Transport Theories

A complete description of the neutron population must specify simultaneously the distribution in space, energy, time and direction of motion. The primary objective of this chapter is to develop techniques and computational procedures for determining the angular distribution of neutrons. For this purpose we introduce the following function which gives the most general description of the neutron density  

Thus 4 (T,v Qft) is the total track length traveled per unit time by the neutrons described in (7.1). In most of the analysis presented in this chapter it will be convenient to work with this function rather than with the neutron density. 

We will lead up to the calculation of the neutron flux defined by (7.2) by introducing first a somewhat more limited description of the neutron population, namely, the one-velocity model. The appropriate function is (r,O,0. By omitting, for a time, the problem of describing the distribution of neutrons in energy space, we can focus attention on the directional properties of the neutron motion. Thus our first objective is to construct a picture of the neutron population which gives its dis- 

The analytical method employed in both of these presentations involves the use of expansions in spherical harmonics. In each case the original integrodifferential equation, which states the neutron-balance condition, is reduced to an infinite set of coupled equations in the various harmonics of the function t . By truncating the series expansion for the flux, this set can be further reduced to finite size (depending on the accuracy desired), and it is shown that in first approximation the resulting equations yield the diffusion theory and Fermi age models. 

A second general technique for treating the angular distribution of the neutron flux is presented in Sec. 7.4. This is the method of integral equations. Solutions for the directed flux 0(r,Q) are derived on the basis of the one-velocity model for various media of infinite extent. The application of these solutions for the infinite medium to systems of finite size is demonstrated in the case of the homogeneous slab and sphere. 

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In 1872, Boltzmaim introduced the basic equation of transport theory for dilute gases. His equation detemiines the time-dependent position and velocity distribution fiinction for the molecules in a dilute gas, which we have denoted by /(r,v,0- Here we present his derivation and some of its major consequences, particularly the so-called //-tlieorem, which shows the consistency of the Boltzmann equation with the irreversible fomi of the second law of themiodynamics. We also briefly discuss some of the famous debates surrounding the mechanical foundations of this equation. 

The final chapters, 11 and 12, are concerned with the particular application of transport theory to which this monograph is principally directed, namely the modeling of porous catalyst pellets. The behavior of a porous catalyst is described by differencial equations obtained from material and... 

The mathematical model most widely used for steady-state behavior of a reactor is diffusion theory, a simplification of transport theory which in turn is an adaptation of Boltzmann s kinetic theory of gases. By solving a differential equation, the flux distribution in space and time is found or the conditions on materials and geometry that give a steady-state system are determined. 

The same k p scheme has been extended to the study of transport properties of CNTs. The conductivity calculated in the Boltzmann transport theory has shown a large positive magnetoresistance [18], This positive magnetoresistance has been confirmed by full quantum mechanical calculations in the case that the mean free path is much larger than the circumference length [19]. When the mean free path is short, the transport is reduced to that in a 2D graphite, which has also interesting characteristic features [20]. 

Very little work has been done in this area. Even electrolyte transport has not been well characterized for multicomponent electrolyte systems. Multicomponent electrochemical transport theory [36] has not been applied to transport in lithium-ion electrolytes, even though these electrolytes consist of a blend of solvents. It is easy to imagine that ions are preferentially solvated and ion transport causes changes in solvent composition near the electrodes. Still, even the most sophisticated mathematical models [37] model transport as a binary salt. 

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. 

Boltzmann s H-Theorem. —One of the most striking features of transport theory is seen from the result that, although collisions are completely reversible phenomena (since they are based upon the reversible laws of mechanics), the solutions of the Boltzmann equation depict irreversible phenomena. This effect is most clearly seen from a consideration of Boltzmann s IZ-function, which will be discussed here for a gas in a uniform state (no dependence of the distribution function on position and no external forces) for simplicity. 

Transformations in Hilbert space, 433 Transition probabilities of negatons in, external fields, 626 Transport theory, 1 Transportation problems, 261,296 Transversal amplitude, 552 Transversal vector, 554 Transverse gauge, 643 Triangular factorization, 65 Tridiagonal form, 73 Triple product ensemble, 218 Truncation error, 52 Truncation of differential equations/ 388... 

Lustig, SR Camthers, JM Peppas, NA, Continuum Thermodynamics and Transport Theory for Polymer-Fluid Mixtures, Chemical Engineering Science 12, 3037, 1992. 

A combination of continuum transport theory and the Poisson distribution of solution charges has been popular in interpreting transport of ions or conductivity of electrolytes. Assuming zero gradient in pressure and concentration of other species, the flux of an ion depends on the concentration gradient, the electrical potential gradient, and a convection... 

If it can be shown that the prefactor is the identity matrix plus a matrix linear in x, then this is, in essence, Onsager s regression hypothesis [10] and the basis for linear transport theory. 

Membrane transport represents a major application of mass transport theory in the pharmaceutical sciences [4], Since convection is not generally involved, we will use Fick s first and second laws to find flux and concentration across membranes in this section. We begin with the discussion of steady diffusion across a thin film and a membrane with or without aqueous diffusion resistance, followed by steady diffusion across the skin, and conclude this section with unsteady membrane diffusion and membrane diffusion with reaction. 

Volume 171. Biomembranes (Part R Transport Theory Cells and Model Membranes)... 

B. Davison, Neutron Transport Theory, Clarendon Press, Oxford, UK, 1957, Chap. VI, Section 1. 

Frederiksen T, Paulsson M, Brandbyge M, Jauho A-P (2007) Inelastic transport theory from first principles methodology and application to nanoscale devices. Phys Rev B 75(20) 205413-205422... 

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