A reactive BGK-type model: influence of elastic collisions and chemical interactions

A BGK-type model for a reactive multicomponent gas undergoing chemical bimolecular reactions is here presented. The mathematical and physical consistency of the model is stated in detail. The relaxation process towards local Maxwellians, depending on mass and numerical densities of each species, as well as on common mean velocity and temperature, is investigated with respect to chemical equilibrium. Such a trend is numerically tested within the hydrogen-air reaction mechanism. INTRODUCTION A large piece of research works has been addressed to simplified kinetic models of the Boltzmann equation, stimulated by the mathematical complexity of the true collision operator. A wide literature evidentiates the relevance of BGK-type models and their reliability also for computing gas transport properties far from equilibrium, assuming relaxation of the distribution function towards either a local Maxwellian or an anisotropic Gaussian [1, 2]. Extensions of BGK-type models to multicomponent systems can be found in [3], and more recently in [4] where a model satisfying the main properties of the true Boltzmann collision operator is presented. On this line, it seems to be a new interesting topic to deal with a BGK approximation of the extended Boltzmann equation for chemically reacting gases. The general structure of chemistry source terms, conservation and equilibrium properties in kinetic equations have been widely focused in [5]. More in detail considering bimolecular reactions, the exact kinetic equations which will be referred to in this work are those derived in paper [6]. A first attempt to build a BGK-type model for a mixture of four gas species with bimolecular chemical reaction of type A1 A2 A3 A4 (1) has been performed in paper [7], where the kinetic equations have been written in the form ∂ fi ∂ t v ∇ fi νi fi v fi v Ri f f v i 1 4 (2) In Equation (2) fi denotes the local Maxwellian distribution of species i depending on the number densities n i of each i-species, common mean velocity u and temperature T , i. e. fi v ni mi 2πkBT 3 2 exp mi v u 2 2kBT (3) The term Ri approximates the true reactive operator R i, introduced in Ref. [6], which includes the effects of the inelastic chemical process. In Ri the chemical gain term involves mechanical equilibrium and chemical disequilibrium in such a way that the model verifies the indifferentiability principle and conservation of mass, momentum and total energy (kinetic plus internal chemical bond energy). Moreover the H theorem holds true under a suitable hypothesis. In the last part of the paper the behavior of the model is numerically tested with respect to its trend to equilibrium for different initial conditions, in order to show the influence of elastic collisions and reactive interactions. The numerical experiments are performed for the elementary reaction occurring in the Hydrogen-Air reaction mechanism, namely H2O H OH H2 which is typical in Hydrogen combustion applications. KINETIC MODEL With reference to Eq. (2), the microscopic state of a mixture of four gas species, say A i , i 1 4, is defined by the one–particle distribution function f i fi t x v t IR x IR3 v IR3 for each species Ai with molecular mass mi and internal energy Ei such that m1 m2 m3 m4. In addition, setting ∆E E3 E4 E1 E2 0 the forward reaction turns out to be the endothermic one. The time and space dependence, if useless, will be omitted in the sequel. Exact equations. The exact kinetic equations for the reactive gas mixture are given by ∂ fi ∂ t v ∇ fi Ji f v Ri f v f f1 f4 (4) Ji f v Gi f v Li f v Ri f v Gi f v Li f v (5) The gain and loss terms Gi, Li due to elastic collisions, and Gi, Li due to chemical reactions, are not reported here for brevity, but can be recovered in paper [6]. They satisfy the following properties IR Ji f v dv 0 i 1 4 (6) IR R1 f v dv IR R2 f v dv IR R3 f v dv IR R4 f v dv (7) The former is typical of the elastic collision operator and accounts for conservation of the particle numbers of each species, the latter is due to the fact that the evolution of each species is predicted by chemical exchanges according to the bimolecular reaction (1). For what concerns mechanical equilibrium, each J i f v vanishes when f is a Maxwellian given by (3). Conversely, for what concerns chemical equilibrium, each R i f v vanishes if the distribution functions are Maxwellian and, in addition, the following condition holds m3m4 3 f1 v f2 w m1m2 3 f3 v1 f4 w1 (8) In Eq. (8), v, w and v1, w1 are the pre and post-collisional velocities, respectively. The post-collisional velocities, Ω being the unit vector of the relative post collisional velocity, are given by v1 r1v r2w r4VΩ w1 r1v r2w r3VΩ ri mi m1 m2 V w v 2 μ 2∆E r1m2 μ m3m4 m1m2 (9) Condition (8), as it can be easily seen, implies the mass–action–law of chemical equilibrium n1n2 n3n4 μ 3 2 exp ∆E kBT (10) Approximated equations. Following the procedure adopted in paper [7], the BGK-type approximation of the exact equations (4) consists in inserting in both elastic and inelastic gain terms Maxwellian distributions with parameters n i which do not satisfy condition (10) and, thus, do not imply chemical equilibrium. Such a procedure is justified since, in general [8], the relaxation time of elastic collision is of some order of magnitude smaller than the one of chemical interactions. The BGK equations then read ∂ fi ∂ t v ∇ fi Ji f f v Ri f f v i 1 4 (11) where Ji f f v and Ri f f v approximate the true collision operators Ji and Ri, according to the above said conjecture and assuming that the distribution functions satisfy conditions IR φi v fi v dv IR φi v fi v dv (12) with φi alternatively equal to mi, miv or 2 miv 2 Ei . Conditions (12) imply that the distributions f i and fi , which possess the same moments n ρu T (ρ being the total density), will be different only for what deals with the computation of higher moments. The explicit expressions of Ji and Ri will be here reported from paper [7], avoiding the details of their deduction. Adopting constant cross sections of Maxwell molecules, α i j , the BGK-type approximation of the elastic collision gain and loss terms, leads to the expression Ji f f v νi fi v fi v νi 4π 4 ∑ j 1 αi jn j (13) Conversely, the inelastic collision gain and loss terms have the expressions R1 f f v γ T n4 f3 η T n2 f1 R2 f f v γ T n3 f4 η T n1 f2 R3 f f v η T n2 f1 γ T n4 f3 R4 f f v η T n1 f2 γ T n3 f4 (14) where γ T 4πβ θ S T kBT η T γ T μ 3 2 exp ∆E kBT θ m3m4 m1 m2 S T ξ π 2πkBT θ exp θ ξ 2 2kBT kBT θ ξ 2 1 er f θ 2kBT ξ (15) ξ being the exothermic threshold velocity, β a scalar factor and k B the Boltzmann constant. Expressions (14) have been derived using the proper cross sections, related to exothermic and endothermic reactions, proposed in paper [9]. CONSISTENCY OF THE MODEL On the line of standard procedures and notations of classical kinetic theory [10], and by using assumption (12), the following properties can be proven. Property 1 The approximated elastic collision term Ji is such that IR 3 Ji f f v dv 4π 4 ∑ j 1 αi jn j IR 3 fi v fi v dv 0 i 1 4 (16) This property means that elastic collisions only, when modeled by the BGK equations (2), do not modify the species of the incoming particles. In fact, from condition (16), it results IR ∂ fi ∂ t v ∇ fi elast dv 0 dni dt elast 0 (17) Property 2 The approximated reactive collision terms Ri, i 1 4, are such that IR 3 R1 f f v dv IR 3 R2 f f v dv IR 3 R3 f f v dv IR 3 R4 f f v dv (18) This property implies dn1 dt inelast dn2 dt inelast dn3 dt inelast dn4 dt inelast (19) which agrees with the fact that through inelastic collisions with chemical reaction (1), if one particle of A 1 species is lost then also an A2 particle vanishes with creation of two particles of A3 and A4 species, and viceversa. Moreover, Eqs. (19) assures that the BGK approximated model reproduces the laws of chemical kinetics. Conservation laws. Conservation of mass, momentum and total energy can be stated through the following Property 3 For every choice of collision invariants φ i v , namely mi, miv or 2 miv 2 Ei , the BGK-type collision operator verifies the following equality ∑ i IR 3 Ji f f v Ri f f v φi v dv 0 (20) Entropy inequality. Elastic collisions and chemical reactions contribute to increase the entropy of the system, according to the next Property 4 Let H and H be the functionals defined by H x t ∑ i IR 3 fi log fi mi dv H x t ∑ i IR 3 fi log fi mi v dv (21) Then ∂H ∂ t x t divH x t 0 (22) provided that 4 ∑ i 2 IR 3 log fi fi Ri f f v dv 0 (23) Moreover one has ∂H ∂ t x t divH x t 0 if and only if fi fi and m3m4 3 f1 v f2 w m1m2 3 f3 v1 f4 w1 (24) The constraint (23) is purely mathematical. Nevertheless in the numerical experiments of the next Section this constraint has been checked finding a relative error whose order is 10 4 at most. It obviously converges to zero approaching to equilibrium. Indifferentiability principle. When all gas species are assumed to have same mass m and frequency ν , it is straightforward to show that the total distribution f ∑i fi verifies the single species BGK equation. NUMERICAL EXPERIMENTS In this section some numerical tests for the proposed model, in the spatial homogeneous case and for the Hydrogen-Air reversible reaction, are presented in order to evaluate the trend to thermodynamical equilibrium and the influence of elastic collisions towards inelastic interactions. With reference to Fig.1, non-symmetric bimodal distributions are assumed as initial data for f 1 f4. The corresponding macroscopic quantities (in mole /l for number densities and Kelvin degrees for temperature) are n1 0 0375 n2 0 0075 n3 0 225 n4 0 3375 u 0 T 1200 We set αi j α 1 and β 15; for such values the ratio of the elastic and inelastic collision frequencies, that is ω T 4πα γ T , ranges between 50 and 75. In Figs. 1a-1d the distributions f1 and f4 ( f2 and f3 behave analogously) are plotted versus v at different successive times. Since the reaction is prevalently exothermic f 1 and f4 correspond, respectively, to distribution of product and reactant of the chemical process. It can be observed that, due to the assumptions on the BGK-type mechanism of collisions, the product distribution assumes a “Maxwellian” shape rather quickly, whereas the reactant distribution converges slower to such a shape. In the last picture d both product and reactant have reached the equilibrium configuration which prescribes, with respect to initial data, a loss of the reactant H2 and a gain of the product H2O.