June  2017, 10(2): 513-539. doi: 10.3934/krm.2017020

On modified simple reacting spheres kinetic model for chemically reactive gases

1. 

Department of Mathematics, California State University, Northridge, California 91330, USA

2. 

Centre of Mathematics, University of Minho, 4710-057 Braga, Portugal

* Corresponding author: Jacek Polewczak

Received  December 2015 Revised  May 2016 Published  November 2016

We consider the modified simple reacting spheres (MSRS) kinetic model that, in addition to the conservation of energy and momentum, also preserves the angular momentum in the collisional processes. In contrast to the line-of-center models or chemical reactive models considered in [23], in the MSRS (SRS) kinetic models, the microscopic reversibility (detailed balance) can be easily shown to be satisfied, and thus all mathematical aspects of the model can be fully justified. In the MSRS model, the molecules behave as if they were single mass points with two internal states. Collisions may alter the internal states of the molecules, and this occurs when the kinetic energy associated with the reactive motion exceeds the activation energy. Reactive and non-reactive collision events are considered to be hard spheres-like. We consider a four component mixture $A$, $B$, $A^*$, $B^*$, in which the chemical reactions are of the type $A+B\rightleftharpoons A^*+B^*$, with $A^*$ and $B^*$ being distinct species from $A$ and $B$. We provide fundamental physical and mathematical properties of the MSRS model, concerning the consistency of the model, the entropy inequality for the reactive system, the characterization of the equilibrium solutions, the macroscopic setting of the model and the spatially homogeneous evolution. Moreover, we show that the MSRS kinetic model reduces to the previously considered SRS model (e.g., [21], [27]) if the reduced masses of the reacting pairs are the same before and after collisions, and state in the Appendix the more important properties of the SRS system.

Citation: Jacek Polewczak, Ana Jacinta Soares. On modified simple reacting spheres kinetic model for chemically reactive gases. Kinetic and Related Models, 2017, 10 (2) : 513-539. doi: 10.3934/krm.2017020
References:
[1]

M. Bisi and M. J. Cáceres, A BGK relaxation model for polyatomic gas mixtures, Commun. Math. Sci., 14 (2016), 297-325.  doi: 10.4310/CMS.2016.v14.n2.a1.

[2]

F. Carvalho, J. Polewczak and A. J. Soares, On the kinetic systems for simple reacting spheres: modeling and linearized equations, in From Particle Systems and Partial Differential Equations Ⅰ, Springer Proceedings in Mathematics & Statistics, 75 (2014), 251-267. doi: 10.1007/978-3-642-54271-8_12.

[3]

F. Carvalho, J. Polewczak and A. J. Soares, Kinetic theory of simple reacting spheres: an application to coloring processes, in From Particle Systems and Partial Differential Equations Ⅱ, Springer Proceedings in Mathematics & Statistics, 129 (2015), 153-172. doi: 10.1007/978-3-319-16637-7_4.

[4] C. Cercignani, The Boltzmann Equation and its Applications, Springer, New York, 1988.  doi: 10.1007/978-1-4612-1039-9.
[5]

A. S. Cukrowski, The role of products and a reverse reaction in analysis of nonequilibrium effects in a bimolecular chemical reaction in a dilute gas, Physica A, 275 (2000), 134-151.  doi: 10.1016/S0378-4371(99)00410-0.

[6]

J. S. Dahler and L. Qin, Nonequilibrium Statistical Mechanics of Chemically Reactive Fluids, J. Chem. Physics, 118 (2003), 8396-8404.  doi: 10.1063/1.1565331.

[7]

L. DesvillettesR. Monaco and F. Salvarani, A kinetic model allowing to obtain the energy law of polytropic gases in the presence of chemical reactions, Eur. J. Mech. B Fluids, 24 (2005), 219-236.  doi: 10.1016/j.euromechflu.2004.07.004.

[8]

A. Ern and V. Giovangigli, Kinetic theory of reactive gas mixtures with application to combustion, Transp. Theory Stat. Phys., 32 (2003), 657-677.  doi: 10.1081/TT-120025071.

[9]

V. Giovangigli and M. Massot, Entropic structure of multicomponent reactive flows with partial equilibrium reduced chemistry, Math. Meth. Appl. Sci., 27 (2004), 739-768.  doi: 10.1002/mma.429.

[10] V. Giovangigli, Multicomponent Flow Modeling, Birkhäuser, Boston, 1999.  doi: 10.1007/978-1-4612-1580-6.
[11]

M. Groppi and J. Polewczak, On two kinetic models for chemical reactions: Comparisons and existence results, J. Stat. Physics, 117 (2004), 211-241.  doi: 10.1023/B:JOSS.0000044059.59066.a9.

[12]

M. Groppi and G. Spiga, Kinetic approach to chemical reactions and inelastic transitions in a rarefied gas, J. Math. Chem., 26 (1999), 197-219. 

[13]

M. Groppi and G. Spiga, A Bhatnagar-Gross-Krook-type approach for chemically reacting gas mixtures, Phys. Fluids, 16 (2004), 4273-4284.  doi: 10.1063/1.1808651.

[14]

G. M. Kremer, M. P. Bianchi and A. J. Soares, A relaxation kinetic model for transport phenomena in a reactive flow Phys. Fluids 18 (2006), 037104, 15pp. doi: 10.1063/1.2185691.

[15]

G. M. KremerM. P. Bianchi and A. J. Soares, Analysis of the trend to equilibrium of a chemically reacting system, J. Phys. A: Math. Theor., 40 (2007), 2553-2571.  doi: 10.1088/1751-8113/40/10/020.

[16]

J. Polewczak, The kinetic theory of simple reacting spheres: Ⅰ. Global existence result in a dilute-gas case, J. Stat. Physics, 100 (2000), 327-362.  doi: 10.1023/A:1018608216136.

[17]

J. Polewczak and A. J. Soares, Kinetic theory of simple reacting spheres Ⅰ, in 27th International Symposium on Rarefied Gas Dynamics, 2010, Pacific Grove. AIP Conference Proceedings, 1333 (2011), 117-122. doi: 10.1063/1.3562635.

[18]

R. D. Present, On the velocity distribution in a chemically reacting gas, J. Chem. Phys., 31 (1959), 747-750.  doi: 10.1063/1.1730456.

[19]

I. Prigogine and M. Mahieu, Sur lapPerturbation de la distribution de Maxwell par des réactions chimiques en phase gazeuse, Physica, XVI (1950), 51-64. 

[20]

I. Prigogine and E. Xhrouet, On the perturbation of Maxwell distribution function by chemical reaction in gases, Physica, 15 (1949), 913-932.  doi: 10.1016/0031-8914(49)90057-9.

[21]

L. Qin and J. S. Dahler, The kinetic theory of a simple, chemically reactive fluid: Scattering functions and relaxation processes, J. Chem. Physics, 103 (1995), 725-750.  doi: 10.1063/1.470106.

[22]

J. Ross and P. Mazur, Some deductions from a formal statistical mechanical theory of chemical kinetics, J. Chem. Phys., 35 (1961), 19-28.  doi: 10.1063/1.1731889.

[23]

A. Rossani and G. Spiga, A note on the kinetic theory of chemically reacting gases, Physica A, 272 (1999), 563-573.  doi: 10.1016/S0378-4371(99)00336-2.

[24]

B. Shizgal and M. Karplus, Nonequilibrium contributions to the rate of reaction. Ⅰ. Perturbation of the velocity distribution function, J. Chem. Phys., 52 (1969), 4262-4278. 

[25]

A. W. SilvaG. M. Alves and G. M. Kremer, Transport phenomena in a reactive quaternary gas mixture, Physica A, 374 (2007), 533-548.  doi: 10.1016/j.physa.2006.07.039.

[26]

H. Van Beijeren and M. H. Ernst, The modified Enskog equation, Physica, 68 (1973), 437-456.  doi: 10.1016/0031-8914(73)90372-8.

[27]

N. Xystris and J. S. Dahler, Kinetic theory of simple reacting spheres, J. Chem. Phys., 68 (1978), 387-401.  doi: 10.1063/1.435772.

show all references

References:
[1]

M. Bisi and M. J. Cáceres, A BGK relaxation model for polyatomic gas mixtures, Commun. Math. Sci., 14 (2016), 297-325.  doi: 10.4310/CMS.2016.v14.n2.a1.

[2]

F. Carvalho, J. Polewczak and A. J. Soares, On the kinetic systems for simple reacting spheres: modeling and linearized equations, in From Particle Systems and Partial Differential Equations Ⅰ, Springer Proceedings in Mathematics & Statistics, 75 (2014), 251-267. doi: 10.1007/978-3-642-54271-8_12.

[3]

F. Carvalho, J. Polewczak and A. J. Soares, Kinetic theory of simple reacting spheres: an application to coloring processes, in From Particle Systems and Partial Differential Equations Ⅱ, Springer Proceedings in Mathematics & Statistics, 129 (2015), 153-172. doi: 10.1007/978-3-319-16637-7_4.

[4] C. Cercignani, The Boltzmann Equation and its Applications, Springer, New York, 1988.  doi: 10.1007/978-1-4612-1039-9.
[5]

A. S. Cukrowski, The role of products and a reverse reaction in analysis of nonequilibrium effects in a bimolecular chemical reaction in a dilute gas, Physica A, 275 (2000), 134-151.  doi: 10.1016/S0378-4371(99)00410-0.

[6]

J. S. Dahler and L. Qin, Nonequilibrium Statistical Mechanics of Chemically Reactive Fluids, J. Chem. Physics, 118 (2003), 8396-8404.  doi: 10.1063/1.1565331.

[7]

L. DesvillettesR. Monaco and F. Salvarani, A kinetic model allowing to obtain the energy law of polytropic gases in the presence of chemical reactions, Eur. J. Mech. B Fluids, 24 (2005), 219-236.  doi: 10.1016/j.euromechflu.2004.07.004.

[8]

A. Ern and V. Giovangigli, Kinetic theory of reactive gas mixtures with application to combustion, Transp. Theory Stat. Phys., 32 (2003), 657-677.  doi: 10.1081/TT-120025071.

[9]

V. Giovangigli and M. Massot, Entropic structure of multicomponent reactive flows with partial equilibrium reduced chemistry, Math. Meth. Appl. Sci., 27 (2004), 739-768.  doi: 10.1002/mma.429.

[10] V. Giovangigli, Multicomponent Flow Modeling, Birkhäuser, Boston, 1999.  doi: 10.1007/978-1-4612-1580-6.
[11]

M. Groppi and J. Polewczak, On two kinetic models for chemical reactions: Comparisons and existence results, J. Stat. Physics, 117 (2004), 211-241.  doi: 10.1023/B:JOSS.0000044059.59066.a9.

[12]

M. Groppi and G. Spiga, Kinetic approach to chemical reactions and inelastic transitions in a rarefied gas, J. Math. Chem., 26 (1999), 197-219. 

[13]

M. Groppi and G. Spiga, A Bhatnagar-Gross-Krook-type approach for chemically reacting gas mixtures, Phys. Fluids, 16 (2004), 4273-4284.  doi: 10.1063/1.1808651.

[14]

G. M. Kremer, M. P. Bianchi and A. J. Soares, A relaxation kinetic model for transport phenomena in a reactive flow Phys. Fluids 18 (2006), 037104, 15pp. doi: 10.1063/1.2185691.

[15]

G. M. KremerM. P. Bianchi and A. J. Soares, Analysis of the trend to equilibrium of a chemically reacting system, J. Phys. A: Math. Theor., 40 (2007), 2553-2571.  doi: 10.1088/1751-8113/40/10/020.

[16]

J. Polewczak, The kinetic theory of simple reacting spheres: Ⅰ. Global existence result in a dilute-gas case, J. Stat. Physics, 100 (2000), 327-362.  doi: 10.1023/A:1018608216136.

[17]

J. Polewczak and A. J. Soares, Kinetic theory of simple reacting spheres Ⅰ, in 27th International Symposium on Rarefied Gas Dynamics, 2010, Pacific Grove. AIP Conference Proceedings, 1333 (2011), 117-122. doi: 10.1063/1.3562635.

[18]

R. D. Present, On the velocity distribution in a chemically reacting gas, J. Chem. Phys., 31 (1959), 747-750.  doi: 10.1063/1.1730456.

[19]

I. Prigogine and M. Mahieu, Sur lapPerturbation de la distribution de Maxwell par des réactions chimiques en phase gazeuse, Physica, XVI (1950), 51-64. 

[20]

I. Prigogine and E. Xhrouet, On the perturbation of Maxwell distribution function by chemical reaction in gases, Physica, 15 (1949), 913-932.  doi: 10.1016/0031-8914(49)90057-9.

[21]

L. Qin and J. S. Dahler, The kinetic theory of a simple, chemically reactive fluid: Scattering functions and relaxation processes, J. Chem. Physics, 103 (1995), 725-750.  doi: 10.1063/1.470106.

[22]

J. Ross and P. Mazur, Some deductions from a formal statistical mechanical theory of chemical kinetics, J. Chem. Phys., 35 (1961), 19-28.  doi: 10.1063/1.1731889.

[23]

A. Rossani and G. Spiga, A note on the kinetic theory of chemically reacting gases, Physica A, 272 (1999), 563-573.  doi: 10.1016/S0378-4371(99)00336-2.

[24]

B. Shizgal and M. Karplus, Nonequilibrium contributions to the rate of reaction. Ⅰ. Perturbation of the velocity distribution function, J. Chem. Phys., 52 (1969), 4262-4278. 

[25]

A. W. SilvaG. M. Alves and G. M. Kremer, Transport phenomena in a reactive quaternary gas mixture, Physica A, 374 (2007), 533-548.  doi: 10.1016/j.physa.2006.07.039.

[26]

H. Van Beijeren and M. H. Ernst, The modified Enskog equation, Physica, 68 (1973), 437-456.  doi: 10.1016/0031-8914(73)90372-8.

[27]

N. Xystris and J. S. Dahler, Kinetic theory of simple reacting spheres, J. Chem. Phys., 68 (1978), 387-401.  doi: 10.1063/1.435772.

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