Kinetic theory of transport phenomena

Schematic Maxwellian velocity distribution Up) du is the fraction of the molecules with velocities between v and v + dv. Values of the rms speed u and the mean speed c are shown. 

Another very important concept in kinetic theory is the average distance a molecule travels between collisions—the so-called mean free path. On the basis of a very simple conception of molecular collisions, the following equation for the mean free path A can be derived  

We shall consider in detail the predictions of the hard-sphere model for the viscosity, thermal conductivity, and diffusion of gases indeed, the kinetic theory treatment of these three transport properties is very similar. But first let us consider the simpler problem of molecular effusion. 

The theory of effusion (Knudsen flow) is quite straightforward since only molecular flow is involved i.e., in the process of effusing, the molecules act independently of each other. For a Maxwellian distribution of velocities it can be shown that the number Zof molecular impacts on a unit area of wall surface in unit time is 

TABLE 1 Mean free paths of various gases at 25°C 

---

Gray P (1968) The kinetic theory of transport phenomena in simple liquids. In Temperley HNV, Rowlinson JS, Rushbrook GS (eds) Physics of simple liquids. North-Holland, Amsterdam, pp 507—562... 

The kinetic theory of transport phenomena is the most elementary and perhaps the first step toward understanding more complex transport theories [1], Consider a plane z, across which particles travel carrying mass and kinetic energy. Consider two fictitious planes at z + t and... 

In this section we discuss five assumptions that have traditionally been introduced in order to continue with the development of the kinetic theory of transport phenomena and get useful results. Some of these assumptions are made because we do not at present know enough about the distribution functions that appear in the expressions in Tables 1 and 2, in particular the pair distribution function and the momentum-space distnbution function. Other assumptions are introduced in order to simplify the subsequent problem-solving for specific molecular models. All the assumptions we present here can be challenged some of them should be modified, and some of them may ultimately be eliminated. 

The main objective of performing kinetic theory analyzes is to explain physical phenomena that are measurable at the macroscopic level in a gas at- or near equilibrium in terms of the properties of the individual molecules and the intermolecular forces. For instance, one of the original aims of kinetic theory was to explain the experimental form of the ideal gas law from basic principles [65]. The kinetic theory of transport processes determines the transport coefficients (i.e., conductivity, diffusivity, and viscosity) and the mathematical form of the heat, mass and momentum fluxes. Nowadays the kinetic theory of gases originating in statistical mechanics is thus strongly linked with irreversible- or non-equilibrium thermodynamics which is a modern held in thermodynamics describing transport processes in systems that are not in global equilibrium. 

In this article we review the theory of transport phenomena in polymeric liquids in the framework of the phase-space kinetic theory for models with no internal constraints. That is, we restrict ourselves to bead-spring models, but we do allow for any kind of connectivity hence the treatment here allows for chains, rings, stars, combs, and branched chains. This presentation includes formally both mixtures and polydisperse systems. We have expanded the coverage here to include the heat-flux vector, which was not investigated in [5], [7],... 

Introduction.—Statistical physics deals with the relation between the macroscopic laws that describe the internal state of a system and the dynamics of the interactions of its microscopic constituents. The derivation of the nonequilibrium macroscopic laws, such as those of hydrodynamics, from the microscopic laws has not been developed as generally as in the equilibrium case (the derivation of thermodynamic relations by equilibrium statistical mechanics). The microscopic analysis of nonequilibrium phenomena, however, has achieved a considerable degree of success for the particular case of dilute gases. In this case, the kinetic theory, or transport theory, allows one to relate the transport of matter or of energy, for example (as in diffusion, or heat flow, respectively), to the mechanics of the molecules that make up the system. 

Analogies between the three transport phenomena are evident and far-reaching [2]. For instance, consider a low pressure, ideal gas, and assume that the kinetic theory of gases holds. It is shown that ... 

It is worth noting at this point that the various scientific theories that quantitatively and mathematically formulate natural phenomena are in fact mathematical models of nature. Such, for example, are the kinetic theory of gases and rubber elasticity, Bohr s atomic model, molecular theories of polymer solutions, and even the equations of transport phenomena cited earlier in this chapter. Not unlike the engineering mathematical models, they contain simplifying assumptions. For example, the transport equations involve the assumption that matter can be viewed as a continuum and that even in fast, irreversible processes, local equilibrium can be achieved. The paramount difference between a mathematical model of a natural process and that of an engineering system is the required level of accuracy and, of course, the generality of the phenomena involved. 

There are general relationships of transport phenomena based on phenomenological theory, i.e., on the correlations between macroscopically measurable quantities. The molecular theories explain the mechanism of transport processes taking into account the molecular structure of the given medium, applying the kinetic-statistical theory of matter. The hydrodynamic theories are also applied especially to describe - convection. 

Statistical mechanics can provide phenomenological descriptions of nonequilibrium processes. An alternative approach based on kinetic theory is favorable especially in describing the transport and rate phenomena. A kinetic theory of nonequilibrium systems has been developed for dilute monatomic gases at low pressure. Substantial progress has also been achieved in extending the theory to dense gases, real gases, and liquids. 

This chapter concerns the structures and propagation velocities of the deflagration waves defined in Chapter 2. Deflagrations, or laminar flames, constitute the central problem of combustion theory in at least two respects. First, the earliest combustion problem to require the simultaneous consideration of transport phenomena and of chemical kinetics was the deflagration problem. Second, knowledge of the concepts developed and results obtained in laminar-flame theory is essential for many other studies in combustion. Attention here is restricted to the steadily propagating, planar laminar flame. Time-dependent and multidimensional effects are considered in Chapter 9. 

The thermodynamics of irreversible processes should be set up from the scratch as a continuum theory, treating the state parameters of the theory as field variables [32]. This is also the way in which classical fluid mechanic theory is formulated. Therefore, in the computational fluid dynamics literature, the transport phenomena and the extensions of the classical thermodynamic relations are both interpreted as closures of the fluid dynamic theory. The validity of the thermodynamic relations for fluid dynamic systems has been approached from the viewpoint of the kinetic theory of gases [13]. However, any Arm distinction between irreversible thermodynamics and fluid mechanics... 

As in all mathematical descriptions of transport phenomena, the theory of polydisperse multiphase flows introduces a set of dimensionless numbers that are pertinent in describing the behavior of the flow. Depending on the complexity of the flow (e.g. variations in physical properties due to chemical reactions, collisions, etc.), the set of dimensionless numbers can be quite large. (Details on the physical models for momentum exchange are given in Chapter 5.) As will be described in detail in Chapter 4, a kinetic equation can be derived for the number-density function (NDF) of the velocity of the disperse phase n t, X, v). Also in this example, for clarity, we will assume that the problem has only one particle velocity component v and is one-dimensional in physical space with coordinate x at time t. Furthermore, we will assume that the NDF has been normalized (by multiplying it by the volume of a particle) such that the first three velocity moments are... 

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. 

Section 4.2 considered the kinetic theory of the transport properties of dilute, pure fluids in some detail in this section the theory is extended to gas mixtures. Naturally, the theory of mixtures shares many features with that of pure species so that the same pattern will be adopted for the presentation, although duplication will be avoided whenever possible. For these reasons the semiclassical kinetic theory description is immediately adopted here, and similar consequences for the description of some phenomena as they pertain to pure gases are accepted. 

Cohen, E. G. D. (1969). The kinetic theory of moderately dense gases, in Transport Phenomena in Fluids, ed. H. J. M. Hanley, pp. 157-207. New York Marcel Dekker. 

---