Boltzmann equation, multicomponent

The derivation of the mixture-balance laws has been given by Chapman and Cowling for a binary mixture. Its generalization to multicomponent mixtures, as in Equation 5-1, uses a determination of the invariance of the Boltzmann equation. This development has been detailed by Hirschfelderet These derivations were summarized in the notes of Theodore von Karmin s Sorbonne lectures given in 1951-1952, and the results of his summaries were stated in Pinner s monograph. For turbulent flow, the species-balance equation can be represented in the Boussinesq approximation as ... 

Following Kerkhof and Geboers (2005), an approximation of the Boltzmann equation for a multicomponent monatomic... 

The equations for conservation of mass, momentum, and energy for a one-component continuum are well known and are derived in standard treatises on fluid mechanics [l]-[3]. On the other hand, the conservation equations for reacting, multicomponent gas mixtures are generally obtained as the equations of change for the summational invariants arising in the solution of the Boltzmann equation (see Appendix D and [4] and [5]), One of several exceptions to the last statement is the analysis of von Karman [6], whose results are quoted in [7] and are extended in a more recent publication [8] to a point where the equivalence of the continuum-theory and kinetic-theory results becomes apparent [9]. This appendix is based on material in [8]. 

Muckenfuss, C., Stefan-Maxwell Relations for Multicomponent Diffusion and the Chapman Enskog Solution of the Boltzmann Equations, J. Chem. Phys., 59, 1747-1752 (1973). 

The rigorous Fickian multicomponent mass diffusion flux formulation is derived from kinetic theory of dilute gases adopting the Enskog solution of the Boltzmann equation (e.g., [17] [18] [19] [89] [5]). This mass flux is defined by the relation given in the last line of (2.281) ... 

It is required to express either pair of molecular velocities before and after the interaction in terms of the other pair, and of two independent geometrical variables b and (j>) in order to complete the specification of the encounter. The original Boltzmann equation derivation considers elastic collisions in free space between two spin-less molecules of equal mass. However, due to the major interest in multicomponent mixtures, the theory outline consider elastic collisions between two spin-less mono-atomic molecules in an ideal gas mixture. The theory may be useful even if the molecules are not mono-atomic, provided that their states of internal motion (i.e., rotation and vibration) are not affected by the collisions. The two molecules under consideration are treated as point particles with respective masses m and m2. In the laboratory frame, the incoming molecule positions are denoted by ri and r2, and the particle velocities are indicated by ci and C2. The corresponding positions and velocities after the encounter are r j, and c, c, respectively. The classical trajectories for two interacting molecules presented in the laboratory system frame are viewed in Fig. 2.1. It is supposed that the particle interaction is determined by conservative potential interaction forces only. Any external forces which might act on the molecules are considered negligible compared to the potential forces involved locally in the collision. The relative position vectors in the laboratory frame are defined by ... 

The Boltzmann factor in the denominator of this equation corresponds to coupling a distinguished molecule of component a to the solution. This result is reminiscent of local composition free energy models that are widely used to calculate fluid-phase equilibria for multicomponent mixtures of nonelectrolytes. We note that > 1 corresponds to less favorable interactions in the mixtures, and 1 as 1. 

Multicomponent diffusion in pores is described by the dusty-gas model (DGM) [38,44,46 8]. This model combines molecular diffusion, Knudsen diffusion, viscous flux, and surface diffusion. The DGM is suitable for any model of porous structure. It was developed by Mason et al. [42] and is based on the Maxwell-Stefan approach for dilute gases, itself an approximation of Boltzmann s equation. The diffusion model obtained is called the generalized Maxwell-Stefan model (GMS). Thermal diffusion, pressmn diffusion, and forced diffusion are all easily included in the GMS model. This model is based on the principle that in order to cause relative motion between individual species in a mixture, a driving force has to be exerted on each of the individual species. The driving force exerted on any particular species i is balanced by the friction this species experiences with all other species present in the mixture. Each of these friction contributions is considered to be proportional to the corresponding differences in the diffusion velocities. 

The lattice Boltzmann method is a mesoscopic simulation method for complex fluid systems. The fluid is modeled as fictitious particles, and they propagate and coUide over a discrete lattice domain at discrete time steps. Macroscopic continuum equations can be obtained from this propagation-colhsion dynamics through a mathematical analysis. The particulate nature and local d3mamics also provide advantages for complex boundaries, multiphase/multicomponent flows, and parallel computation. 

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