Giampiero Spiga

A note on the kinetic theory of chemically reacting gases

A kinetic approach to a mixture of gases whose particles interact by elastic scattering and by a bimolecular chemical reaction is analyzed by means of the relevant Boltzmann-like equation. Conservation laws and moment equations are considered, and, improving previous results in the literature, equilibria under collision are studied in detail, by providing in particular a strict stability result, which shows relaxation starting from any physically meaningful initial condition.

Kinetic approach to chemical reactions and inelastic transitions in a rarefied gas

A kinetic approach is presented for the analysis of a gas mixture with two kinds of nonconservative interactions. In a bimolecular chemical reaction, mass transfer and energy of chemical link arise, and in inelastic mechanical encounters, molecules get excited or de‐excited due to their quantized structure. Molecules undergo transitions between energy levels also by absorption and emission of photons of the self‐consistent radiation field. From the kinetic Boltzmann‐type equations, the problem of equilibria and of their stability is addressed. A detailed balance principle is proved and a Lyapunov functional is constructed; mass action law and Plancks law of radiation are recovered.

Step response of the crossflow heat exchanger with finite wall capacitance

Abstract The step response of a single-pass crossflow heat exchanger is investigated. The two-dimensional transient temperature distributions in the core wall and in both the unmixed fluids are analytically determined, considering finite wall heat capacity (i.e. finite propagation speed of disturbances). The solutions are deduced by the local energy balance equations, resorting to the Green functions and to a threefold Laplace transform. The dimensionless temperatures are presented in terms of the governing parameters (number of transfer units, capacity ratios, heat transfer resistance ratio). Assuming constant initial conditions and entrance temperature of the hot fluid subjected to a sudden step increase, the transient temperature field is computed with extremely low computational time; the results are plotted for a wide range of the governing parameters.

A Bhatnagar-Gross-Krook-type approach for chemically reacting gas mixtures

A recently proposed consistent approach for elastically scattering gas mixtures, of the type introduced by Bhatnagar, Gross, and Krook (BGK), has been extended to deal with a four species gas undergoing reversible bimolecular chemical reactions. The single BGK collision operator introduced for each species must take into account also transfer of mass and of energy of chemical bond. Suitable auxiliary fields have then to be introduced not only for temperatures and velocities, but also for densities, in order to fulfill correctly balance equations for mass, momentum, and total energy. The exact collision equilibrium, satisfying the mass action law of chemistry, is also recovered, and the proper choice of collision frequencies is discussed. Preliminary numerical results for the relaxation problem in space-homogeneous conditions are reported and briefly commented on.

Grad’s equations and hydrodynamics for weakly inelastic granular flows

We introduce and discuss Grad’s moment equations for dilute granular systems of hard spheres with dissipative collisions and variable coefficient of restitution, under the assumption of weak inelasticity. An important by-product is that in this way we obtain the hydrodynamic description of a system of nearly elastic particles by a direct procedure from the Boltzmann equation, without resorting to any homogeneous cooling state assumption. Several crucial results of the pertinent literature are recovered in the present physical context in which deviation from elastic scattering is of the same order as the Knudsen number. In particular, the correlation function plays a fundamental role in the decay of the temperature, and the latter is described asymptotically, in space homogeneous conditions, by a corrected Haff’s law.

A generalized BKW solution of the nonlinear Boltzmann equation with removal

The nonlinear Boltzmann equation for Maxwell molecules is considered in the case when the collision between two particles may lead not only to elastic scattering, but also to a removal event, with the disappearance of both incident molecules. An exact particular solution is found, which generalizes the well‐known BKW mode, valid in the conservative case with no removal.

The dissipative linear Boltzmann equation

We introduce and discuss a linear Boltzmann equation describing dissipative interactions of a gas of test particles with a fixed background. For a pseudo-Maxwellian collision kernel, it is shown that, if the initial distribution has finite temperature, the solution converges exponentially for large time to a Maxwellian profile drifting at the same velocity as field particles and with a universal nonzero temperature which is lower than the given background temperature.

Inelastic scattering models in transport theory and their small mean free path analysis

In this paper we perform an asymptotic analysis of a singularly perturbed linear Boltzmann equation with inelastic scattering operator in the Lorentz gas limit, when the parameter corresponding to the mean free path of particles is small. The physical model allows for two-level field particles (ground state and excited state), so that scattering test particles trigger either excitation or de-excitation processes, with corresponding loss or gain of kinetic energy. After examining the main properties of the collision mechanism, the compressed Chapman–Enskog expansion procedure is applied to find the asymptotic equation when the collisions are dominant. A peculiarity of this inelastic process is that the collision operator has an infinite dimensional null-space. On the hydrodynamic level this is reflected in the small mean free path approximation being rather a family of diffusion equations than a single equation, as is the case in classical transport theory. Also the appropriate hydrodynamic quantity turns out to be different from the standard macroscopic density. Copyright

On the equivalence between the probabilistic, kinetic, and scattering kernel formulations of the Boltzmann equation

Abstract We prove the equivalence between the probabilistic, kinetic, and scattering kernel formulations of the Boltzmann equation, and we discuss in more detail the features of the scattering kernel formulation in both the deterministic and the stochastic versions.

Shock structure analysis in chemically reacting gas mixtures by a relaxation-time kinetic model

The shock structure in a gas mixture undergoing a bimolecular chemical reaction is studied by means of a reactive kinetic relaxation model. The relevant nonlinear integrodifferential equations are numerically solved in one space dimension with upstream and downstream asymptotic equilibrium conditions satisfying the reactive Rankine–Hugoniot relations and entropy condition. Numerical results are presented, emphasizing the role of Mach number, upstream concentration fractions, and change in the chemical composition across the shock.