An internal state kinetic model for chemically reacting mixtures of monatomic and polyatomic gases

Marzia Bisi; Thomas Borsoni; Maria Groppi

doi:10.3934/krm.2023023

Article Contents

2024, Volume 17, Issue 2: 276-311. Doi: 10.3934/krm.2023023

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An internal state kinetic model for chemically reacting mixtures of monatomic and polyatomic gases

1.
Department of Mathematical, Physical and Computer Sciences, University of Parma, I-43121 Parma, Italy
2.
Sorbonne Université, CNRS, Université Paris Cité, Laboratoire Jacques-Louis Lions (LJLL), F-75005 Paris, France

^*Corresponding author: Maria Groppi
^*Corresponding author: Maria Groppi

Received: December 2022

Revised: May 2023

Early access: May 30, 2023

Published online: May 30, 2023

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Abstract

We propose a general kinetic framework for the description of a mixture of non-relativistic mono- or polyatomic gases undergoing any number of bimolecular reactions. The description of the internal structure of the molecules is kept general, in order to model it according to the physical properties of the considered species. The possibility of keeping separate rotational and vibrational energies of polyatomic particles is thus included in the present framework. Moreover, activation energy for chemical reactions is taken into account, allowing to accurately describe the kinetics of the chemical process. We prove the validity of the H-Theorem, recover the expected Maxwellian equilibrium distributions joint with the mass-action laws of chemical reactions, and obtain an explicit formula for the rate constant of a given chemical reaction. The system of Euler equations in the case of moderately slow chemical reactions is also derived.

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Mathematics Subject Classification: 82B40, 76P05.

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Figure 1. Reaction coordinate diagram

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Figure 2. Reaction coordinate diagram with reference energy thresholds

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Figure 3. Plot of $ \mathit{K}_{ij}^{kl} $ given in (81) and $ \mathit{K}_{Arr} $ as functions of $ \frac{k_B T}{E_a} $, with $ \beta = 1 $

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Table 1. Quantities related to internal energy modelling of a single species

	Fundamental energy of the species $ i $	$ \varepsilon^0_i = \mathrm{inf \, ess}_{\mu_i} \big\{\varepsilon_i \big\} $
	Grounded internal energy function of the species $ i $	$ \bar{\varepsilon}_i = \varepsilon_i - \varepsilon^0_i : \mathcal{E}_i \to \mathbb{R}_+ $
	Energy law of the species $ i $	$ \mu_i^{\bar{\varepsilon}_i} $ is the image measure of $ \mu_i $ by $ \bar{\varepsilon}_i $
	Partition function of the species $ i $	$ Z_i(\beta) = \int_{ \mathcal{E}_i} e^{-\beta \, \bar{\varepsilon}_i(\zeta)} \, \mathrm{d} \mu_i(\zeta) = \int_{ \mathbb{R}_+} e^{-\beta I} \, \mathrm{d} \mu_i^{\bar{\varepsilon}_i} (I) $
	Gibbs measure on the space of internal states of the species $ i $ at temperature $ T >0 $	$ \mathrm{d} \nu^i_T(\zeta) = Z \left( \frac{1}{k_B T} \right)^{-1} \, \exp \left(- \frac{\bar{\varepsilon}_i(\zeta)}{k_B T} \right) \, \mathrm{d} \mu_i(\zeta) $
	Gibbs measure on the space of internal energies of the species $ i $ at temperature $ T>0 $	$ \mathrm{d} \widetilde{\nu}^i_T(I) = Z \left( \frac{1}{k_B T} \right)^{-1} \, \exp \left(- \frac{I}{k_B T} \right) \, \mathrm{d} \mu_i^{\bar{\varepsilon}_i} (I) $
	Number of internal degrees of freedom of the species $ i $ at temperature $ T>0 $	$ \delta_i(T) = - \frac{2}{k_B T} \big( \log(Z_i) \big)' \left( \frac{1}{k_B T} \right) = 2 \, \mathbb{E}_{\nu^i_{T}} \left[ \frac{\bar{\varepsilon}_i}{k_B T} \right] $
	Heat capacity at constant volume of the species $ i $ at temperature $ T>0 $	$ \begin{array}{c} c_V^i(T) = \frac{3 + D_i(T)}{2}, \\D_i(T) = \frac{ \mathrm{d} (T \delta_i(T))}{ \mathrm{d} T} = \frac{2}{(k_B T)^2} \big( \log(Z_i) \big)'' \left( \frac{1}{k_B T} \right) = 2 \, \mathrm{Var}_{\nu^i_{T}} \left[ \frac{\bar{\varepsilon}_i}{k_B T} \right]\end{array} $

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Table 2. Fundamental quantities involved in interactions between different species

	Reduced and total mass of species $ i $ and $ j $	$ m_{ij}^* = \frac{m_i m_j}{m_i + m_j}, \qquad M_{ij} = m_i + m_j $
	Energy consumed by the chemical reaction $ \big\{ (i,j) \rightarrow (k,l) \big\} $	$ (\Delta \varepsilon^0)_{ij}^{kl} = \varepsilon^0_k + \varepsilon^0_l - \varepsilon^0_i - \varepsilon^0_j $
	Activation energy of $ \big\{ (i,j) \rightarrow (k,l) \big\} $	$ E_a = \kappa_{ij}^{kl} - \varepsilon^0_i - \varepsilon^0_j $
	Averaged collision kernel	$ \begin{array}{c} \bar{B}_{ij}^{kl}(g,\zeta,\zeta_) = \iint_{ \mathcal{E}_k \times \mathcal{E}_l} \int_{ \mathbb{S}^2} \widetilde{B}_{ij}^{kl}(g,\zeta,\zeta_,\zeta',\zeta'_,\omega) \, \mathrm{d} \omega \, \mathrm{d} \mu_k(\zeta') \, \mathrm{d} \mu_l(\zeta'_), \\ \widetilde{B}_{ij}^{kl}(v-v_,\zeta,\zeta_,\zeta',\zeta'_,\omega) = B_{ij}^{kl}(v,v_,\zeta,\zeta_,\zeta',\zeta'_,\omega) \end{array} $

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Table 3. Quantities related to mechanical or global collision equilibrium

	Relative kinetic Gibbs probability measure on $ \mathbb{R}^3 $, of the species $ i $ and $ j $ at temperature $ T>0 $	$ \mathrm{d} \lambda^{ij}_T(g) = \left( \frac{m_{ij}^}{2 \pi k_B T} \right)^{3/2} \exp \left(-\frac{m_{ij}^ \|g\|^2}{2 k_B T} \right) \, \mathrm{d} g $
	Rate constant of the chemical reaction $ \left\{(i,j) \to (k,l) \right\} $ at temperature $ T>0 $	$ \mathit{K}_{ij}^{kl}(T) = \mathbb{E}_{\lambda^{ij}_T \otimes \nu^i_T \otimes \nu^j_T} \left[\bar{B}_{ij}^{kl} \right] $
	Rate of the chemical reaction $ \left\{(i,j) \to (k,l) \right\} $ at temperature $ T>0 $ and with respective densities $ n_i,n_j,n_k $ and $ n_l $	$ r_{ij}^{kl} = \mathit{K}^{ij}_{kl}(T) \, n_k \, n_l - \mathit{K}_{ij}^{kl}(T) \, n_i \, n_j $
	Reaction rate of the species $ i $	$ r_i =\underset{\big\{ (i,j) \rightarrow (k,l) \big\} \in \Omega_{chem}}{ \quad \sum_{j,k,l} \quad r_{ij}^{kl}} $
	Equilibrium constant of the chemical reaction $ \left\{(i,j) \to (k,l) \right\} $ at temperature $ T>0 $, with $ \sigma_k = \sigma_l = - \sigma_i = - \sigma_j = 1 $	$ \begin{array}{ccl} \mathcal{K}^{kl}_{ij}(T) = \prod\limits_{p \in \{i,j,k,l\}} \left\{ \left( 2 \pi m_i k_B T \right)^{3 \sigma_p/2} \, Z_p \left( \frac{1}{k_B T} \right)^{\sigma_p} \right\} \\ \times \exp \left( - \frac{(\Delta \varepsilon^{0})_{ij}^{kl}}{k_B T} \right) \end{array} $

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References

[1]	P. Andries, K. Aoki and B. Perthame, A consistent BGK-type model for gas mixtures, J. Stat. Phys., 106 (2002), 993-1018. doi: 10.1023/A:1014033703134.
[2]	K. Aoki and V. Giovangigli, Kinetic theory of chemical reactions on crystal surfaces, Physica A, 565 (2021), 125573. doi: 10.1016/j.physa.2020.125573.
[3]	T. Arima, T. Ruggeri and M. Sugiyama, Rational extended thermodynamics of dense polyatomic gases incorporating molecular rotation and vibration, Philos. Trans. Royal Soc. A, 378 (2020), 20190176. doi: 10.1098/rsta.2019.0176.
[4]	P. W. Atkins and J. de Paula, Atkin's Physical Chemistry, 8^th edition, Oxford University Press, New York, 2006.
[5]	C. Baranger, M. Bisi, S. Brull and L. Desvillettes, On the Chapman-Enskog asymptotics for a mixture of monoatomic and polyatomic rarefied gases, Kinet. Relat. Models, 11 (2018), 821-858. doi: 10.3934/krm.2018033.
[6]	N. Bernhoff, Linearized Boltzmann collision operator: Ⅰ. Polyatomic molecules modeled by a discrete internal energy variable and multicomponent mixtures, Acta Appl. Math., 183 (2023), Paper No. 3, 45 pp. doi: 10.1007/s10440-022-00550-6.
[7]	N. Bernhoff, Linearized Boltzmann collision operator: Ⅱ. Polyatomic molecules modeled by a continuous internal energy variable, to appear, Kinet. Relat. Models.
[8]	M. Bisi, M. Groppi and G. Spiga, A kinetic model for bimolecular chemical reactions, Kinetic Methods for Nonconservative and Reacting Systems, Aracne Editrice, Naples, 16 (2005), 1-143.
[9]	M. Bisi and R. Travaglini, A BGK model for mixtures of monoatomic and polyatomic gases with discrete internal energy, Physica A, 547 (2020), 124441. doi: 10.1016/j.physa.2020.124441.
[10]	A. V. Bobylev, M. Bisi, M. Groppi, G. Spiga and I. F. Potapenko, A general consistent BGK model for gas mixtures, Kinet. Relat. Models, 11 (2018), 1377-1393. doi: 10.3934/krm.2018054.
[11]	C. Borgnakke and P. S. Larsen, Statistical collision model for Monte Carlo simulation of polyatomic gas mixture, J. Comput. Phys., 18 (1975), 405-420.
[12]	T. Borsoni, M. Bisi and M. Groppi, A general framework for the kinetic modelling of polyatomic gases, Commun. Math. Phys., 393 (2022), 215-266. doi: 10.1007/s00220-022-04367-0.
[13]	T. Borsoni, L. Boudin and F. Salvarani, Compactness property of the linearized Boltzmann operator for a polyatomic gas undergoing resonant collisions, J. Math. Anal. Appl., 517 (2023), 126579. doi: 10.1016/j.jmaa.2022.126579.
[14]	F. Bouchut, F. Golse and M. Pulvirenti, Kinetic Equations and Asymptotic Theory, Elsevier, Paris, 2000.
[15]	L. Boudin and F. Salvarani, Compactness of linearized kinetic operators, From particle systems to partial differential equations. III, Springer Proc. Math. Stat., Springer, 162 (2016), 73-97. doi: 10.1007/978-3-319-32144-8_4.
[16]	D. Bothe and W. Dreyer, Continuum thermodynamics of chemically reacting fluid mixtures, Acta Mech., 226 (2015), 1757-1805. doi: 10.1007/s00707-014-1275-1.
[17]	S. Brull, An ellipsoidal statistical model for a monoatomic and a polyatomic gas mixture, Commun. Math. Sci., 19 (2021), 2177-2194. doi: 10.4310/CMS.2021.v19.n8.a5.
[18]	S. Brull and J. Schneider, Derivation of a BGK model for reacting gas mixtures, Commun. Math. Sci., 12 (2014), 1199-1223. doi: 10.4310/CMS.2014.v12.n7.a2.
[19]	S. Brull, M. Shahine and P. Thieullen, Compactness property of the linearized Boltzmann operator for a diatomic single gas model, Netw. Heterog. Media, 17 (2022), 847-861. doi: 10.3934/nhm.2022029.
[20]	C. Cercignani, Rarefied Gas Dynamics: From Basic Concepts to Actual Calculations, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2000.
[21]	F. Conforto, R. Monaco and M. Pandolfi Bianchi, Boltzmann–type equations for chain reactions modeling, Proceedings of Waves and Stability in Continuous Media, World Scientific, Singapore, (2008), 168-176.
[22]	Y. Dauvois, J. Mathiaud and L. Mieussens, An ES-BGK model for polyatomic gases in rotational and vibrational nonequilibrium, Europ. J. Mech. B Fluids, 88 (2021), 1-16. doi: 10.1016/j.euromechflu.2021.02.006.
[23]	E. De Angelis and C. P. Grunfeld, Modeling and analytic problems for a generalized Boltzmann equation for a multicomponent reacting gas, Nonlinear Anal. Real World Appl., 4 (2003), 189-202. doi: 10.1016/S1468-1218(02)00022-6.
[24]	L. Desvillettes, Sur un modèle de type Borgnakke-Larsen conduisant à des lois d'énergie non linéaires en température pour les gaz parfaits polyatomiques, Ann. Fac. Sci. Toulouse Math., 6 (1997), 257-262. doi: 10.5802/afst.864.
[25]	L. Desvillettes, R. Monaco and F. Salvarani, A kinetic model allowing to obtain the energy law of polytropic gases in the presence of chemical reactions, Europ. J. Mech. B Fluids, 24 (2005), 219-236. doi: 10.1016/j.euromechflu.2004.07.004.
[26]	V. Giovangigli, Multicomponent Flow Modeling, Birkhäuser, Boston, 1999. doi: 10.1007/978-1-4612-1580-6.
[27]	F. Golse, Fluid dynamic limits of the kinetic theory of gases, From Particle Systems to Partial Differential Equations, Springer Proceedings in Mathematics & Statistics, 75 (2014), 3-91. doi: 10.1007/978-3-642-54271-8_1.
[28]	P. Griehsnig, F. Schürrer and G. Kügerl, Kinetic theory for particles with internal degrees of freedom, Rarefied Gas Dynamics - Theory and Simulations, Progress in Astronautics and Aeronautics AIAA, 159 (1994), 581-589.
[29]	M. Groppi, Different collision-dominated regimes for chemically reacting gases, Progress in Industrial Mathematics at ECMI 2000, Springer Series Mathematics in Industry, (2002), 554-559.
[30]	M. Groppi and J. Polewczak, On two kinetic models for chemical reactions: Comparisons and existence results, J. Stat. Phys., 117 (2004), 211-241. doi: 10.1023/B:JOSS.0000044059.59066.a9.
[31]	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, 33 (2000), 8819-8833. doi: 10.1088/0305-4470/33/48/317.
[32]	M. Groppi and G. Spiga, Kinetic approach to chemical reactions and inelastic transitions in a rarefied gas, J. Math. Chem., 26 (1999), 197-219.
[33]	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.
[34]	W. J. Hehre, A Guide to Molecular Mechanics and Quantum Chemical Calculations, Wavefunction Irvine, CA, 2003.
[35]	G. Herzberg, Molecular Spectra and Molecular Structure, Van Nostrand Reinold, New York, 1950.
[36]	M. N. Kogan, Rarefied Gas Dynamics, Plenum Press, New York, 1969.
[37]	G. M. Kremer, M. Pandolfi 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.
[38]	J. A. McLennan, Boltzmann equation for a dissociating gas, J. Stat. Phys., 57 (1989), 887-905.
[39]	R. Monaco, M. Pandolfi Bianchi and A. J. Soares, BGK-type models in strong reaction and kinetic chemical equilibrium regimes, J. Phys. A: Math. Gen., 38 (2005), 10413-10431. doi: 10.1088/0305-4470/38/48/012.
[40]	J. Polewczak and A. J. Soares, On modified simple reacting spheres kinetic model for chemically reactive gases, Kinet. Relat. Models, 10 (2017), 513-539. doi: 10.3934/krm.2017020.
[41]	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.
[42]	D. Shear, An analog of the Boltzmann H–theorem (a Liapunov function) for systems of coupled chemical reactions, J. Theor. Biol., 16 (1967), 212-228.