Equations of Transport

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 authors [5] have formulated fractional equation of transport processes, having the following form ... 

Mathematical analysis of this problem has been performed by Belyakov et al. (1979a) on the basis of the equations of transport along the surface with due account for carrier migration in the self-induced electric field S (y). 

Bird, R. B., Stewart, W. E., and Lightfoot, E. N., Transport Phenomena, Wiley, New York, 1960. (Undergraduate and graduate levels. A classic textbook on the formulation and solutions of equations of transport phenomena in engineering. Contains both undergraduate-level and graduate-level topics and problems.)... 

In the cause of logical clarity, it is unfortunate that the equations of transport are so often derived from the Boltzmann integro-differential equation. Their derivation from the Liouville equation is a straightforward exercise in -dimensional calculus,... 

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. 

Dimensionless groups derived from the equations of transport... 

The constitutive equations of transport in porous media comprise both physical properties of components and pairs of components and simplifying assumptions about the geometrical characteristics of the porous medium. Two advanced effective-scale (i.e., space-averaged) models are commonly applied for description of combined bulk diffusion, Knudsen diffusion and permeation transport of multicomponent gas mixtures—Mean Transport-Pore Model (MTPM)—and Dusty Gas Model (DGM) cf. Mason and Malinauskas (1983), Schneider and Gelbin (1984), and Krishna and Wesseling (1997). The molar flux intensity of the z th component A) is the sum of the diffusion Nc- and permeation N contributions,... 

These dimensionless groups of fluid properties play important roles in dimensionless modeling equations of transport processes, and for systems where simultaneous transport processes occur. 

Actually, research by modelling is more and more extensively used in many applications because complex devices models, composed of different elements, can be made by assembling models the solutions of which are frequently available. This behaviour presents an impressive growth and is sustained by the extraordinary developments in numerical calculations and by the implementation of commonly used computers with a high capacity and calculus rate. Nevertheless, modelling based on the equations of transport phenomena cannot be applied to every system, because they can present some limitations, which are summarized here. 

The models based on the equations of transport phenomena and on stochastic models contain an appreciable quantity of mathematics, software creation, computer programming and data processing. 

For KjpCpD — 1, the relation between concentration and temperature. (9.9), is independent of the nature of the flow, either laminar or turbulent. It applies to both the instantaneous and time-averaged concentration and temperature fields, but only in regions in which condensation has not yet occurred. When the equations of transport for the jet flow are reduced to the form used in turbulent flow, the molecular diffusivity and thermal diffusivity are usually neglected in comparison with the turbulent diffusivities. This is acceptable for studies of gross transport and the time-averaged composition and temperature. However, this frequently made assumption is not correct for molecular scale processes like nucleation and condensation, which depend locally on the molecular transport properties. 

Transport of materials in bentonite is controlled by the interlayer structure of montmorillonite lamellae and by interlayer cations. The governing equation of transport is given by... 

There is still another technique, in which dimension analysis is used. The differential equations of transport can be set up sometimes easily, but it is too difficult to solve them under the conditions required. However, if the equations are made dimensionless, then the relevant combinations of dimensionless quantities can be found out. 

Equations 7.2.a-l and 7.2.a-5 are in fact extensions of the continuity equations used in previous chapters, where the flow terms were normally not present. These somewhat detailed derivations have been used to carefully illustrate the development of the equations of transport processes into forms needed to describe chemical reactors. It is seldom that the full equations have to be utilized, and normally only the most important terms will be retained in practical situations. However Eqs. 7.2.a-l or 5 are useful to have available as a fundamental basis. 

The equations of transport and mass balance to describe the desorption of substances from a monolayer packaging (Figure 13.5), whose transport properties are uniform and constant over time, are written in one dimension ... 

The solver, second element of a CFD package, transforms the equations of transport phenomena into a system of algebraic equations via discretization and solves them by an iterative approach. These steps can be carried out either by the FEM or by the FVM that are explained in the following sections. The following balance equation in an arbitrarily chosen element or cell can be written for a general flow variable (p, for example, a velocity component or enthalpy [12] ... 

Step 2. From the above growth rate, all 15 unknowns are calculated from 15 equations of transport and biocatalytic reaction rale. The initial feed concentration of the substrate glucose, [c ]. and the product ethanol, [c ], are used for the calculation. The numerical answers so obtained correspond to time r = 0, and they arc shown in the bracket [—] . 

The three formulas (2.22)-(2.24) are examples of linear transport equations they relate the response of a system (the flux) to a small perturbing force (the gradient). The transport coefficients Z), rj, and x are the parameters of proportionality, to be determined experimentally. A familiar transport equation is Ohm s law. Here voltage is the force, current the response, and conductivity (the reciprocal of resistance) the transport coefficient. In general, equations of transport are not as simple as these. In a two-component system with a temperature gradient, Fourier s law states that there is only heat flow. However, if the masses of the components... 

This simple procedure, requiring only the application of a single and established expression, is the exception rather than the rule in problems involving mass transport. In the vast majority of cases we have to draw on additional tools to complete the mathematical formulation of the process. The tools required comprise various forms of the law of conservation of mass, supplemented by what we term auxiliary relations. These latter relations are largely empirical in nature and include the equations of transport seen in the previous chapter, as well as expressions describing chemical reaction rates and phase equilibria. 

At about the same time, a second major educational revolution was occurring at the University of Wisconsin. Professors Bird, Stewart and Lightfoot, collectively known to future generations of students as BSL, prepared a set of notes (in 1957) and eventually a book (in 1960) entitled Transport PhenomenUy which offered a new approach to the analysis of chemical engineering unit problems. The main lesson of BSL is that there is a strong unifying backbone to apparently different unit operations in the framework of the continuum equations of transport. The necessity for analysis of individual operations or processes does not disappear, but the differential volume and the balance equations become the central theme of this approach. 

This is the equation of transport for the pulse voltage problem. 

---