American Institute of Mathematical Sciences

Advanced
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. 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. doi: 10.1016/0378-4371(90)90203-5. Google Scholar

[3]

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

[4]

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

[5]

S. Chapman and T. G. Cowling, "The Mathematical Theory of Non-Uniform Gases,'', University Press, (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. doi: 10.1002/mma.474. Google Scholar

[7]

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

[8]

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

[9]

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

[10]

I. Glassman, "Combustion,'', Academic Press, (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. doi: 10.1088/0305-4470/33/48/317. Google Scholar

[12]

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

[13]

L. He, Analysis of compressibility effects on Darrieus-Landau instability of deflagration wave,, Europhys. Lett., 49 (2000), 576. 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. 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. doi: 10.1137/0143032. Google Scholar

[16]

K. K. Kuo, "Principles of Combustion,'', Wiley, (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. 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. Google Scholar

[19]

J. R. Mika and J. Banasiak, "Singularly Perturbed Evolution Equations with Applications to Kinetic Theory,'', World, (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), (2007), 144. 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. 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. 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. 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. doi: 10.1063/1.1723464. Google Scholar

[25]

D. H. Wagner, Existence of deflagration waves: Connection to a degenerate critical point,, in, 102 (1985), 187. 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), (1977), 501. 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. 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. 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. doi: 10.1016/0378-4371(90)90203-5. Google Scholar

[3]

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

[4]

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

[5]

S. Chapman and T. G. Cowling, "The Mathematical Theory of Non-Uniform Gases,'', University Press, (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. doi: 10.1002/mma.474. Google Scholar

[7]

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

[8]

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

[9]

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

[10]

I. Glassman, "Combustion,'', Academic Press, (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. doi: 10.1088/0305-4470/33/48/317. Google Scholar

[12]

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

[13]

L. He, Analysis of compressibility effects on Darrieus-Landau instability of deflagration wave,, Europhys. Lett., 49 (2000), 576. 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. 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. doi: 10.1137/0143032. Google Scholar

[16]

K. K. Kuo, "Principles of Combustion,'', Wiley, (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. 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. Google Scholar

[19]

J. R. Mika and J. Banasiak, "Singularly Perturbed Evolution Equations with Applications to Kinetic Theory,'', World, (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), (2007), 144. 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. 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. 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. 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. doi: 10.1063/1.1723464. Google Scholar

[25]

D. H. Wagner, Existence of deflagration waves: Connection to a degenerate critical point,, in, 102 (1985), 187. 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), (1977), 501. 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. Google Scholar

[1]

Marzia Bisi, Maria Groppi, Giampiero Spiga. Flame structure from a kinetic model for chemical reactions. Kinetic & Related Models, 2010, 3 (1) : 17-34. doi: 10.3934/krm.2010.3.17
[2]

Marzia Bisi, Giampiero Spiga. On a kinetic BGK model for slow chemical reactions. Kinetic & Related Models, 2011, 4 (1) : 153-167. doi: 10.3934/krm.2011.4.153
[3]

Arno F. Münster. Simulation of stationary chemical patterns and waves in ionic reactions. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 35-46. doi: 10.3934/dcdsb.2002.2.35
[4]

Julien Coatléven, Claudio Altafini. A kinetic mechanism inducing oscillations in simple chemical reactions networks. Mathematical Biosciences & Engineering, 2010, 7 (2) : 301-312. doi: 10.3934/mbe.2010.7.301
[5]

Congming Li, Eric S. Wright. Modeling chemical reactions in rivers: A three component reaction. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 377-384. doi: 10.3934/dcds.2001.7.373
[6]

Jerry L. Bona, Thierry Colin, Colette Guillopé. Propagation of long-crested water waves. Ⅱ. Bore propagation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5543-5569. doi: 10.3934/dcds.2019244
[7]

Wenxiong Chen, Congming Li, Eric S. Wright. On a nonlinear parabolic system-modeling chemical reactions in rivers. Communications on Pure & Applied Analysis, 2005, 4 (4) : 889-899. doi: 10.3934/cpaa.2005.4.889
[8]

Andrea Picco, Lamberto Rondoni. Boltzmann maps for networks of chemical reactions and the multi-stability problem. Networks & Heterogeneous Media, 2009, 4 (3) : 501-526. doi: 10.3934/nhm.2009.4.501
[9]

Steinar Evje, Aksel Hiorth, Merete V. Madland, Reidar I. Korsnes. A mathematical model relevant for weakening of chalk reservoirs due to chemical reactions. Networks & Heterogeneous Media, 2009, 4 (4) : 755-788. doi: 10.3934/nhm.2009.4.755
[10]

Bogdan Kazmierczak, Zbigniew Peradzynski. Calcium waves with mechano-chemical couplings. Mathematical Biosciences & Engineering, 2013, 10 (3) : 743-759. doi: 10.3934/mbe.2013.10.743
[11]

Niclas Bernhoff. Boundary layers for discrete kinetic models: Multicomponent mixtures, polyatomic molecules, bimolecular reactions, and quantum kinetic equations. Kinetic & Related Models, 2017, 10 (4) : 925-955. doi: 10.3934/krm.2017037
[12]

Darryl D. Holm, Vakhtang Putkaradze, Cesare Tronci. Collisionless kinetic theory of rolling molecules. Kinetic & Related Models, 2013, 6 (2) : 429-458. doi: 10.3934/krm.2013.6.429
[13]

Emmanuel Frénod, Mathieu Lutz. On the Geometrical Gyro-Kinetic theory. Kinetic & Related Models, 2014, 7 (4) : 621-659. doi: 10.3934/krm.2014.7.621
[14]

Guillaume Bal, Tomasz Komorowski, Lenya Ryzhik. Kinetic limits for waves in a random medium. Kinetic & Related Models, 2010, 3 (4) : 529-644. doi: 10.3934/krm.2010.3.529
[15]

Paolo Barbante, Aldo Frezzotti, Livio Gibelli. A kinetic theory description of liquid menisci at the microscale. Kinetic & Related Models, 2015, 8 (2) : 235-254. doi: 10.3934/krm.2015.8.235
[16]

José Antonio Alcántara, Simone Calogero. On a relativistic Fokker-Planck equation in kinetic theory. Kinetic & Related Models, 2011, 4 (2) : 401-426. doi: 10.3934/krm.2011.4.401
[17]

Hung-Wen Kuo. Effect of abrupt change of the wall temperature in the kinetic theory. Kinetic & Related Models, 2019, 12 (4) : 765-789. doi: 10.3934/krm.2019030
[18]

Wenying Feng, Guang Zhang, Yikang Chai. Existence of positive solutions for second order differential equations arising from chemical reactor theory. Conference Publications, 2007, 2007 (Special) : 373-381. doi: 10.3934/proc.2007.2007.373
[19]

José A. Carrillo, M. R. D’Orsogna, V. Panferov. Double milling in self-propelled swarms from kinetic theory. Kinetic & Related Models, 2009, 2 (2) : 363-378. doi: 10.3934/krm.2009.2.363
[20]

Marzia Bisi, Tommaso Ruggeri, Giampiero Spiga. Dynamical pressure in a polyatomic gas: Interplay between kinetic theory and extended thermodynamics. Kinetic & Related Models, 2018, 11 (1) : 71-95. doi: 10.3934/krm.2018004

2018 Impact Factor: 1.38

Tools

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences