Introduction & Background: Alright! Let's find out why the interspiders have spun this place on the interweb for me.

This page is devoted to showing off my balloon modeling project. The project is geared to showing what's happening as a balloon flies and predicting a balloon's flight trajectory.

During the summer of 2009 my professor Dr. Chris Elmer went out to Taylor University to learn how to launch high altitude balloons. We wondered if there was a way to predict where a balloon would land so that it could be found easier and so the balloon modeling project was born!

How about some background information on how balloons fly before we get into the nitty gritty (math, ha). A balloon flies due to the gas inside of the balloon being less dense than the gas it's displacing (air). The total force upward is called buoyant force. The net force upward is what I call free lift force which would be the buoyant force minus the weight of the payload. To be neutrally buoyant the balloon must be neither going up nor down. When I refer to free lift (not force), I am talking about the amount of mass the balloon can lift past its point of neutral buoyancy. As a balloon goes along its journey through the skies the pressure is decreasing. This decrease in pressure causes the gas inside the balloon to expand and the density of the gas to lessen. Eventually the balloon will expand so much that it will pop and come back to earth.

Background Formulae: Hurray! You now have a little background information in how high altitude balloons work. Let's look into this with more depth now!

Buoyant Force
Isn't formulae such a cool word? Anyways. Let's talk about this buoyant force thing a little bit more in depth. As you undoubtedly recall from the Introduction, buoyant force, as it pertains to balloons, is the total upward force of the balloon. There's a cool formula for buoyant force..let me share it with you:

Ideal Gas Law

A neat little equation involving densities is in place! The density for air at 0 Celsius/ at sea level is 1.292 kg/m^3. The density for helium given the same restraints is 0.1786 and hydrogen is at 0.08988. However as was stated in the introduction, the density of gas will decrease as the balloon's altitude increases. Now you should ask how densities of gases change......... Why thank you for asking. A gas's density changes based on two things: temperature and pressure. There is actually a law for this and it's called the ideal gas law. Let me save you a trip to Wikipedia. The ideal gas law states:

This is pretty straightforward except for that gas constant which I've said absolutely nothing about thus far. The gas constant for air is 287, helium: 2077, hydrogen: 4127. I calculated these values by switching the ideal gas law around with the given densities. In case you don't know, 0 degrees Celsius is 273.15 degrees Kelvin. What the gas constant actually is/ how to actually calculate the gas constants from scratch is beyond the scope of this article. I guess if you want to do this then I won't be saving you a trip to Wikipedia after all.

Free Lift

So if you can recall from the Introduction, free lift refers to the amount of mass the balloon can lift past its point of neutral buoyancy. This is going to sort of use the buoyant force, but we're going to take out the acceleration due to gravity. From this we're going to subtract the mass of the payload to get the net upward lift (free lift). Here it is in equation form:

Cool.

Terminal Velocity

A pretty important equation when talking about all this balloon stuff is most certainly the equation for terminal velocity. I'll get into why it's important later on. The equation is:

In case it's hard to read this is stating that the terminal velocity is equal to the square root of ( (2*mg)/(C_d * D_a * A) ). If it helps this can also be written as Math.sqrt( (2*m*g) / (C*D*A) ). Ha. Anyways.. The drag coefficient I'll be using for the balloon is 0.3. When we get to parachute modeling I'll be using 1.5. I've shamelessly stolen these values from the internet because I don't have a giant wind tunnel of my own. The area we shall be using will usually be a circle since thats what the cross sectional shape of a sphere is. Same goes for the parachutes coincidentally.

My Formulae: Hurray you're still awake. Time to get into some stuff I've come up with!

Alright! So while a balloon is going upward, the pressure is decreasing and the temperature is fluctuating. This article on Wikipedia contains the temperature offsets used in my model. The ideal gas law shows that the density of our gas will be changing due to these changes in pressure and temperature so let's see how this affects there change that altitude has on the buoyant force of the balloon. Please note that the ideal gas law can be rewritten in terms of mass and volume instead of density because density = mass / volume. Let's see what substituting in the ideal gas law for the densities and volume gives us:

Note: This section of the site is submitting to some super serious changes. Beware: below is what it beheld before and contains useful information in a less friendly form.

Use at your own risk.

Love,

Management

Modeling the Flight of High Altitude Balloons
by
Robert Auld
Shepherd University
Dr. Chris Elmer - Advisor

This project involves modeling various stages in the flight of high altitude balloons. The modeling of these balloons is actually complex as there are many forces at work including, but certainly not limited to, changes in pressure, temperature, and wind velocities at various levels in altitude. There are multiple directions to model as a balloon is in flight: altitude, east, west, north, and south. The altitude part of the model may be broken up into two portions including: the time when the balloon is let go until the balloon bursts, and the time after burst until the balloon hits the ground.

Many high altitude balloons are flown for a multitude of purposes; the predominant one being research. At the moment there exists prediction software that will, based on results of wind data gathered by the National Weather Service, provide an estimate on where a balloon will land based on its starting GPS coordinates along with a couple other key parameters. This prediction is useful, but it can be improved upon by using the continuous GPS data gathered by the balloon in flight to measure current wind velocity at various altitudes in conjunction with the National Weather Service’s data.

The purpose of my project is to utilize existing formulas and data to derive a mathematical model that will show us exactly what is happening with our balloon during flight. This project allows us insight into the dynamics of high altitude balloon flight and to perhaps create prediction software that anyone in the world can use with their ballooning projects. Based on several inputs such as the amount of gas, what type of gas, balloon maximum filling capacity, starting coordinates, etc. a program based on these models could show a prediction of the flight path of the balloon which is constantly getting better based on incoming data. In the case that radio contact is lost with the balloon; my software could be used to provide an estimate as to where to search for it. This could potentially save several hundreds of dollars in electronic equipment from being lost. Needless to say this software could prove invaluable to balloonists.