Advertisement

Experiments in Fluids

, 61:4 | Cite as

Clustering of paraffin-based hybrid rocket fuels combustion data

  • A. RüttgersEmail author
  • A. Petrarolo
  • M. Kobald
Research Article
  • 41 Downloads

Abstract

Clustering was applied to image data of hybrid rocket combustion tests for a better understanding of the complex flow phenomena. Novel techniques such as hybrid rockets that allow for cost reductions of space transport vehicles are of high importance in space flight. However, the combustion process in hybrid rocket engines is still a matter of ongoing research and not fully understood yet. Recently, combustion tests with different paraffin-based fuels have been performed at the German Aerospace Center (DLR). For a detailed analysis, the combustion process has been captured with a high-speed video camera, which leads to a huge amount of images for each test. In the end, a large data set with a total number of 30,000 images for each combustion test has to be analyzed. To catch the essential flow structures, the combustion data set was clustered with a K-means++ algorithm. Since the algorithm might converge to local optimal solutions, expensive repetitions have been performed to ensure that a global solution is found in the end. Furthermore, a detailed analysis was performed to find an adequate clustering algorithm in the first place and to estimate the number of relevant clusters K in each experiment. As a result, valuable insights into the different combustion phases were obtained and a comparison of the quality of the combustion flame in the different tests could be made. In particular, depending on the fuel formulation and oxidizer mass flow, differences in the transients and flame brightness were found.

Graphic abstract

List of symbols

\(C_i\)

single cluster (−)

I(xy)

grayscale pixel intensity (−)

J

objective function (squared error) (−)

K

number of clusters (−)

\(\varvec{x}_j\)

data point j (a single image) (−)

d

problem dimension (resolution of \(\varvec{x}_j\)) (−)

n

number of data points in single test (−)

\(s(\varvec{x}_j)\)

silhouette value of the data point \(\varvec{x}_j\) (−)

\(\bar{x}, \bar{y}\)

image barycenter coordinates (−)

f(K)

evaluation function to determine K (−)

\(\alpha _k\)

weight factor in f(K) (−)

\(\varvec{\mu }_i\)

mean of cluster \(C_i\) (centroid) (−)

Notes

Acknowledgements

This research was carried out under the project Antriebstechnologien und Komponenten für Trägersysteme (ATEK) by the German Aerospace Center (DLR).

References

  1. Arthur D, Vassilvitskii S (2007) K-means++: The advantages of careful seeding. In: Proceedings of the eighteenth annual ACM-SIAM symposium on discrete algorithms, society for industrial and applied mathematics, Philadelphia, PA, USA, SODA 2007, pp 1027–1035. http://dl.acm.org/citation.cfm?id=1283383.1283494
  2. Ciezki HK, Sender J, Clauß W, Feinauer A, Thumann A (2003) Combustion of solid-fuel slabs containing boron particles in step combustor. J Propul Power 19(6):1180–1191.  https://doi.org/10.2514/2.6938 CrossRefGoogle Scholar
  3. Devriendt K, Hook HV, Ceursters B, Petters J (1996) Kinetics of formation of chemiluminescent CH by the elementary reactions of C2H with O and O2: a pulse laser photolysis study. Chem Phys Lett 261:450–456CrossRefGoogle Scholar
  4. Ester M, Kriegel HP, Sander J, Xu X (1996) A density-based algorithm for discovering clusters a density-based algorithm for discovering clusters in large spatial databases with noise. In: Proceedings of the second international conference on knowledge discovery and data mining, AAAI Press, KDD’96, pp 226–231. http://dl.acm.org/citation.cfm?id=3001460.3001507
  5. Hastie T, Tibshirani R, Friedman J (2009) Hierarchical clustering. Elements Stat Learn 2009:2Google Scholar
  6. Jain A (2010) Data clustering: 50 years beyond k-means. Pattern Recogn Lett 31(8):651–666CrossRefGoogle Scholar
  7. Karabeyoglu A, Altman D, Cantwell BJ (2002) Combustion of liquefying hybrid propellants: part 1, general theory. J Propul Power 18(3):610–620.  https://doi.org/10.2514/2.5975 CrossRefGoogle Scholar
  8. Karabeyoglu A, Cantwell B, Altman D (2001) Development and testing of paraffin-based hybrid rocket fuels. In: 37th AIAA/ASME/SAE/ASEE Joint propulsion conference and exhibit, American Institute of Aeronautics and Astronautics, Salt Lake City, Utah.  https://doi.org/10.2514/6.2001-4503
  9. Karabeyoglu A, Stevens J, Geyzel D, Cantwell B, Micheletti D (2011) High performance hybrid upper stage motor. In: 47th AIAA/ASME/SAE/ASEE Joint propulsion conference and exhibit. American Institute of Aeronautics and Astronautics.  https://doi.org/10.2514/6.2011-6025
  10. Kobald M, Petrarolo A, Schlechtriem S (2015) Combustion visualization and characterization of liquefying hybrid rocket fuels. In: 51st AIAA/SAE/ASEE Joint propulsion conference. American Institute of Aeronautics and Astronautics.  https://doi.org/10.2514/6.2015-4137
  11. Krajsek K, Comito C, Götz M, Hagemeier B, Knechtges P, Siggel M (2018) The Helmholtz analytics toolkit (heat): a scientific big data library for hpc. In: Extreme data workshop 2018. https://elib.dlr.de/124422/
  12. Lloyd S (1982) Least squares quantization in pcm. IEEE T Inform Theory 28(2):129–137MathSciNetCrossRefGoogle Scholar
  13. MacQueen J (1967) Some methods for classification and analysis of multivariate observations. In: Proceedings of the fifth berkeley symposium on mathematical statistics and probability, volume 1: statistics, University of California Press, Berkeley, Calif., pp 281–297. https://projecteuclid.org/euclid.bsmsp/1200512992
  14. Ng AY, Jordan MI, Weiss Y (2001) On spectral clustering: analysis and an algorithm. Advances in neural information processing systems. MIT Press, Cambridge, pp 849–856Google Scholar
  15. Pedregosa F, Varoquaux G, Gramfort A, Michel V, Thirion B, Grisel O, Blondel M, Prettenhofer P, Weiss R, Dubourg V, Vanderplas J, Passos A, Cournapeau D, Brucher M, Perrot M, Duchesnay E (2011) Scikit-learn: machine learning in Python. J Mach Learn Res 12:2825–2830MathSciNetzbMATHGoogle Scholar
  16. Petrarolo A, Kobald M (2016) Evaluation techniques for optical analysis of hybrid rocket propulsion. J Fluid Sci Technol 11(4):JFST0028–JFST0028.  https://doi.org/10.1299/jfst.2016jfst0028 CrossRefGoogle Scholar
  17. Petrarolo A, Kobald M (2018) Schlechtriem S (2018) Understanding Kelvin-Helmholtz instability in paraffin-based hybrid rocket fuels. Exp Fluids 59:62.  https://doi.org/10.1007/s00348-018-2516-1 CrossRefGoogle Scholar
  18. Pham D, Dimov S, Nguyen C (2005) Selection of k in k-means clustering. Proc Inst Mech Eng Part C J Mech Eng Sci 219(1):103–119CrossRefGoogle Scholar
  19. Rousseeuw P (1987) Silhouettes: a graphical aid to the interpretation and validation of cluster analysis. J Comput Appl Math 20:53–65CrossRefGoogle Scholar
  20. Schefer RW (1997) Flame sheet imaging using CH chemiluminescence. Combust Sci Technol 126(1–6):255–279.  https://doi.org/10.1080/00102209708935676 CrossRefGoogle Scholar
  21. Sculley D (2010) Web-scale k-means clustering. In: Proceedings of the 19th international conference on World wide web, ACM, pp 1177–1178Google Scholar
  22. Thumann A, Ciezki HK (2002) Combustion of energetic materials, chap. Comparison of PIV and Colour-Schlieren measurements of the combusiton process of boron particle containing soild fuel slabs in a rearward facing step combustor, vol 5, Begell House Inc.  https://doi.org/10.1615/IntJEnergeticMaterialsChemProp.v5.i1-6.770 CrossRefGoogle Scholar
  23. Tibshirani R, Walther G, Hastie T (2001) Estimating the number of clusters in a data set via the gap statistic. J R Statist Soc B 63(2):411–423MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Simulation and Software Technology, High-Performance Computing DepartmentGerman Aerospace Center (DLR)CologneGermany
  2. 2.Institute of Space Propulsion, Propellants DepartmentGerman Aerospace Center (DLR)HardthausenGermany

Personalised recommendations