Abstract
The potential of the Levenberg–Marquardt method combined with an explicit Runge–Kutta method for non-stiff systems, and, an implicit Rosenbrock method for stiff systems to investigate burning velocities using explosion bombs was explored. The implementation of this combination of methods was verified on three benchmark test problems, and, by the application of two integral balance models to laminar hydrogen-air and methane-air explosions. The methodology described here was subsequently applied to quantify the coefficients of a turbulent burning velocity correlation for a methane-air explosion in the decaying flow field of the standard 20-litre explosion sphere. The outcome of this research indicates that the usefulness of the 20-litre sphere can be extended beyond the measurement of practical explosion parameters. When combined with the methodology in this paper, turbulent burning velocity correlations can be assessed in different parts of the Borghi-diagram.
Similar content being viewed by others
References
Popat, N.R., Catlin, C.A., Arntzen, B.J., Lindstedt, R.P., Hjertager, B.H., Solberg, T., Saeter, O., van den Berg, A.C.: Investigations to improve and assess the accuracy of computational fluid dynamic based explosion models. J. Hazard. Mater. 45, 1–15 (1996)
Watterson, J.K., Connell, I.J., Savill, A.M., Dawes, W.N.: A solution-adaptive mesh procedure for predicting confined explosions. Int. J. Numer. Methods Fluids 26, 235–247 (1998)
Molkov, V.V., Makarov, D., Grigorash, A.: Cellular structure of explosion flames: modeling and large-eddy simulation. Combust. Sci. Technol. 176, 851–865 (2004)
Skjold, T., Arntzen, B., Hansen, O.R., Taraldset, O.J., Storvik, I.E., Eckhoff, R.K.: Simulating dust explosions with the first version of DESC. Trans. Inst. Chem. Eng. B, Process. Saf. Environ. Protect. 83, 151–160 (2005)
Skjold, T., Arntzen, B., Hansen, O.R., Storvik, I.E., Eckhoff, R.K.: Simulation of dust explosions in complex geometries with experimental input from standardized tests. J. Loss Prev. Process Ind. 19, 210–217 (2006)
Eckhoff, R.K.: Current status and expected future trends in dust explosion research. J. Loss Prev. Process Ind. 18, 225–237 (2005)
Sathiah, P., Komen, E., Roekaerts, D.: The role of CFD combustion modeling in hydrogen safety management - I: validation based on small scale experiments. Nucl. Eng. Des. 248, 93–107 (2012)
Sathiah, P., van Haren, S., Komen, E., Roekaerts, D.: The role of CFD combustion modeling in hydrogen safety management - II: validation based on homogeneous hydrogen-air experiments. Nucl. Eng. Des. 252, 289–302 (2012)
Law, C.K.: Propagation, structure, and limit phenomena of laminar flames at elevated pressures. Combust. Sci. Technol. 178, 335–360 (2006)
Kuo, K.K.: Principles of Combustion, 2nd edn. Wiley, New York (2005)
Tse, S.D., Zhu, D.L., Law, C.K.: Morphology and burning rates of expanding spherical flames in H2/O2/inert mixtures up to 60 atmospheres. In: Proceedings of the Twenty-Eighth Symposium (International) on Combustion, pp. 1793–1800. The Combustion Institute, Pittsburgh (2000)
Rosenbrock, H.H.: Some general implicit processes for the numerical solution of differential equations. Comput. J. 5, 329–330 (1963)
Hairer, E., Wanner, G.: Solving Ordinary Differential Equations, 2nd edn. II. Stiff and Differential-Algebraic Problems. Springer Series in Computational Mathematics, vol. 14. Springer, Berlin (1996).
Van den Bulck, E.: Closed algebraic expressions for the adiabatic limit value of the explosion constant in closed volume combustion. J. Loss Prev. Process Ind. 18, 35–42 (2005)
Tennekes, H., Lumley, J.L.: A First Course in Turbulence. MIT Press, Cambridge, MA (1972)
Hinze, J.O.: Turbulence, 2nd edn. McGraw-Hill Series in Mechanical Engineering, New York (1975)
Pope, S.B.: Turbulent Flows. Cambridge University Press, Cambridge, UK (2000)
Yakhot, V.: Propagation velocity of premixed turbulent flames. Combust. Sci. Technol. 60, 191–214 (1988)
Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes, The Art of Scientific Computing, 3rd edn. Cambridge University Press, Cambridge, UK (2007)
Knuth, D.E.: The Art of Computer Programming, Seminumerical Algorithms, vol. 2, 3rd edn. Addison-Wesley, Reading, MA (1997)
Box, G.E.P., Muller, M.E.: A note on the generation of random normal deviates. Ann. Math. Stat. 29, 610–611 (1958)
Muller, M.E.: A comparison of methods for generating normal deviates on digital computers. J. ACM 6(3), 376–383 (1959)
Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in C++, The Art of Scientific Computing, 2nd edn. Cambridge University Press, Cambridge, UK (2002)
Enright, W.H., Pryce, J.D.: Two FORTRAN packages for assessing initial value problems. ACM Trans. Math. Softw. 13, 1–27 (1987)
Gear, C.W.: The automatic integration of stiff ordinary differential equations. Inf. Process. 68, 187–193 (1969)
Lapidus, L., Aiken, R.C., Liu, Y.A.: The occurrence and numerical solution of physical and chemical systems having widely varying time constants. In: Willoughby, R.A. (ed.) Stiff Differential Systems, pp. 187–200. Plenum, New York (1974)
Luss, D., Amundson, N.R.: Stability of batch catalytic fluidized beds. AIChE J., Part A 14, 211–221 (1968)
Dahoe, A.E., Zevenbergen, J.F., Lemkowitz, S.M., Scarlett, B.: Dust explosions in spherical vessels: the role of flame thickness in the validity of the ‘cube-root-law’. J. Loss Prev. Process Ind. 9(1), 33–44 (1996)
Dahoe, A.E., de Goey, L.P.H.: On the determination of the laminar burning velocity of closed vessel explosions. J. Loss Prev. Process Ind. 16, 457–478 (2003)
Dahoe, A.E.: Laminar burning velocities of hydrogen-air mixtures from closed vessel gas explosions. J. Loss Prev. Process Ind. 18, 152–166 (2005)
Dahoe, A.E., Cant, R.S., Scarlett, B.: On the decay of turbulence in the 20-liter explosion sphere. Flow Turbul. Combust. 67, 159–184 (2001)
Dahoe, A.E., van der Nat, K., Braithwaite, M., Scarlett, B.: On the effect of turbulence on the maximum explosion pressure of a dust deflagration. KONA – Powder Part. 19, 178–196 (2001)
Dahoe, A.E.: Dust explosions: a study of flame propagation. Ph.D. thesis, Delft University of Technology, Delft, The Netherlands (2000)
Bradley, D., Lau, A.K.C., Lawes, M.: Flame stretch as a determinant of turbulent burning velocity. Philos. Trans. R. Soc. Lond. Ser. A: Math. Phys. Sci. 338, 359–387 (1992)
Miller, J.A., Mitchell, R.E., Smooke, M.D., Kee, R.J.: Toward a comprehensive chemical kinetic mechanism for the oxidation of acetylene: comparison of model predictions with results from flame and shock tube experiments. In: Proceedings of the Nineteenth Symposium (International) on Combustion, pp. 181–196. The Combustion Institute, Pittsburgh (1982)
Andrews, G.E., Bradley, D., Lwakabamba, S.B.: Turbulence and turbulent flame propagation – a critical appraisal. Combust. Flame 24, 285–304 (1975)
Gülder, Ö.L.: Turbulent premixed flame propagation models for different combustion regimes. In: Proceedings of the Twenty-Third Symposium (International) on Combustion, pp. 743–750. The Combustion Institute, Pittsburgh (1990)
Lipatnikov, A.N., Chomiak, J.: Turbulent flame speed and thickness: phenomenology, evaluation, and application in multi-dimensional simulations. Prog. Energy Combust. Sci. 28, 1–74 (2002)
Dixon-Lewis, G., Wilson, J.G.: A method for the measurement of the temperature distribution in the inner cone of a Bunsen flame. Trans. Faraday Soc. 46, 1106–1114 (1951)
Janisch, G.: Geschwindigkeits- und Temperaturverteilung in einer ebenen laminaren Flammenfront. Chem. Ing. Tech. 43, 561–565 (1971)
Andrews, G.E., Bradley, D.: The burning velocity of methane-air mixtures. Combust. Flame 19, 275–288 (1972)
Damköhler, G.: Der einfluss der Turbulenz auf die Flammengeschwindigkeit in Gasgemischen. Z. Elektrochem. 46, 601–626 (1940)
Damköhler, G.: The effect of turbulence on the flame velocity in gas mixtures. Technical Memorandum NACA TM 1112. National Advisory Committee for Aeronautics, Washington (1947)
Schelkin, K.I.: On combustion in a turbulent flow. Sov. Phys. – Tech. Phys. 13, 520–530 (1943)
Schelkin, K.I.: On combustion in a turbulent flow. NACA Technical Memorandum 1110. National Advisory Committee for Aeronautics, Washington (1947). Original: J. Tech. Phys. (USSR) 13(9–10), 520–530 (1943)
Karlovitz, B., Denniston, D.W., Wells, F.E.: Investigation of turbulent flames. J. Chem. Phys. 19(5), 541–547 (1951)
Taylor, G.I.: Diffusion by continuous movements. Proc. Lond. Math. Soc. 20, 196–212 (1922)
Leason, D.B.: Turbulence and flame propagation in premixed gases. Fuel 30, 233–239 (1951)
Tucker, M.: Interaction of a free flame front with a turbulence field. Technical Note NACA TN 3407. National Advisory Committee for Aeronautics, Washington (1955)
Clavin, P., Williams, F.A.: Theory of premixed-flame propagation in large-scale turbulence. J. Fluid Mech. 90, 589–604 (1979)
Clavin, P., Williams, F.A.: Effects of molecular diffusion and of thermal expansion on the structure and dynamics of premixed flames in turbulent flows of large scale and low turbulence. J. Fluid Mech. 116, 251–282 (1982)
Klimov, A.M.: Premixed turbulent flames – interplay of hydrodynamic and chemical phenomena. Prog. Astronaut. Aeronaut. 88, 133–146 (1983)
Abdel-Gayed, R.G., Ali-Khishali, K.J., Bradley, D.: Turbulent burning velocity and flame straining in explosions. Philos. Trans. R. Soc. Lond. Ser. A: Math. Phys. Sci. 391, 393–414 (1984)
Pope, S.B., Anand, M.S.: Flamelet and distributed combustion in premixed turbulent flames. In: Proceedings of the Twentieth Symposium (International) on Combustion, pp. 403–410. The Combustion Institute, Pittsburgh (1984)
Kerstein, A.R.: Pair-exchange model of turbulent premixed flame propagation. In: Proceedings of the Twenty-First Symposium (International) on Combustion, pp. 1281–1289. The Combustion Institute, Pittsburgh (1986)
Gouldin, F.C.: An application of fractals to modeling premixed turbulent flames. Combust. Flame 68, 249–266 (1987)
Lovejoy, S.: The area-perimeter relation for rain and cloud areas. Science 216, 185–187 (1982)
Hentschel, H.G.E., Procaccia, I.: Relative diffusion in turbulent media: the fractal dimension of clouds. Phys. Rev. A 29, 1461–1470 (1984)
Sreenivasan, K.R., Meneveau, C.: The fractal facets of turbulence. J. Fluid Mech. 173, 357–186 (1986)
Abdel-Gayed, R.G., Bradley, D.: Derivation of turbulent transport coefficients from turbulent parameters in isotropic turbulence. Trans. ASME J. Fluids Eng. 99, 732–736 (1977)
Peters, N.: Laminar flamelet concepts in turbulent combustion. In: Proceedings of the Twenty-First Symposium (International) on Combustion, pp. 1231–1250. The Combustion Institute, Pittsburgh (1988)
Peters, N.: Length scales in laminar and turbulent flames. Prog. Astronaut. Aeronaut. 135, 155–182 (1991)
Kerstein, A.R.: Simple derivation of Yakhot’s turbulent premixed flamespeed formula. Combust. Sci. Technol. 60, 163–165 (1988)
Poinsot, T., Veynante, D.: Theoretical and Numerical Combustion, 2nd edn. Edwards, Philadelphia (2005)
Kerstein, A.R., Ashurst, W., Williams, F.A.: Field equation for interface propagation in an unsteady homogeneous flow field. Phys. Rev. A 37, 2728–2731 (1988)
Liu, Y., Lenze, B.: The influence of turbulence on the burning velocity of premixed CH4–H2 flames with different laminar burning velocities. In: Proceedings of the Twenty-Second Symposium (International) on Combustion, pp. 747–754. The Combustion Institute, Pittsburgh (1988)
Leuckel, W., Nastoll, W., Zarzalis, N.: Experimental investigation of the influence of turbulence on the transient premixed flame propagation inside closed vessels. In: Proceedings of the Twenty-Third Symposium (International) on Combustion, pp. 729–734. The Combustion Institute, Pittsburgh (1990)
Abdel-Gayed, R.G., Bradley, D., Lawes, M.: Turbulent burning velocities: a general correlation in terms of straining rates. Philos. Trans. R. Soc. Lond. Ser. A: Math. Phys. Sci. 414, 389–413 (1987)
Zimont, V.L., Lipatnikov, A.N.: A numerical model of premixed turbulent combustion of gases. Chem. Phys. Rep. 14, 993–1025 (1995)
Bray, K.N.C., Libby, P.A., Moss, J.B.: Flamelet crossing frequencies and mean reaction rates in premixed turbulent combustion. Combust. Sci. Technol. 41, 143–172 (1984)
Bray, K.N.C., Libby, P.A., Moss, J.B.: Unified modelling approach for premixed turbulent combustion – part I: general formulation. Combust. Flame 61, 87–102 (1985)
Bray, K.N.C., Libby, P.A.: Passage times and flamelet crossing frequencies in premixed turbulent combustion. Combust. Sci. Technol. 47, 253 (1986)
Shy, S.S., Lin, W.J., Wei, J.C.: An experimental correlation of turbulent burning velocities for premixed turbulent methane-air combustion. Philos. Trans. R. Soc. Lond. Ser. A: Math. Phys. Sci. 456, 1997–2019 (2000)
Hamberger, P., Schneider, H., Jamois, D., Proust, C.: Correlation of turbulent burning velocity and turbulence intensity for starch dust air mixtures. In: Proceedings of the Third European Combustion Meeting, 11–13 April 2007, Chania, Greece (2007)
Bray, K.N.C.: Studies of the turbulent burning velocity. Philos. Trans. R. Soc. Lond. Ser. A: Math. Phys. Sci. 431, 315–335 (1990)
Schneider, H., Proust, C.: Determination of turbulent burning velocities of dust air mixtures with the open tube method. J. Loss Prev. Process Ind. 20, 470–476 (2007)
Dahoe, A.E., van Velzen T.J., Sluijs, L.P., Neervoort, F.J., Leschonski, S., Lemkowitz, S.M., van der Wel, P.G.J., Scarlett, B.: Construction and operation of a 20-litre dust explosion sphere at and above atmospheric conditions. In: Mewis, J.J., Pasman, H.J., De Rademaeker, E.E. (eds.) Loss Prevention and Safety Promotion in the Process Industries, Proceedings of the 8th International Symposium, vol. 2, pp. 285–302. European Federation of Chemical Engineering (EFCE), Elsevier Science (1995)
Dahoe, A.E., van der Wel, P.G.J., Lemkowitz, S.M., Leschonski, S., Scarlett, B.: Effects of turbulence on dust explosions at elevated initial pressures. In: Leschonski, K. (ed.) PARTEC 95, 6th European Symposium Particle Characterization, pp. 257–266. European Federation of Chemical Engineering (EFCE), Nürnberg Messe GmbH (1995)
Dahoe, A.E., Zevenbergen, J.F., Lemkowitz, S.M., Scarlett, B.: Dust explosion testing with the strengthened 20-litre sphere. In: The Seventh International Colloquium on Dust Explosions, pp. 7.30–7.46. Christian Michelsen Research AS (1996)
Siwek, R.: 20-l laborapparatur für die Bestimmung der explosionskenngrößen brennbarer Stäube. Ph.D. thesis, Technical University of Winterthur, Winterthur, Switzerland (1977)
Borghi, R.: On the structure and morphology of turbulent premixed flames. In: Casci, C. (ed.) Recent Advances in the Aerospace Sciences, pp. 117–138. Plenum Publishing Corporation (1985)
Goodwin, D.G.: Cantera C++ User’s Guide. California Institute of Technology (2002)
Marquardt, D.W.: An algorithm for least-squares estimation of nonlinear parameters. SIAM J. Appl. Math. 11, 431–441 (1963)
Williams, F.A.: Combustion Theory: The Fundamental Theory of Chemically Reacting Flow Systems. Combustion Science and Engineering Series, 2nd edn. The Benjamin/Cummings Publishing Company, Menlo Park, California (1985)
Lawson, C.L., Hanson, R.J.: Solving Least Squares Problems. SIAM Classics in Applied Mathematics, vol. 15. Society for Industrial and Applied Mathematics, Philadelphia (1995)
Dennis, J.E., Schnabel, R.B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. SIAM Classics in Applied Mathematics, vol. 16. Society for Industrial and Applied Mathematics, Philadelphia (1996)
Butcher, J.C.: Numerical Methods for Ordinary Differential Equations. Wiley, Chichester, England (2003)
Lambert, J.D.: Numerical methods for ordinary differential systems: the initial value problem. Wiley, Chichester (1991)
Hairer, E., Norsett, S.P., Wanner, G.: Solving ordinary differential equations. I. Nonstiff problems. In: Springer Series in Computational Mathematics, vol. 8, 2nd edn. Springer, Berlin (1993)
Butcher, J.C.: Implicit Runge–Kutta processes. Math. Comput. 18, 50–64 (1964)
Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in C, The Art of Scientific Computing, 2nd edn. Cambridge University Press, Cambridge, UK (1992)
Cash, J.R., Karp, A.H.: A variable order Runge–Kutta method for initial value problems with rapidly varying right-hand sides. ACM Trans. Math. Softw. 16, 201–222 (1990)
Shampine, L.F.: Implementation of Rosenbrock methods. ACM Trans. Math. Softw. 8, 93–113 (1982)
Spiegel, M.R., Stephens, L.J.: Theory and Problems of Statistics. Schaum’s Outline Series, 3rd edn. McGraw-Hill, New York (1999)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dahoe, A.E., Skjold, T., Roekaerts, D.J.E.M. et al. On the Application of the Levenberg–Marquardt Method in Conjunction with an Explicit Runge–Kutta and an Implicit Rosenbrock Method to Assess Burning Velocities from Confined Deflagrations. Flow Turbulence Combust 91, 281–317 (2013). https://doi.org/10.1007/s10494-013-9462-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10494-013-9462-z