48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition 4-7 January 2010, Orlando, Florida AIAA 2010-37 Near Space Balloon Performance Predictions Joseph P. Conner, Jr. 1 and Andrew S. Arena, Jr. 2 Oklahoma State University, Stillwater,OK, 74078 The objective of this paper is to discuss the development of a performance predictor for near space balloon systems during the ascent phase of the mission. During the ascent phase, the major parameters that are involved are the drag coefficient and the expansion of the balloons volume. These parameters will be determined by using a combination of real flight data and experiments ran in a lab. Once these parameters are found, a numerical simulation of the ascent phase is then compared to the results obtained from actual flights. Nomenclature A = cross-sectional area R = gas constant B = buoyancy Re = Reynolds Number C d = drag coefficient T = temperature c(i) = coefficient of polynomial V = velocity D = drag component V = volume of balloon g = gravity W = system weight m = mass of system ρ = density P = pressure Subscripts air = air alt = at altitude gnd = ground elevation He = Helium I. Introduction s exploration of near-space grows, a need for an accurate prediction of the mission has also grown. In the Acourse of an investigation into the development of a real-time performance predictor for near space balloon missions it became apparent that there was a need to fully understand the dynamics that are encountered during a mission. One of the most misunderstood phases of flight was the dynamics that occur during the ascent phase. In order to fully understand the behavior that is observed during ascent, it was required to gain an understanding of how the diameter of the balloon as well as its drag coefficient changed through out the mission. The mission typically ranges from a height of approximately 300 to 30,000 meters, with balloon volumes of less then 600 cubic feet at launch. In order to determine the drag coefficient on the balloon, it was necessary to establish the relationship of how the balloon s volume changed as a function of altitude. Once this relationship was found, flight data from several missions was used to experimentally determine the drag coefficient acting on the balloon. After the relationship was determined on how the balloon s performance changed as a function of altitude, a dynamic model was then created to aid in the prediction of future missions. II. Method The first issue that had to be addressed was the relationship of the balloon s volume as a function of altitude. The dynamic of the system during the ascent phase is governed by 1 2 Eq. ( 1) 1 Graduate Student, School of Mechanical and Aerospace Engineering, 218 Engineering North, AIAA Member. 2 L. Andrew Maciula Professor in Engineering, School of Mechanical and Aerospace Engineering, 218 Engineering North, AIAA Associate Fellow. 1 Copyright 2010 by Joseph P. Conner Jr. Published by the, Inc., with permission.
The generated buoyancy force is a function of the volume of the balloon, and the difference between the density of air and helium at altitude. The overall weight of the entire system includes the weight of the balloon, helium, and payloads. Finally, the drag term in this equation contains the drag coefficient, cross-sectional area of the balloon, and the balloon s velocity. Now that the governing equation for the balloon and payload have been developed, one needs to determine the maximum altitude that the system can traverse before the balloon has increased its volume to the point of bursting. The maximum volume that a balloon can increase is governed by the design and construction of the balloon. For the purpose of this paper we are going to assume that the information provided by the manufacturer of the balloon is correct. To determine the expansion of the balloon it will be assumed that it is governed by the ideal gas law. Since the volume, pressure, and temperature are known at the time of launch, one can then determine the conditions where the balloon will reach its maximum diameter as specified by the manufacturer. In order to test the assumption that the balloon can be modeled using the ideal gas law, we will need to verify that the balloon is near, or has the behavior of a zero-pressure balloon. A zero-pressure balloon is designed to expand such that the pressure difference between the inside and outside of the balloon remains the same and thus there is a zero-pressure difference between the inner and outer walls of the balloon. The balloons used for this experiment were designed and built by Kaymont. The balloons have been designed to maintain a spherical shape at inflation and expand at such a rate to maintain nearly zero pressure difference until burst. An experiment with the aim of verifying the zeropressure assumption was conducted by monitoring the pressure difference of a Kaymont KCI 30 balloon during inflation. During the experiment the fill volume of air was monitored by a gas sampling test meter which allow measurement of the volume of air to a resolution of 0.1 cubic feet. The pressure was monitored by use of a gas pressure sensor, which has a pressure range 0 to 210 kpa with a resolution of 0.05kPa. Finally the size of the balloon was recorded on camera with a 48 metal ruler in the foreground. The experimental setup is shown in Figure 1. Ball-Value Flow Control The objective of this experiment was to inflate the balloon and then record its volume and internal pressure. Once the balloon had reached a fixed volume, the fill process was paused and the interior pressure and a picture of the balloon were recorded. The fill then proceeded, stopping at fixed intervals until burst. The results of the incremental test, shown in Figure 2, indicated that while the pressure did vary as the balloon was filling, the change in pressure as a whole did not seem to vary dramatically. After reaching a volume of approximately 5 cubic feet the differential pressure required to continue to inflate the balloon actually started to decrease. This behavior was discussed by Fox in an article written for Physics Education. As the balloon is inflated, its skin becomes thinner and exerts less pressure on the air 2 Fill Nozzle Pressure Transducer Figure 1 Balloon Fill Experimental Setup Figure 2 Extended Pressure Fill Test for a Kaymont KCI 30 Flow Volume Meter
inside, thus taking less pressure to fill the. This is one of the main reasons why when you first start to inflate a balloon by mouth you have to blow very hard to stretch the rubber. As you continue to inflate balloon, it becomes easier. An example of a typical run with the different fill volumes is shown in Figure 3. One item that should be noted is that during inflation the balloon takes on a spherical shape and then continues to maintain this shape until burst. This spherical shape allows the cross-sectional area to be directly determined as function of the balloon s diameter. 1 cubic foot 5 cubic feet 10 cubic feet 15 cubic feet 20 cubic feet 22 cubic feet 24 cubic feet 26 cubic feet 28 cubic feet 30 cubic feet Figure 3 Images of Balloon During Pressure Test After repeating this experiment for a total of five times, the data was processed and it was observed that the interior pressure of the balloon remained at or near 99.98 kpa (14.5 psi) with a standard deviation of 0.074 kpa, which results in a pressure differential between interior and external pressures of approximately 0.95 kpa (0.139 psi). The results of the five runs shown in Figure 4, illustrate the typical internal pressures seen during inflation. One drawback with this figure is that the results are plotted with respect to absolute pressure, and as can be seen in the zero volume state, there was a difference in atmospheric pressure over the several days the experiments were ran. To alleviate this problem, it was decided to plot the differential pressure between the interior and exterior of the balloon. The resulting differential pressure, shown in Figure 5, indicates that the pressure difference of the balloon remained at or near 0.938 kpa (0.136psi), with a standard deviation of 0.0485kPa. This small difference supports the assumption that the balloon can be approximated as a zero-pressure balloon. Since the balloon can now safely be modeled as a zero-pressure balloon, it s expansion rate should be governed by the Ideal gas law. Figure 4 Pressure Fill Test for Several Kaymont KCI 30 Balloons Figure 5 Differential of Interior and Exterior Pressure Test 3
With the balloon now established as a near zero-pressure balloon, the expansion of the balloon as a function of altitude can safely be assumed to be governed by the ideal gas law. Occurring to the Ideal gas law, the state of an amount of gas is determined by its pressure, volume, and temperature according to the equation Eq. ( 2) Once the balloon is filled with a known volume of helium, the volume of the balloon at altitude can be found with Eq. ( 3) Using Eq. ( 3) along with the standard atmospheric model, a graph of balloon expansion as a function of altitude was determined. In order to obtain the results, it was assumed that the balloon was filled with approximately 550 cubic feet of helium at launch and that the helium expands at a rate governed by the ideal gas law. To determine how well this method works, information obtained from a camera on ASTRO-04 was used to determine the size of the balloon during the flight. To accomplish this, a payload was designed that held a digital camera which was focused toward the balloon. The fill volume of the balloon was determined to 550 cubic feet of helium by monitoring the pressure remaining in the two pressure tanks used during fill. Using the resulting pictures shown in Figure 6, a comparison between results obtained with the equation developed above and the pictures were created to indicate how the balloons diameter increased as a function of altitude. The results indicate that the model of balloon expansion developed above, matches well with data obtained from the ASTRO-04 flight. Figure 6. Expansion of Balloon during ASTRO-04 Now that there is an understanding on how the balloon s diameter changes with respect to time, one can now establish how the drag changes as a function of altitude. The total drag on the system can be determined with 1 2 Eq. ( 4) In order to determine the drag coefficient acting on the balloon, it was assumed that at any given time a quasi-steady solution to Eq. ( 1) would be satisfied. In order to verify that a quasi-steady solution is valid, the flight results for several flights were used to determine the acceleration during the ascent phase of the mission. The results obtained from nine missions indicate that while there is some acceleration, it is typically well below 1/100 th of gravity. 4
g/100 Figure 7. Acceleration of system during Ascent Phase Figure 8. Real Flight Data Showing Altitude as a Function of Mission Time The acceleration profiles used above where not obtained directly but from fitting a 4 th order polynomial line through mission data of altitude as a function of mission time, as seen in Figure 8. In order to obtain the altitude information, a packet radio along with a GPS receiver onboard the balloon was used. This radio transmitted the location of the balloon on a 13 second interval and then the signal was decoded and stored for use in real-time tracking as well as post flight analysis. Once this data was obtained, a 4th order polynomial was fit to the data and then plotted against the real data. Examples of this curve fitting are shown in Figure 9 and illustrate the results of the curve fit compared to real flight data for several flights. a)astro-12 b) ASTRO-09 c) ASTRO-06 d) ASTRO-04 Figure 9. Real Flight Data Compared to 4 th Order Polynomial Fit 5
The polynomial equations were then differentiated to obtain velocity as a function of altitude and then differentiated once more to obtain the acceleration graphs discussed above. Now with the capability of determining velocity during the ascent phase at any given altitude, one can focus on determining the drag coefficient as a function of either altitude or Reynolds number. 2 Eq. ( 5) Using Eq. ( 5) along with flight data from nine different ASTRO flights drag coefficient as a function of Reynolds number was found. Since the balloon is designed to maintain a spherical shape the results are plotted against the classic drag coefficient of a sphere 2 and are shown in Figure 10. The drag coefficient results of the flights were then fit with a 4th order polynomial to obtain a model of the drag coefficient over the flight regimes that had been encountered; this result is then used during the ascent phase to aid in the prediction of flight performance during the ascent phase. The resulting balloon drag model is shown in Figure 10 with the classic spherical drag shown as a reference. The results show that while the balloons drag coefficient does exhibit the behavior of transition from low to high as Reynolds number decreases, it does so with a smoother transition. Where the classic drag on a sphere has a major transition from approximately 0.10 to 0.4 over a Reynolds number range of 250,000 to 400,000, the balloon transition over this same drag coefficient range occurs in a Reynolds number range of 200,000 to 1,200,000. Figure 10. Reynolds Number Versus Drag Coefficient Figure 11. Drag Model Compared with Flight Data Figure 12 Drag Model Compared to Classic Spherical Drag 6
Results Once the drag coefficient was found, a full numerical simulation of the ascent phase could be performed using Eq. ( 1). In order to determine how well this numerical simulation works, actual results from several ASTRO flights were used. It was decided to plot the results of ASTRO-09 and ASTRO-11 against numerical simulation of both flights. The reasoning behind selecting these two flights is that with information collected to date ASTRO-11 represents one of the fastest ascent rates and ASTRO-09 one of the slowest. As can be seen in Figure 13, the results of the simulation and the flight data agree. a) ASTRO-11 b) ASTRO-09 Figure 13. Ascent Simulation Compared with Flight Results While the slower flight, Figure 13, ASTRO-09 appears to be a constant ascent rate this is actually not the case. In fact, below 10,000 meters the balloons ascent rate is slowly increasing. As the balloon passes 10,000 meters its ascent rate decreases until approximately 25,000 meters where its ascent rate starts to increase again. This resulting change in ascent rate can be seen in Figure 14. Figure 14. Ascent Velocity as a Function of Altitude Figure 15. Drag Coefficient as a Function of Altitude This change in flight speed has been noted by other ballooning programs but few have taken the time to fully understand why this happens. If we take a look at drag coefficient as a function of altitude, we note that as the balloon is approaching 10,000 meters the drag coefficient increases rapidly. Then as the balloon passes 25,000 meters, its drag coefficient then decreases, as can be seen in Figure 15. This change in drag coefficient then directly reflects the changes in ascent rate that was observed. 7
Conclusion This paper presented the development of a balloon drag model which took on the form c i Re where: c 0 7.119E 01 c 1 2.568E 06 c 2 4.707E 12 c 3 4.040E 18 c 4 1.309E 24 Eq. ( 6) In the development of this model, it was determined that the Kaymont balloon could be modeled as a zero-pressure balloon and as such its volume was directly a function of the Ideal gas law. Finally, using the information develop, it was found that one could predict the performance of the balloon and system during the ascent phase and only requires knowledge of the initial weight of the system and fill volume of the balloon. Acknowledgments The author s thanks are due to NASA for funding provided through EPSCoR and Oklahoma Space Grant Consortium which made this project possible. References 1 Conner, J.P., Development of a Real-Time Performance Predictor and an Investigation of a Return to Point Vehicle for High Altitude Ballooning, Oklahoma State University, Stillwater, OK, 2009 2 Hoerner, S. F. Fluid Dynamic Drag. Bakersfield : authors family Hoerner Fluid Dynamics, 1965. 3 Fox, John N. The Baffling balloons! Physic Education. 1993. 8