Binary Mixture Of Ideal Gases Research Articles

Equilibrium and stability properties of detonation waves in the hydrodynamic limit of a kinetic model

A shock wave structure problem, like 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 . 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 balance laws. With respect to other approaches known in the kinetic literature for detonation problems with a reversible reaction, this paper aims 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. Finally, 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 model 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.

Shock structure and temperature overshoot in macroscopic multi-temperature model of mixtures

The paper discusses the shock structure in macroscopic multi-temperature model of gaseous mixtures, recently established within the framework of extended thermodynamics. The study is restricted to weak and moderate shocks in a binary mixture of ideal gases with negligible viscosity and heat conductivity. The model predicts the existence of temperature overshoot of heavier constituent, like more sophisticated approaches, but also puts in evidence its non-monotonic behavior not documented in other studies. This phenomenon is explained as a consequence of weak energy exchange between the constituents, either due to large mass difference, or large rarefaction of the mixture. In the range of small Mach number it is also shown that shock thickness (or equivalently, the inverse of Knudsen number) decreases with the increase of Mach number, as well as when the mixture tends to behave like a single-component gas (small mass difference and/or presence of one constituent in traces).

Frame dependence of stationary heat transfer in an inert mixture of ideal gases

This paper is devoted to the study of a stationary heat transfer problem in an inert binary mixture of ideal gases. The mixture is enclosed between two coaxial cylinders, kept at two different constant temperatures and at rest in a non-inertial frame. In particular, the frame is assumed to rotate with a constant angular velocity around the cylinder axes. The phenomenon is described by the linearized equations of extended thermodynamics with 13 moments. Integration of the field equations requires the assignment of non-controllable boundary values, which are determined through a fluctuation principle. It turns out that the temperature field presents boundary layers. Moreover, in agreement with the predictions of the kinetic theory, the tangential components of the heat and of the diffusion fluxes cannot vanish. We also show that the mixture and its constituents cannot be at rest in the rotating frame. An analysis is proposed concerning the combined effects of thermo-diffusion and rotation in the separation of mixture components. A comparison between classical and extended thermodynamics predictions is also proposed.

Governing equations in non-isothermal diffusion

Generalizations of Fick's law for the diffusion flux are often considered in the literature by analogy with those for the heat flux. The paper reviews the balance equations for a fluid mixture and provides the equations for the diffusion fluxes. As a consequence, the mass densities are shown to satisfy a system of hyperbolic equations. Moreover, for a binary mixture of ideal gases in stationary conditions, Fick's law is recovered. Next, diffusion fluxes are regarded as constitutive functions and a whole set of thermodynamic restrictions are determined which account for diffusion, heat conduction, viscosity and inhomogeneities. Hyperbolic models for diffusion and heat fluxes are established which involve the co-rotational derivative. The driving term of diffusion turns out to be the gradient of chemical potential rescaled by the temperature.

On inelastic reactive collisions in kinetic theory of chemically reacting gas mixtures

A kinetic theory for a simple reversible reaction–characterized by a binary mixture of ideal gases whose constituents denoted by A and B undergo a reaction of the type A + A ⇌ B + B –is developed by considering the reactive collisions as inelastic ones. The geometry of the collision is taken into account in the line-of-centers differential cross section by allowing that a chemical reaction may occur only when the energy of the relative velocity in the direction of the line which joins the centers of the molecules at collision is larger than the activation energy. It is shown that the restitution coefficients: (i) depend explicitly on the reaction heat and on the relative translational energy in the direction of the line which joins the centers of the molecules during an inelastic collision; (ii) vanish when the reaction heat is zero; (iii) are larger or smaller than one depending on the direction of the reaction and on the sign of the reaction heat. First approximations to the distribution functions are determined from the system of Boltzmann equations for the last stage of a chemical reaction. It is shown that the deviations from the Maxwellian distribution functions and the production terms of the particle number densities: (i) vanish when the reaction heat is zero provided that the affinity is close to zero and (ii) are negative or positive depending on the sign of the reaction heat and on the direction of the reaction.

On the heat and mass transfer analogy for natural convection of non-dilute binary mixtures of ideal gases in cavities

An extension of a slightly compressible flow model to double-diffusive convection of a binary mixture of ideal gases enclosed in a cavity is presented. The problem formulation is based on a low Mach number approximation and the impermeable surface assumption is not invoked. The main objectives of this paper are the statement of a new problem formulation, and the analysis of some significant results showing the influence of density variations on transient solutions for pure thermal or pure solutal convection. At steady-state, it is shown that the heat and mass transfer analogy may be applied for non-dilute mixtures at parameter ranges larger than those usually considered. To cite this article: H. Sun, G. Lauriat, C. R. Mecanique 337 (2009).

Temperature in ideal gas mixtures in Couette flow: A maximum-entropy approach

A binary mixture of ideal gases in Couette flow is studied in the framework of information theory. It is shown that if the total shear viscous pressure is imposed as a constraint, there is equipartition of energy between the different chemical species but not amongst the different spatial directions. In contrast, if a same shear rate is imposed on both species, there is no equipartition between species neither amongst spatial directions. In both situations, the thermodynamic temperature, defined from the Lagrange multiplier conjugated to the internal energy, is the same for both species, independently of the equality or not of their local-equilibrium temperature.

Quasi-steady diffusion-controlled droplet evaporation and condensation

This theoretical investigation deals with the vaporization or condensation rate of a motionless liquid droplet using the quasi-steady diffusion-controlled model. A single-component liquid droplet is considered to be undergoing a phase change within a binary mixture of ideal gases (vapor plus noncondensable gas). Droplet vaporization rates corresponding to specified ambient conditions have been calculated by numerical solution of the variable-property governing equations. Results are presented for water and a series of pure hydrocarbon liquids for a range of ambient conditions of interest. A dimensionless correlation is given for the hydrocarbon vaporization and condensation rates. The pressure variation in the region surrounding a droplet undergoing vaporization or condensation has been investigated by numerical integration of the momentum equation. The resulting calculations indicate that the pressure decreases with increasing distance from the droplet for condensation as well as vaporization. Finally, criteria are given for estimating when the pressure gradient and viscous dissipation may be of significance in the energy equation.

Acoustic wave propagation in a binary mixture of inviscid fluids

Linearized equations governing the thermo-mechanical behaviour of a binary mixture of inviscid fluids are derived. Restrictions which are sufficient for the equations to have a unique solution are imposed on some of the material constants. The propagation of plane harmonic waves of small amplitude in the mixture is examined and the inequalities are shown to ensure a physically reasonable response. As an application of the theory properties of acoustic waves in a binary mixture of ideal gases are evaluated numerically.