I hope this post was insightful to you and happy tinkering!