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March  2011, 4(1): 259-276. doi: 10.3934/krm.2011.4.259

Kinetic approach to deflagration processes in a recombination reaction

Fiammetta Conforto 1, , Maria Groppi 2, , Roberto Monaco 3, and  Giampiero Spiga 4,

1. 

Dipartimento di Matematica, Università di Messina, Viale F. Stagno d'Alcontres 31 - 98166 Messina, Italy
2. 

Dipartimento di Matematica, Università di Parma, V.le G.P. Usberti 53/A, 43100 Parma
3. 

Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24 - 10129 Torino, Italy
4. 

Dipartimento di Matematica, Università di Parma, V.le G.P. Usberti 53/A, 43124 Parma

Received  July 2010 Revised  October 2010 Published  January 2011

Steady one-dimensional flame structure is investigated in a binary gas mixture made up by diatomic molecules and atoms, which undergo an irreversible exothermic two--steps reaction, a recombination process followed by inelastic scattering (de-excitation). A kinetic model at the Boltzmann level, accounting for chemical encounters as well as for mechanical collisions, is proposed and its main features are analyzed. In the case of collision dominated regime with slow recombination and fast de-excitation, the model is the starting point for a consistent derivation, via suitable asymptotic expansion of Chapman-Enskog type, of reactive fluid-dynamic Navier-Stokes equations. The resulting set of ordinary differential equations for the smooth steady deflagration profile is investigated in the frame of the qualitative theory of dynamical systems, and numerical results for the flame eigenvalue and for the main macroscopic observables are presented and briefly commented on for illustrative purposes.
Keywords: Kinetic theory; Irreversible chemical reactions; Deflagration waves..
Mathematics Subject Classification: 82C40, 80A25, 76V0.
Citation: Fiammetta Conforto, Maria Groppi, Roberto Monaco, Giampiero Spiga. Kinetic approach to deflagration processes in a recombination reaction. Kinetic & Related Models, 2011, 4 (1) : 259-276. doi: 10.3934/krm.2011.4.259
References:
[1]

M. Bisi, M. Groppi and G. Spiga, Flame structure from a kinetic model for chemical reactions, Kinetic and Related Models, 3 (2010), 17-34. doi: 10.3934/krm.2010.3.17.  Google Scholar

[2]

V. C. Boffi, V. Protopopescu and G. Spiga, On the equivalence between the probabilistic, kinetic, and scattering kernel formulations of the Boltzmann equation, Physica A, 164 (1990), 400-410. doi: 10.1016/0378-4371(90)90203-5.  Google Scholar

[3]

C. Cercignani, "The Boltzmann Equation and its Applications,'' Springer, New York, 1988.  Google Scholar

[4]

C. Cercignani, "Rarefied Gas Dynamics. From Basic Concepts to Actual Calculations,'' University Press, Cambridge, 2000.  Google Scholar

[5]

S. Chapman and T. G. Cowling, "The Mathematical Theory of Non-Uniform Gases,'' University Press, Cambridge, 1990.  Google Scholar

[6]

R. M. Colombo and A. Corli, Sonic and kinetic phase transitions with applications to Chapman–Jouguet deflagrations, Math. Meth. Appl. Sci., 27 (2004), 843-864. doi: 10.1002/mma.474.  Google Scholar

[7]

S. R. De Groot and P. Mazur, "Non-Equilibrium Thermodynamics,'' North-Holland, Amsterdam, 1963.  Google Scholar

[8]

W. Fickett and W. C. Davis, "Detonation, Theory and Experiment,'' Dover, New York, 1979. Google Scholar

[9]

V. Giovangigli, "Multicomponent Flow Modeling,'' Birkhäuser, Boston, 1999.  Google Scholar

[10]

I. Glassman, "Combustion,'' Academic Press, New York, 1987. Google Scholar

[11]

M. Groppi, A. Rossani and G. Spiga, Kinetic theory of a diatomic gas with reactions of dissociation and recombination through a transition state, J. Phys. A: Math. Gen., 33 (2000), 8819-8833. doi: 10.1088/0305-4470/33/48/317.  Google Scholar

[12]

D. Jacob, "Introduction to Atmosperic Chemistry,'' University Press, Princeton, 1999. Google Scholar

[13]

L. He, Analysis of compressibility effects on Darrieus-Landau instability of deflagration wave, Europhys. Lett., 49 (2000), 576-582. doi: 10.1209/epl/i2000-00189-8.  Google Scholar

[14]

L. Kagan, On the transition from deflagration to detonation in narrow channels, Math. Model. Nat. Phenom., 2 (2007), 40-55. doi: 10.1051/mmnp:2008018.  Google Scholar

[15]

A. K. Kapila, B. J. Matkowsky and A. van Harten, An asymptotic theory of deflagrations and detonations. I. The steady solutions, SIAM J. Appl. Math., 43 (1983), 491-519. doi: 10.1137/0143032.  Google Scholar

[16]

K. K. Kuo, "Principles of Combustion,'' Wiley, Hoboken (New Jersey), 2005. Google Scholar

[17]

S. B. Margolis and M. R. Baer, A singular-perturbation analysis of the burning-rate eigenvalue for a two-temperature model of deflagrations in confined porous energetic materials, SIAM J. Appl. Math., 62 (2001), 627-663. doi: 10.1137/S0036139900377780.  Google Scholar

[18]

J. Menkes, On the stability of a plane deflagration wave, Proc. Roy. Soc. London. Ser. A, 253 (1959), 380-389.  Google Scholar

[19]

J. R. Mika and J. Banasiak, "Singularly Perturbed Evolution Equations with Applications to Kinetic Theory,'' World, Singapore, 1995.  Google Scholar

[20]

I. Müller, Flame structure in ordinary and extended thermodynamics, in "Asymptotic Methods in Nonlinear Wave Phenomena'' (eds. T. Ruggeri and M. Sammartino), World, Singapore, (2007), 144-153. doi: 10.1142/9789812708908_0013.  Google Scholar

[21]

L. Pan and W. Sheng, The scalar Zeldovich-von Neumann-Doering combustion model (II) interactions of shock and deflagration, Nonlinear Analysis: Real World Applications, 10 (2009), 449-457. doi: 10.1016/j.nonrwa.2007.10.006.  Google Scholar

[22]

W. C. Sheng and D. C. Tan, Weak deflagration solutions to the simplest combustion model, Journal of Differential Equations, 107 (1994), 207-230. doi: 10.1006/jdeq.1994.1009.  Google Scholar

[23]

S. Takata and K. Aoki, Two-surface problems of a multicomponent mixture of vapors and non condensable gases in the continuum limit in the light of kinetic theory, Phys. Fluids, 11 (1999), 2743-2756. doi: 10.1063/1.870133.  Google Scholar

[24]

S. Takata, Kinetic theory analysis of the two-surface problem of a vapor-vapor mixture in the continuum limit, Phys. Fluids, 16 (2004), 2182-2198. doi: 10.1063/1.1723464.  Google Scholar

[25]

D. H. Wagner, Existence of deflagration waves: Connection to a degenerate critical point, in "Lecture Notes in Pure and Appl. Math.," 102, Dekker, New York, (1985), 187-197.  Google Scholar

[26]

Y. Yoshizawa, Wave structures of chemically reacting gas by the kinetic theory of gases, in "Rarefied Gas Dynamics'' (ed. J.L. Potter), A.I.A.A., New York, (1977), 501-517. Google Scholar

[27]

P. Zhang and T. Zhang, The Riemann problem for scalar CJ-combustion model without convexity, Discrete and Cont. Dynamical Systems, 1 (1995), 195-206.  Google Scholar

show all references

References:
[1]

M. Bisi, M. Groppi and G. Spiga, Flame structure from a kinetic model for chemical reactions, Kinetic and Related Models, 3 (2010), 17-34. doi: 10.3934/krm.2010.3.17.  Google Scholar

[2]

V. C. Boffi, V. Protopopescu and G. Spiga, On the equivalence between the probabilistic, kinetic, and scattering kernel formulations of the Boltzmann equation, Physica A, 164 (1990), 400-410. doi: 10.1016/0378-4371(90)90203-5.  Google Scholar

[3]

C. Cercignani, "The Boltzmann Equation and its Applications,'' Springer, New York, 1988.  Google Scholar

[4]

C. Cercignani, "Rarefied Gas Dynamics. From Basic Concepts to Actual Calculations,'' University Press, Cambridge, 2000.  Google Scholar

[5]

S. Chapman and T. G. Cowling, "The Mathematical Theory of Non-Uniform Gases,'' University Press, Cambridge, 1990.  Google Scholar

[6]

R. M. Colombo and A. Corli, Sonic and kinetic phase transitions with applications to Chapman–Jouguet deflagrations, Math. Meth. Appl. Sci., 27 (2004), 843-864. doi: 10.1002/mma.474.  Google Scholar

[7]

S. R. De Groot and P. Mazur, "Non-Equilibrium Thermodynamics,'' North-Holland, Amsterdam, 1963.  Google Scholar

[8]

W. Fickett and W. C. Davis, "Detonation, Theory and Experiment,'' Dover, New York, 1979. Google Scholar

[9]

V. Giovangigli, "Multicomponent Flow Modeling,'' Birkhäuser, Boston, 1999.  Google Scholar

[10]

I. Glassman, "Combustion,'' Academic Press, New York, 1987. Google Scholar

[11]

M. Groppi, A. Rossani and G. Spiga, Kinetic theory of a diatomic gas with reactions of dissociation and recombination through a transition state, J. Phys. A: Math. Gen., 33 (2000), 8819-8833. doi: 10.1088/0305-4470/33/48/317.  Google Scholar

[12]

D. Jacob, "Introduction to Atmosperic Chemistry,'' University Press, Princeton, 1999. Google Scholar

[13]

L. He, Analysis of compressibility effects on Darrieus-Landau instability of deflagration wave, Europhys. Lett., 49 (2000), 576-582. doi: 10.1209/epl/i2000-00189-8.  Google Scholar

[14]

L. Kagan, On the transition from deflagration to detonation in narrow channels, Math. Model. Nat. Phenom., 2 (2007), 40-55. doi: 10.1051/mmnp:2008018.  Google Scholar

[15]

A. K. Kapila, B. J. Matkowsky and A. van Harten, An asymptotic theory of deflagrations and detonations. I. The steady solutions, SIAM J. Appl. Math., 43 (1983), 491-519. doi: 10.1137/0143032.  Google Scholar

[16]

K. K. Kuo, "Principles of Combustion,'' Wiley, Hoboken (New Jersey), 2005. Google Scholar

[17]

S. B. Margolis and M. R. Baer, A singular-perturbation analysis of the burning-rate eigenvalue for a two-temperature model of deflagrations in confined porous energetic materials, SIAM J. Appl. Math., 62 (2001), 627-663. doi: 10.1137/S0036139900377780.  Google Scholar

[18]

J. Menkes, On the stability of a plane deflagration wave, Proc. Roy. Soc. London. Ser. A, 253 (1959), 380-389.  Google Scholar

[19]

J. R. Mika and J. Banasiak, "Singularly Perturbed Evolution Equations with Applications to Kinetic Theory,'' World, Singapore, 1995.  Google Scholar

[20]

I. Müller, Flame structure in ordinary and extended thermodynamics, in "Asymptotic Methods in Nonlinear Wave Phenomena'' (eds. T. Ruggeri and M. Sammartino), World, Singapore, (2007), 144-153. doi: 10.1142/9789812708908_0013.  Google Scholar

[21]

L. Pan and W. Sheng, The scalar Zeldovich-von Neumann-Doering combustion model (II) interactions of shock and deflagration, Nonlinear Analysis: Real World Applications, 10 (2009), 449-457. doi: 10.1016/j.nonrwa.2007.10.006.  Google Scholar

[22]

W. C. Sheng and D. C. Tan, Weak deflagration solutions to the simplest combustion model, Journal of Differential Equations, 107 (1994), 207-230. doi: 10.1006/jdeq.1994.1009.  Google Scholar

[23]

S. Takata and K. Aoki, Two-surface problems of a multicomponent mixture of vapors and non condensable gases in the continuum limit in the light of kinetic theory, Phys. Fluids, 11 (1999), 2743-2756. doi: 10.1063/1.870133.  Google Scholar

[24]

S. Takata, Kinetic theory analysis of the two-surface problem of a vapor-vapor mixture in the continuum limit, Phys. Fluids, 16 (2004), 2182-2198. doi: 10.1063/1.1723464.  Google Scholar

[25]

D. H. Wagner, Existence of deflagration waves: Connection to a degenerate critical point, in "Lecture Notes in Pure and Appl. Math.," 102, Dekker, New York, (1985), 187-197.  Google Scholar

[26]

Y. Yoshizawa, Wave structures of chemically reacting gas by the kinetic theory of gases, in "Rarefied Gas Dynamics'' (ed. J.L. Potter), A.I.A.A., New York, (1977), 501-517. Google Scholar

[27]

P. Zhang and T. Zhang, The Riemann problem for scalar CJ-combustion model without convexity, Discrete and Cont. Dynamical Systems, 1 (1995), 195-206.  Google Scholar

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