Mathematical model and simulation of free balloon liftoff in the presence of surface winds
Abstract
A mathematical model of free balloon launches in windy conditions is based on the conservation of the linear momentum in horizontal and vertical axes. Linear momentum conservation equations are represented by a set of four nonlinear first-order ODEs. Some ODEs were solved analytically, while the nonlinear Riccati ODE with variable coefficients for the vertical acceleration was solved using numerical ODE solvers. Transient aerodynamic lift and horizontal drag are caused by the slip flow over the balloon envelope. It takes free balloon ten half times to reach 90.9% of the wind velocity in a step function response. A launch condition was developed in terms of the minimum required envelope temperature for which the net aerostatic lift overcomes inert weight of a balloon. Perturbation analysis was used to explore changes in the net aerostatic lift. Simulations were performed to cases with and without envelope distortion and enhanced cooling due to forced convection. Since all balloon takeoffs are performed downwind, obstacle clearance becomes an issue due to rapid loss of aerodynamic lift. Balloons may stop climbing and even start descending shortly after liftoff despite intense heating representing real hazard.
Keyword : aerostatic lift, slip flow, transient aerodynamic lift, transient aerodynamic drag, envelope distortion, enhanced envelope cooling, numerical ODE solvers
How to Cite
Daidzic, N. E. (2022). Mathematical model and simulation of free balloon liftoff in the presence of surface winds. Aviation, 26(1), 22–31. https://doi.org/10.3846/aviation.2022.16621
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
Aaron, K. M., Heun, M. K., & Nock, K. T. (2002). A method for balloon trajectory control. Advances in Space Research, 30(5), 1227–1232. https://doi.org/10.1016/S0273-1177(02)00526-4
Abell, M. L., & Braselton, J. P. (2010). Introductory differential equations with boundary value problems (3rd ed.). Elsevier (Academic Press).
Anderson, J. D. Jr. (1991). Fundamentals of aerodynamics (2nd ed.). McGraw-Hill.
Cameron, D. (1980). Ballooning handbook. Pelham Books (Penguin Group).
Cameron, J., Smith, I. S., Cutts, J. A., Raque, S., Jones, J., & Wu, J. (1999, 28 June–1 July). Versatile modeling and simulation of Earth and planetary balloon systems. In AIAA-99-3876, 13th AIAA Lighter-Than-Air Systems Technology Conference. Norfolk, Virginia. https://doi.org/10.2514/6.1999-3876
Carlson, L. A., & Horn, W. J. (1983). New thermal and trajectory model for high-altitude balloons. Journal of Aircraft, 20(6), 500–507. https://doi.org/10.2514/3.44900
Carnahan, B., Luther, H. A., & Wilkes, J. O. (1969). Applied numerical methods. John Wiley & Sons.
Chapra, S. C., & Canale, R. P. (2006). Numerical methods for engineers (5th ed.). McGraw-Hill.
Daidzic, N. E. (2014). Could we colonize Venus? Professional Pilot, 48(3), 92–96.
Daidzic, N. E. (2015). Efficient general computational method for estimation of standard atmosphere parameters. International Journal of Aviation Aeronautics, and Aerospace, 2(1), 1–35. https://doi.org/10.15394/ijaaa.2015.1053
Daidzic, N. E. (2021). Mathematical model of hot-air balloon steady-state vertical flight performance. Aviation, 25(3), 149–158. https://doi.org/10.3846/aviation.2021.15330
Das, T., Mukherjee R., & Cameron, J. (2003). Optimal trajectory planning for hot-air balloons in linear wind fields. Journal of Guidance, Control, and Dynamics, 26(3), 416–424. https://doi.org/10.2514/2.5079
Davis, H. T. (1962). Introduction to nonlinear differential and integral equations. Dover.
Dorrington, G. E. (2013). Buoyancy estimation of a Montgolfière in the atmosphere of Titan. Aeronautical Journal, 177(1195), 1–15. https://doi.org/10.1017/S0001924000008605
Du, H., Li, J., Qu, Z., Zhang, L., & Lv, M. 2019. Flight performance simulation and station-keeping endurance analysis for stratospheric super-pressure balloon in real wind field. Aerospace Science and Technology, 86, 1–10. https://doi.org/10.1016/j.ast.2019.01.001
Furfaro, R., Lunine, J. I., Elfes, A., & Reh, K. (2008). Wind-based navigation of a hot-air balloon on titan: A feasibility study. In Proceedings of SPIE – the International Society for Optical Engineering, 6960. Orlando, FL. https://doi.org/10.1117/12.777654
Granger, R. A. (1995). Fluid mechanics. Dover.
Houghton, E. L., & Carpenter, P. W. (1993). Aerodynamics for engineering students (4th ed.). Edward Arnold, Hodder & Stoughton.
Jackson, D. D. (1980). The aeronauts. Time-Life Books.
Jackson, J. P., & Dichtl, R. J. (1977). The science and art of hot-air ballooning. Garland Publishing.
Kayhan, O., & Hastaoglu, M. A. (2014). Modelling of stratospheric balloon using transport phenomena and gas compress-release system. Journal of Thermophysics and Heat Transfer, 28(3), 534–541. https://doi.org/10.2514/1.T4271
Khoury, G. A., & Gillett, J. D. (1999). Airship technology (Cambridge Aerospace Series 10). Cambridge University Press.
Kreider, J. F. (1975). Mathematical modeling of high-altitude balloon performance. In the 5th Aerodynamic Declaration Systems Conference, AIAA Paper 75-1385. https://doi.org/10.2514/6.1975-1385
Kreith, F. (1965). Principles of heat transfer (2nd. ed). International Textbook Company.
Kreith, F., & Kreider, J. F. (1974). Numerical prediction of the performance of high altitude balloons (NCAR-TN/STR-65 1971, revised 1974). National Center for Atmospheric Research (NCAR).
Lally, V. E. (1971). Superpressure balloons for horizontal soundings of the atmosphere (NCAR-TN-28). National Center for Atmospheric Research (NCAR).
McCormick, B. W. (1995). Aerodynamics, aeronautics, and flight mechanics (2nd ed.). John Wiley & Sons.
Moran, J. (2003). An introduction to theoretical and computational aerodynamics. Dover.
Morris, A. L. (1975). Scientific ballooning handbook (NCAR-TN/IA-99). National Center for Atmospheric Research (NCAR).
Nayfeh, A. H. (2004). Perturbation methods. Wiley-VCH Verlag GmbH & Co. KGaA.
Press, W. H, Teulkolsky, S. A., Vetterling, W. T., & Flannery, B. P. (1992). Numerical recipes in FORTRAN: The art of scientific computing (2nd ed.). University Press.
Roth, C. H. (1917). A short course on the theory and operation of the free balloon (2nd ed.) The Goodyear Tire and Rubber Company.
Shi, H., Song, B., & Yao, Q. (2009). Thermal performance of stratospheric airships during ascent and descent. Journal of Thermophysics and Heat Transfer, 23(4), 816–821. https://doi.org/10.2514/1.42634
Spiegel, M. R. (1981). Applied differential equations (3rd ed.). Prentice-Hall.
Stefan, K. (1979). Performance theory for hot air balloons. Journal of Aircraft, 16(8), 539–542. https://doi.org/10.2514/3.58561
Stefan, B. (1997, 3 June–5 June). Theoretical aerodynamics and thermal endurance for a hot air balloon. In the 12th AIAA Lighter-Than-Air Systems Technology Conference, AIAA-97-1490. San Francisco, California. https://doi.org/10.2514/6.1997-1490
Taylor, J. A. (2014). Principles of Aerostatics: The theory of lighter-than-air flight. CreateSpace Independent Publishing Platform.
US Department of Transportation, Federal Aviation Administration. (1981). Balloon safety tips: Powerlines and thunderstorms (FAA-P-8740-34, AFO-800-0581). FAA. Washington, DC.
US Department of Transportation, Federal Aviation Administration. (1982). Balloon safety tips: False lift, shear & rotors (FAA-P-8740-39, AFO-800-0582). FAA. Washington, DC.
US Department of Transportation, Federal Aviation Administration. (1996a). Operations of hot air balloons with airborne heater (AC 91-71, 6/13/96 AFS-820). FAA. Washington, DC.
US Department of Transportation, Federal Aviation Administration. (1996b). Private Pilot Practical Test Standards for Lighter-than-Air (FAA-S-8081-17). FAA. Washington, DC.
US Department of Transportation, Federal Aviation Administration. (1997). Commercial Pilot Practical Test Standards for Lighter-than-Air (FAA-S-8081-18). FAA. Washington, DC.
US Department of Transportation, Federal Aviation Administration. (1999). Part 31, Airworthiness Standards: Manned Free Balloons. FAA. Washington, DC.
US Department of Transportation, Federal Aviation Administration. (2008). Balloon flying handbook (FAA-H-8083-11A). FAA. Washington, DC.
Weber, H. J., & Arfken, G. B. (2004). Essential mathematical methods for physicists. Elsevier (Academic Press).
Zwillinger, D. (1992). Handbook of differential equations (2nd ed.). Academic Press. https://doi.org/10.1016/B978-0-12-784391-9.50085-3
Abell, M. L., & Braselton, J. P. (2010). Introductory differential equations with boundary value problems (3rd ed.). Elsevier (Academic Press).
Anderson, J. D. Jr. (1991). Fundamentals of aerodynamics (2nd ed.). McGraw-Hill.
Cameron, D. (1980). Ballooning handbook. Pelham Books (Penguin Group).
Cameron, J., Smith, I. S., Cutts, J. A., Raque, S., Jones, J., & Wu, J. (1999, 28 June–1 July). Versatile modeling and simulation of Earth and planetary balloon systems. In AIAA-99-3876, 13th AIAA Lighter-Than-Air Systems Technology Conference. Norfolk, Virginia. https://doi.org/10.2514/6.1999-3876
Carlson, L. A., & Horn, W. J. (1983). New thermal and trajectory model for high-altitude balloons. Journal of Aircraft, 20(6), 500–507. https://doi.org/10.2514/3.44900
Carnahan, B., Luther, H. A., & Wilkes, J. O. (1969). Applied numerical methods. John Wiley & Sons.
Chapra, S. C., & Canale, R. P. (2006). Numerical methods for engineers (5th ed.). McGraw-Hill.
Daidzic, N. E. (2014). Could we colonize Venus? Professional Pilot, 48(3), 92–96.
Daidzic, N. E. (2015). Efficient general computational method for estimation of standard atmosphere parameters. International Journal of Aviation Aeronautics, and Aerospace, 2(1), 1–35. https://doi.org/10.15394/ijaaa.2015.1053
Daidzic, N. E. (2021). Mathematical model of hot-air balloon steady-state vertical flight performance. Aviation, 25(3), 149–158. https://doi.org/10.3846/aviation.2021.15330
Das, T., Mukherjee R., & Cameron, J. (2003). Optimal trajectory planning for hot-air balloons in linear wind fields. Journal of Guidance, Control, and Dynamics, 26(3), 416–424. https://doi.org/10.2514/2.5079
Davis, H. T. (1962). Introduction to nonlinear differential and integral equations. Dover.
Dorrington, G. E. (2013). Buoyancy estimation of a Montgolfière in the atmosphere of Titan. Aeronautical Journal, 177(1195), 1–15. https://doi.org/10.1017/S0001924000008605
Du, H., Li, J., Qu, Z., Zhang, L., & Lv, M. 2019. Flight performance simulation and station-keeping endurance analysis for stratospheric super-pressure balloon in real wind field. Aerospace Science and Technology, 86, 1–10. https://doi.org/10.1016/j.ast.2019.01.001
Furfaro, R., Lunine, J. I., Elfes, A., & Reh, K. (2008). Wind-based navigation of a hot-air balloon on titan: A feasibility study. In Proceedings of SPIE – the International Society for Optical Engineering, 6960. Orlando, FL. https://doi.org/10.1117/12.777654
Granger, R. A. (1995). Fluid mechanics. Dover.
Houghton, E. L., & Carpenter, P. W. (1993). Aerodynamics for engineering students (4th ed.). Edward Arnold, Hodder & Stoughton.
Jackson, D. D. (1980). The aeronauts. Time-Life Books.
Jackson, J. P., & Dichtl, R. J. (1977). The science and art of hot-air ballooning. Garland Publishing.
Kayhan, O., & Hastaoglu, M. A. (2014). Modelling of stratospheric balloon using transport phenomena and gas compress-release system. Journal of Thermophysics and Heat Transfer, 28(3), 534–541. https://doi.org/10.2514/1.T4271
Khoury, G. A., & Gillett, J. D. (1999). Airship technology (Cambridge Aerospace Series 10). Cambridge University Press.
Kreider, J. F. (1975). Mathematical modeling of high-altitude balloon performance. In the 5th Aerodynamic Declaration Systems Conference, AIAA Paper 75-1385. https://doi.org/10.2514/6.1975-1385
Kreith, F. (1965). Principles of heat transfer (2nd. ed). International Textbook Company.
Kreith, F., & Kreider, J. F. (1974). Numerical prediction of the performance of high altitude balloons (NCAR-TN/STR-65 1971, revised 1974). National Center for Atmospheric Research (NCAR).
Lally, V. E. (1971). Superpressure balloons for horizontal soundings of the atmosphere (NCAR-TN-28). National Center for Atmospheric Research (NCAR).
McCormick, B. W. (1995). Aerodynamics, aeronautics, and flight mechanics (2nd ed.). John Wiley & Sons.
Moran, J. (2003). An introduction to theoretical and computational aerodynamics. Dover.
Morris, A. L. (1975). Scientific ballooning handbook (NCAR-TN/IA-99). National Center for Atmospheric Research (NCAR).
Nayfeh, A. H. (2004). Perturbation methods. Wiley-VCH Verlag GmbH & Co. KGaA.
Press, W. H, Teulkolsky, S. A., Vetterling, W. T., & Flannery, B. P. (1992). Numerical recipes in FORTRAN: The art of scientific computing (2nd ed.). University Press.
Roth, C. H. (1917). A short course on the theory and operation of the free balloon (2nd ed.) The Goodyear Tire and Rubber Company.
Shi, H., Song, B., & Yao, Q. (2009). Thermal performance of stratospheric airships during ascent and descent. Journal of Thermophysics and Heat Transfer, 23(4), 816–821. https://doi.org/10.2514/1.42634
Spiegel, M. R. (1981). Applied differential equations (3rd ed.). Prentice-Hall.
Stefan, K. (1979). Performance theory for hot air balloons. Journal of Aircraft, 16(8), 539–542. https://doi.org/10.2514/3.58561
Stefan, B. (1997, 3 June–5 June). Theoretical aerodynamics and thermal endurance for a hot air balloon. In the 12th AIAA Lighter-Than-Air Systems Technology Conference, AIAA-97-1490. San Francisco, California. https://doi.org/10.2514/6.1997-1490
Taylor, J. A. (2014). Principles of Aerostatics: The theory of lighter-than-air flight. CreateSpace Independent Publishing Platform.
US Department of Transportation, Federal Aviation Administration. (1981). Balloon safety tips: Powerlines and thunderstorms (FAA-P-8740-34, AFO-800-0581). FAA. Washington, DC.
US Department of Transportation, Federal Aviation Administration. (1982). Balloon safety tips: False lift, shear & rotors (FAA-P-8740-39, AFO-800-0582). FAA. Washington, DC.
US Department of Transportation, Federal Aviation Administration. (1996a). Operations of hot air balloons with airborne heater (AC 91-71, 6/13/96 AFS-820). FAA. Washington, DC.
US Department of Transportation, Federal Aviation Administration. (1996b). Private Pilot Practical Test Standards for Lighter-than-Air (FAA-S-8081-17). FAA. Washington, DC.
US Department of Transportation, Federal Aviation Administration. (1997). Commercial Pilot Practical Test Standards for Lighter-than-Air (FAA-S-8081-18). FAA. Washington, DC.
US Department of Transportation, Federal Aviation Administration. (1999). Part 31, Airworthiness Standards: Manned Free Balloons. FAA. Washington, DC.
US Department of Transportation, Federal Aviation Administration. (2008). Balloon flying handbook (FAA-H-8083-11A). FAA. Washington, DC.
Weber, H. J., & Arfken, G. B. (2004). Essential mathematical methods for physicists. Elsevier (Academic Press).
Zwillinger, D. (1992). Handbook of differential equations (2nd ed.). Academic Press. https://doi.org/10.1016/B978-0-12-784391-9.50085-3