Fundamental transport equations

For the constant-density flows considered in this work,27 the fundamental governing equations are the Navier-Stokes equation for the fluid velocity U (Bird et al. 2002)  

In interpreting these expressions, the usual summation rules for roman indices apply, e.g., 

The fluid density appearing in (1.27) is denoted by p and is assumed to be constant. The molecular-transport coefficients appearing in the governing equations are the kinematic 

Although this choice excludes combustion, most of the modeling approaches can be directly extended to non-constant-density flows with minor modifications. 

This expression is found from (1.27) using the continuity equation for a constant-density 

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This is the fundamental transport equation for the species i, which does not depend on any assumption other than mass conservation. 

The laws of physics may all be expressed as relations between numerics and are in their simplest form when thus expressed. The use of dimensionless expressions is of particular value in dealing with phenomena too complicated for a complete treatment in terms of the fundamental transport equations of mass, energy and angular momentum. Most of the physical problems in the process industry are of this complicated nature and the combination of variables in the form of dimensionless groups can always be regarded as a safe start in the investigation of new problems. 

Using these assumptions, the fundamental transport equations for solid-liquid fluidized beds are given next. 

Another structure/function transport model, often referred to as the capillary or electroki-netic model, predefines the microlevel structure of an ion-exchange membrane as an array of pores of known dimensions with a specified distribution of ion-exchange sites on the pore walls. Equations describing solute and solvent transport and theories for molecularlevel ion/solvent and ion-membrane interactions are then generated, based on this pore structure [151], The fundamental transport equation for the molar flux of ionic species is the Nemst-Planck equation... 

Rate equations are used to describe interphase mass transfer in batch systems, packed beds, and other contacting devices for sorptive processes and are formulated in terms of fundamental transport properties of adsorbent and adsorbate. 

In this chapter the fundamental characteristics of osmotic systems are explored. The basic transport equations are developed. Examples of the various types of systems available are discussed and related to these transport equations. The relative advantages for each approach are delineated. Also, various aspects of the manufacture of these systems are reviewed. It is important to emphasize that much of the relevant work in this area has been published only in the patent literature. An excellent review of this work for patents published through 1993 is available [5],... 

The material covered in the appendices is provided as a supplement for readers interested in more detail than could be provided in the main text. Appendix A discusses the derivation of the spectral relaxation (SR) model starting from the scalar spectral transport equation. The SR model is introduced in Chapter 4 as a non-equilibrium model for the scalar dissipation rate. The material in Appendix A is an attempt to connect the model to a more fundamental description based on two-point spectral transport. This connection can be exploited to extract model parameters from direct-numerical simulation data of homogeneous turbulent scalar mixing (Fox and Yeung 1999). 

The CSTR model can be derived from the fundamental scalar transport equation (1.28) by integrating the spatial variable over the entire reactor volume. This process results in an integral for the volume-average chemical source term of the form ... 

A fundamental fuel cell model consists of five principles of conservation mass, momentum, species, charge, and thermal energy. These transport equations are then coupled with electrochemical processes through source terms to describe reaction kinetics and electro-osmotic drag in the polymer electrolyte. Such convection—diffusion—source equations can be summarized in the following general form... 

We could perform our discretization on the governing partial differential equations, which are different for each case, but there is no need to go beyond the most fundamental equation of mass transport, equation (2.1). 

The initial chapters of this book (through Chapter 7) concentrate on fluid mechanics, with an emphasis on establishing the fundamental conservation equations that are needed to formulate and solve chemically reacting flow problems. In these chapters, however, details of the chemistry and the molecular transport are treated fairly simply. The following five chapters (Chapters 8 through 12) provide much more depth on thermodynamics, chemical kinetics, and molecular transport. With the physical-chemistry background established,... 

BOLTZMANN TRANSPORT EQUATION. The fundamental equa tion describing the conservation of particles which are diffusing in a scattering, absorbing, and multiplying medium. It states that the lime rate of change of particle density is equal to the rate of production, minus the rate of leakage and the rate of absorption, in the form of a partial differential equation sucli as... 

The equations of this section show that resolution and peak capacity are inversely proportional to a and w (usually reflected in H and N). These equations illustrate how the capacity for separation is diminished, using any reasonable measure, by increases in zone width. This conclusion reemphasizes our deep concern with zone spreading phenomena and the fundamental transport processes that underlie them. 

Plasmas typical of C02 laser discharges operate over a pressure range from 1 Torr to several atmospheres with degrees of ionization, that is, nJN (the ratio of electron density to neutral density) in the range from 10-8 to 10-8. Under these conditions the electron energy distribution function is highly non-Maxwellian. As a consequence it is necessary to solve the Boltzmann transport equation based on a detailed knowledge of the electron collisional channels in order to establish the electron distribution function as a function of the ratio of the electric field to the neutral gas density, E/N, and species concentration. Development of the fundamental techniques for solution of the Boltzmann equation are presented in detail by Shkarofsky, Johnston, and Bachynski [44] and Holstein [45]. 

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