Transport selectivity equation

Let us see how to represent changes in properties for a system volume to property changes for a control volume. Select a control volume (CV) to be identical to volume V t) at time t, but to have a different velocity on its surface. Call this velocity, w. Hence, the volume will move to a different location from the system volume at a later time. For example, for fluid flow in a pipe, the control volume can be selected as stationary (w = 0) between locations 1 and 2 (shown in Figure 3.4, but the system moves to a new location later in time. Let us apply the Reynolds transport theorem, Equation (3.9), twice once to a system volume, V(t), and second to a control volume, CV, where CV and V are identical at time t. Since Equation (3.9) holds for any well-defined volume and surface velocity distribution, we can write for the system... 

In contrast to moment closures, the models used to close the conditional fluxes typically involve random processes. The choice of the models will directly affect the evolution of the shape of the PDF, and thus indirectly affect the moments of the PDF. For example, once closures have been selected, all one-point statistics involving U and 0 can be computed by deriving moment transport equations starting from the transported PDF equation. Thus, in Section 6.4, we will look at the relationship between (6.19) and RANS transport equations. However, we will first consider the composition PDF transport equation. 

The purpose of this entry is to provide an introduction to the morphology of porous media, the behavior of fluids in porous media, and modeling techniques. While selected equations are presented, the emphasis is on phenomenological behavior, the aim being to give the reader a basic foundation in the physics of transport before a more detailed study might be undertaken. Many important topics have been omitted due to space constraints see the Conclusions section for recommended texts on these topics. 

Let us consider the selectivity parameter on the example of metal ions separation. According to the transport model equations the selectivity of two solutes, for example, two metal species, Sm,/M2> is determined by relation ... 

Microkinetics concerns the chemical and physical kinetics at the scale of the particles (catalyst particles, liquid films and droplets, gas bubbles...) and their coupling in order to get the apparent reaction rate and selectivity equations. Macrokinetics refers to the transport of momentiam, mass and heat at the scale of the reactor, and its goal is the establishment of a model of the reactor. Combination of this model with the apparent reaction rate and selectivity equations allows to write the equations of the reactor. 

When the transport is limited by diffusion, the initial transport selectivity is predicted from the results of single-ion transport experiments (27). The flux ratio can be expressed by Equation 29. 

The excess functions of equation (7.13) are not unique but depend strongly on both the functional form and the optimized parameters of the enhancement terms. The optimized parameters of both terms depend on the selected equation of state used to determine the density (especially in the critical region) as well as the compressibility and heat capacity, which are required in the theoretically based critical enhancement models. It is necessary to have data over an extremely wide range of temperature and pressure to resolve any temperature dependence of the transport property excess functions. Detailed examples of such pure fluid correlations are presented in Chapter 14. 

Although microporous membranes are a topic of research interest, all current commercial gas separations are based on the fourth type of mechanism shown in Figure 36, namely diffusion through dense polymer films. Gas transport through dense polymer membranes is governed by equation 8 where is the flux of component /,andare the partial pressure of the component i on either side of the membrane, /is the membrane thickness, and is a constant called the membrane permeability, which is a measure of the membrane s ability to permeate gas. The ability of a membrane to separate two gases, i and is the ratio of their permeabilities,a, called the membrane selectivity (eq. 9). 

In electrochemical cells we often find convective transport of reaction components toward (or away from) the electrode surface. In this case the balance equation describing the supply and escape of the components should be written in the general form (1.38). However, this equation needs further explanation. At any current density during current flow, the migration and diffusion fluxes (or field strength and concentration gradients) will spontaneously settle at values such that condition (4.14) is satisfied. The convective flux, on the other hand, depends on the arbitrary values selected for the flow velocity v and for the component concentrations (i.e., is determined by factors independent of the values selected for the current density). Hence, in the balance equation (1.38), it is not the total convective flux that should appear, only the part that corresponds to the true consumption of reactants from the flux or true product release into the flux. This fraction is defined as tfie difference between the fluxes away from and to the electrode ... 

It is probable that capillary flow of water contributes to transport in the soil. For example, a rate of 7 cm/year would yield an equivalent water velocity of 8 x 10-6 m/h, which exceeds the water diffusion rate by a factor of four. For illustrative purposes we thus select a water transport velocity or coefficient U6 in the soil of 10 x 10 6 m/h, recognizing that this will vary with rainfall characteristics and soil type. These soil processes are in parallel with boundary layer diffusion in series, so the final equations are... 

In Section 3.3, we will use (3.16) with the Navier-Stokes equation and the scalar transport equation to derive one-point transport equations for selected scalar statistics. As seen in Chapter 1, for turbulent reacting flows one of the most important statistics is the mean chemical source term, which is defined in terms of the one-point joint composition PDF +(+x, t) by... 

Multi-environment presumed PDF models can also be easily extended to treat cases with more than two feed streams. For example, a four-environment model for a flow with three feed streams is shown in Fig. 5.24. For this flow, the mixture-fraction vector will have two components, 2 and 22- The micromixing functions should thus be selected to agree with the variance transport equations for both components. However, in comparison with multi-variable presumed PDF methods for the mixture-fraction vector (see Section 5.3), the implementation of multi-environment presumed PDF models in CFD calculations of chemical reactors with multiple feed streams is much simpler. 

The stability, growth, and transport of voids during composite processing is reviewed. As a framework for this model, the autoclave process was selected, but the concepts and equations may be applied equally effectively in a variety of processes, including resin transfer molding, compression molding, and filament winding. In addition, the problem of resin transport and its intimate connection with void suppression are analyzed. 

This book treats a selection of topics in electro-diffusion—a nonlinear transport process whose essence is diffusion of charged particles, combined with their migration in a self-consistent electric field. Basic equations of electro-diffusion were formulated about 100 years ago by Nernst and Planck in the ionic context [1]—[3]. Sixty years later Van Roosbroeck applied these equations to treat the transport of holes and electrons in semiconductors [4]. Correspondingly, major applications of the theory of electro-diffusion still lie in the realms of chemical and electrical engineering, related to ion separation and semiconductor device technology. Some aspects of electrodiffusion are relevant for electrophysiology. 

Hints and Help Estimate the transport time by assuming that the chemicals diffuse from a reservoir of constant concentration into the liner that at time t = 0 is assumed to be uncontaminated. Equation 25-41 may give you an idea how to calculate the effective diffusion coefficient through the liner. Select the most critical compound among the four. What is the criterion ... 

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