Our
study identifies a subtle deviation from Newton’s third law in the derivation of the ideal rocket
equation, also known as the Tsiolkovsky Rocket Equation (TRE). TRE can be
derived using a 1D elastic collision model of the momentum exchange between the
differential propellant mass element (dm)
and the rocket final mass (m1), in
which dm initially travels forward to
collide with m1 and rebounds to exit
through the exhaust nozzle with a velocity that is known as the effective exhaust
velocity ve. We observe that such a
model does not explain how dm was
able to acquire its initial forward velocity without the support of a reactive
mass traveling in the opposite direction. We show instead that the initial
kinetic energy of dm is generated
from dm itself by a process of
self-combustion and expansion. In our ideal rocket with a single particle dm confined inside a hollow tube with
one closed end, we show that the process of self-combustion and expansion of dm will result in a pair of differential
particles each with a mass dm/2, and
each traveling away from one another along the tube axis, from the center of
combustion. These two identical particles represent the active and reactive
sub-components of dm, co-generated in
compliance with Newton’
References
[1]
Tsiolkovsky, K.E. (1975) Study of Outer Space by Reaction Devices. https://archive.org/details/nasa_techdoc_19750021068/page/n1/mode/2up
[2]
Sutton, G.P. and Biblarz, O. (2016) Rocket Propulsion Elements. John Wiley & Sons, New York.
[3]
Taylor, T.S. (2017) Introduction to Rocket Science and Engineering. CRC Press, Boca Raton.
[4]
Goodger, E.M. (2013) Principles of Spaceflight Propulsion. Elsevier, Cham.
[5]
Curtis, H.D. (2013) Orbital Mechanics for Engineering Students. Butterworth-Heinemann, Oxford. https://doi.org/10.1016/B978-0-08-097747-8.00006-2
[6]
Sellers, J.J., Astore, W.J., Giffen, R.B. and Larson, W.J. (2000) Understanding Space: An Introduction to Astronautics. Primis, Hong Kong.
[7]
Kluever, C.A. (2018) Space Flight Dynamics. John Wiley & Sons, New York.
[8]
Cohen, I.B., Whitman, A. and Budenz, J (1999) The Principia: The Authoritative Translation and Guide: Mathematical Principles of Natural Philosophy. University of California Press, California.
[9]
McAllister, S., Chen, J.Y. and Fernandez-Pello, A.C. (2011) Fundamentals of Combustion Processes. Springer, New York. https://doi.org/10.1007/978-1-4419-7943-8
[10]
Afanasenkov, A.N. (2002) Retonation Wave upon Shock-Wave Initiation of Detonation of Solid Explosives. Combustion, Explosion and Shock Waves, 38, 470-472. https://doi.org/10.1023/A:1016271501868
[11]
Jing, Q., Huang, J.X., Liu, Q.M., Chen, X., Wang, Z.S., Liu, C.Q., et al. (2021) The Flame Propagation Characteristics and Detonation Parameters of Ammonia/Oxygen in a Largescale Horizontal Tube: As a Carbon-Free Fuel and Hydrogen-Energy Carrier. International Journal of Hydrogen Energy, 46, 19158-19170. https://doi.org/10.1016/j.ijhydene.2021.03.032
[12]
Oppenheim, A.K., Laderman, A.J. and Urtiew, P.A. (1962) The Onset of Retonation. Combustion and Flame, 6, 193-197. https://doi.org/10.1016/0010-2180(62)90089-5
[13]
Blomshield, F. (2007) Lessons Learned in Solid Rocket Combustion Instability. In 43rd AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, Cincinnati, 8-11 July 2007, 5803. https://doi.org/10.2514/6.2007-5803
[14]
Shima, S., Nakamura, K., Gotoda, H., Ohmichi, Y. and Matsuyama, S. (2021) Formation Mechanism of High-Frequency Combustion Oscillations in a Model Rocket Engine Combustor. Physics of Fluids, 33, Article ID: 064108. https://doi.org/10.1063/5.0048785
[15]
Larsen, C.E. (2008) Nasa Experience with Pogo in Human Spaceflight Vehicles. NATO RTO Symposium ATV-152 on Limit-Cycle Oscillations and Other Amplitude-Limited, Self-Excited Vibrations, number RTO-MP-AVT-152.
[16]
Poinsot, T. (2017) Prediction and Control of Combustion Instabilities in Real Engines. Proceedings of the Combustion Institute, 36, 1-28. https://doi.org/10.1016/j.proci.2016.05.007
[17]
Connelly, T.A. and Kyritsis, D.C. (2015) Experimental Investigation of Flame Propagation in Long, Narrow, and Open Tubes. Journal of Energy Engineering, 141, C4014016. https://doi.org/10.1061/(ASCE)EY.1943-7897.0000230
[18]
Kerampran, S., Desbordes, D., Veyssiere, B. and Bauwens, L. (2001) Flame Propagation in a Tube from Closed to Open End. 39th Aerospace Sciences Meeting and Exhibit, Reno, 8-12 January 2001, 1082. https://doi.org/10.2514/6.2001-1082
[19]
Ciccarelli, G. and Dorofeev, S. (2008) Flame Acceleration and Transition to Detonation in Ducts. Progress in Energy and Combustion Science, 34, 499-550. https://doi.org/10.1016/j.pecs.2007.11.002
[20]
Bennewitz, J.W. and Frederick, R.A. (2013) Overview of Combustion Instabilities in Liquid Rocket Engines-Coupling Mechanisms & Control Techniques. 49th AIAA/ASME/SAE/ASEE Joint Propulsion Conference, San Jose, 15-17 July 2013, 4106. https://doi.org/10.2514/6.2013-4106
[21]
White, F.M. (2008) Fluid Mechanics. The McGraw Hill Companies, New York.
[22]
Dunlap, R., Blackner, A.M., Waugh, R.C., Brown, R.S. and Willoughby, P.G. (1990) Internal Flow Field Studies in a Simulated Cylindrical Port Rocket Chamber. Journal of Propulsion and Power, 6, 690-704. https://doi.org/10.2514/3.23274
[23]
Candel, S.M. (1992) Combustion Instabilities Coupled by Pressure Waves and Their Active Control. Symposium (International) on Combustion, 24, 1277-1296. https://doi.org/10.1016/S0082-0784(06)80150-5
[24]
de Jong, C.A.F. (1995) Analysis of Pulsations and Vibrations in Fluid-Filled Pipe Systems. Proceedings of the ASME 1995 Design Engineering Technical Conferences Collocated with the ASME 1995 15th International Computers in Engineering Conference and the ASME 1995 9th Annual Engineering Database Symposium. Volume 3B: 15th Biennial Conference on Mechanical Vibration and Noise—Acoustics, Vibrations, and Rotating Machines, Boston, 17-20 September 1995, 829-834. https://doi.org/10.1115/DETC1995-0478
[25]
Li, X., Zhou, N., Liu, X.Y., Huang, W.Q., Chen, B. and Rasouli, V. (2020) Numerical Simulation of the Influence of Pipe Length on Explosion Flame Propagation in Open-Ended and Close-Ended Pipes. Science Progress, 103, 1-24. https://doi.org/10.1177/0036850420961607
[26]
Kao, S. and Shepherd, J.E. (2008) Numerical Solution Methods for Control Volume Explosions and ZND Detonation Structure. GALCIT Report FM2006.007, 1-46. https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=10362e38f68752e9f79768ba1ef36ad7a22b0879
