A gaseous mixture of four constituents undergoing a reversible bimolecular reaction is modeled by means of a Bhatnagar, Gross, and Krook ͑BGK͒-type equation in a flow regime close to chemical equilibrium. In the proposed relaxation method, elastic and chemistry collision terms are approximated separately, introducing different reference distribution functions which assure the correct balance laws. A Chapman-Enskog procedure is applied in order to provide explicitly the transport coefficients of diffusion, shear viscosity and thermal conductivity in dependence on elastic and reactive collision frequencies, mass concentrations of each species and temperature of the whole mixture. The closure of the balance equations is performed at the Navier-Stokes level and plane wave solutions are characterized. For the ͑H 2 , Cl, HCl, H͒ system, transport coefficients, as well as the Prandtl number of the mixture, are represented as functions of the temperature and compared with the inert case in order to discuss the influence of chemical reaction. Moreover, the thermal conductivity for nondiffusive and homogeneous mixtures are compared. For the problem of longitudinal wave propagation the phase velocity, attenuation coefficient and affinity are analyzed as functions of the wave frequency.
A shock wave structure problem, as the one which can be formulated for the planar detonation wave, is analyzed here for a binary mixture of ideal gases undergoing the symmetric reaction A 1 + A 1 A 2 + A 2 . The problem is studied at the hydrodynamic Euler limit of a kinetic model of the reactive Boltzmann equation. The chemical rate law is deduced in this frame with a second-order reaction rate, in a chemical regime such that the gas flow is not far away from the chemical equilibrium. The caloric and the thermal equations of state for the specific internal energy and temperature are employed to close the system of the balance laws. With respect to other approaches known in the kinetic literature for detonation problems with a reversible reaction, this paper is addressed to improve some aspects of the wave solution. Within the mathematical analysis of the detonation model, the equation of the equilibrium Hugoniot curve of the final states is explicitly derived for the first time and used to define the correct location of the equilibrium Chapman-Jouguet point in the Hugoniot diagram. The parametric space is widened to investigate the response of the detonation solution to the activation energy of the chemical reaction. At last, the mathematical formulation of the linear stability problem is given for the wave detonation structure via a normal-mode approach, when bidimensional disturbances perturb the steady solution. The stability equations with their boundary conditions and the radiation condition of the considered modeling are explicitly derived for small transversal deviations of the shock wave location. The paper shows how a second-order chemical kinetics description, derived at the microscopic level, and an analytic deduction of the equilibrium Hugoniot curve, lead to an accurate picture of the steady detonation with reversible reaction, as well as to a proper bidimensional linear stability analysis.
In this present paper, a quaternary gaseous reactive mixture, for which the chemical reaction is close to its final stage and the elastic and reactive frequencies are comparable, is modelled within the Boltzmann equation extended to reacting gases. The main objective is a detailed analysis of the non-equilibrium effects arising in the reactive system A1 + A2 ⇌ A3 + A4, in a flow regime which is considered not far away from thermal, mechanical and chemical equilibrium. A first-order perturbation solution technique is applied to the macroscopic field equations for the spatially homogeneous gas system, and the trend to equilibrium is studied in detail. Adopting elastic hard-spheres and reactive line-of-centres cross sections and an appropriate choice of the input distribution functions—which allows us to distinguish the two cases where the constituents are either at same or different temperatures—explicit computations of the linearized production terms for mass, momentum and total energy are performed for each gas species. The departures from the equilibrium states of densities, temperatures and diffusion fluxes are characterized by small perturbations of their corresponding equilibrium values. For the hydrogen–chlorine system, the perturbations are plotted as functions of time for both cases where the species are either at the same or different temperatures. Moreover, the trend to equilibrium of the reaction rates is represented for the forward and backward reaction H2 + Cl ⇌ HCl + H.
We propose a new discrete velocity model of the Boltzmann equation for a mixture of four gases admitting particle elastic collisions and bi-molecular chemical reactions. We first prove an H-theorem and determine the thermodynamical equilibrium state. A Chapman-Enskog expansion on the kinetic equations is then performed, deriving both the Euler and the Navier-Stokes equations of the model. Finally the transport coefficients of diffusivity and viscosity are provided as well.
The analysis of linear stability of a steady detonation wave is formulated for the first time at the kinetic level in the frame of the Boltzmann equation extended to reacting gases. Within this context and for a reversible reaction, the stability problem is carried out, in agreement with most classical papers on gas detonation, through a normal mode approach for the one-dimensional disturbances of the steady wave solution, and an acoustic radiation condition at the final equilibrium as closure condition. The proposed modelling leads to an initial value problem, constituted by the linearized reactive Euler equations in the perturbed shock frame with related Rankine-Hugoniot conditions, which can be solved by means of a proper numerical technique. An application is provided for an elementary bimolecular reaction.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.