The Kinetic Theory of Dense Gases

Our discussion of transport phenomena in gases has been based on the Boltzmann equation and is therefore restricted in application to dilute gases where only binary collisions need to be taken into account. To describe transport processes in dense gases we must generalize the Boltzmann equation to higher order in the density so as to take into account dynamical processes that involve three, four, or more particles. 

The first important attempt to extend the Boltzmann equation to higher densities was made by Enskog in 1922. He modified the Boltzmann equation by incorporating into the collision integral the difference in position 

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Choh, S., and Uhlenbeck, G., The Kinetic Theory of Dense Gases, University of Michigan, 1958. 

Further development of the kinetic theory of dense gases and liquids. 

For a dense system of hard, smooth, and elastic spherical particles, a transport theorem based on the analogy of the kinetic theory of dense gases [Reif, 1965] may be derived. Define an ensemble average of any property xjr of a particle as... 

The kinetic theory of dense gases began with the work of Enskog, who in 1922, generalized Boltzmann s derivation of the transport equation to apply it to a dense gas of hard spheres. Enskog showed that for dense gases there is a mechanism for the transport of momentum and energy by means of the intermolecular potential, which is not taken into account by the Boltzmann equation at low densities, and he derived expressions for the transport coeffi-... 

Yet even this systematic theory is not free of difficulties, and their resolution remains one of the central problems in the kinetic theory of dense gases. ... 

Problems Connected with the Kinetic Theory of Dense Gases... 

All of the problems discussed above in connection with the Boltzmann equation carry over to the kinetic theory of dense gases. In addition to the obvious analogs of the problems mentioned above, the following problems need to be solved. 

The equations of motion for granular flows have been derived by adopting the kinetic theory of dense gases. This approach involves a statistical-mechanical treatment of transport phenomena rather than the kinematic treatment more commonly employed to derive these relationships for fluids. The motivation for going to the formal approach (i.e., dense gas theory) is that the stress field consists of static, translational, and collisional components and the net effect of these can be better handled by statistical mechanics because of its capability for keeping track of collisional trajectories. However, when the static and collisional contributions are removed, the equations of motion derived from dense gas theory should (and do) reduce to the same form as the continuity and momentum equations derived using the traditional continuum fluid dynamics approach. In fact, the difference between the derivation of the granular flow equations by the kinetic approach described above and the conventional approach via the Navier Stokes equations is that, in the latter, the material properties, such as viscosity, are determined by experiment while in the former the fluid properties are mathematically deduced by statistical mechanics of interparticle collision. 

For the exact evaluation of the two-monomer contribution Bf the Rainwater-Friend theory includes results from the kinetic theory of dense gases (see, for example, Cohen 1969), which means that direct kinetic as well as collisional transfer effects are considered. In order to calculate reliable values for B Rainwater (1984) extended earlier results of Snider Curtiss (1958), Hoffman Curtiss (1965) and Bennett Curtiss (1969) so that the dynamics of the two particles for a more realistic potential (here the Lennard-Jones (12-6) potential) were included. At the same time, the effects of bound states have been excluded from the evaluation of the complicated set of integrals that defines the two-monomer contribution (see Rainwater 1984). 

The collisional pressure tensor represents the instantaneous momentum transfer at binary particle collisions, over the distance separating the centers of the two colliding particles. The pressure tensor closure is derived based on an extension of the kinetic theory of dense gases. The collisional pressure tensor is thus the second out of the two pressure tensor components that is calculated by use of the KTGF. 

The difficulties inherent in the Enskog theory of dense gases were resolved with the derivation of the Boltzmann equation from the Liouville equation by Bogoliubov, Cohen, and Green. For the first time, a systematic kinetic theory for dense gases whose particles interact with short-range forces was developed. 

Hoffman, D. K. Curtiss, C. F. (1965). Kinetic theory of dense gases. V. Evaluation of the second transport virial coefficients. Phys. Fluids, 8,890-895. 

Ci j and V j respectively. In accordance with the theoretical basis of the kinetic theory of dilute gases, these properties are transported by short-term collisional contacts. For this reason, the KTGF models including the total particle-particle drag model, Fij = Ffj + f(J represent the solid flow dynamics for dilute binary fluidized beds. However, for dense beds, additional frictional effects need to be considered. An extra frictional particle-particle drag term, f(P must be taken into account. The... 

The Eulerian continuum approach, based on a continuum assumption of phases, provides a field description of the dynamics of each phase. The Lagrangian trajectory approach, from the study of motions of individual particles, is able to yield historical trajectories of the particles. The kinetic theory modeling for interparticle collisions, extended from the kinetic theory of gases, can be applied to dense suspension systems where the transport in the particle phase is dominated by interparticle collisions. The Ergun equation provides important flow relationships, which are useful not only for packed bed systems, but also for some situations in fluidized bed systems. 

When the gas-solid flow in a multiphase system is dominated by the interparticle collisions, the stresses and other dynamic properties of the solid phase can be postulated to be analogous to those of gas molecules. Thus, the kinetic theory of gases is adopted in the modeling of dense gas-solid flows. In this model, it is assumed that collision among particles is the only mechanism for the transport of mass, momentum, and energy of the particles. The energy dissipation due to inelastic collisions is included in the model despite the elastic collision condition dictated by the theory. 

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