One-dimensional transport, equation

Figure 5.18. The diffusion flamelet can be approximated by a one-dimensional transport equation that describes the change in the direction normal to the stoichiometric surface. The rate of change in the tangent direction is assumed to be negligible since the flamelet thickness is small compared with the Kolmogorov length scale. The flamelet approximation is valid when the reaction separates regions of unmixed fluid. Thus, the boundary conditions on each side are known, and can be uniquely expressed in terms of .

If we consider Oj" and h as carriers, we refer, e.g. to chemical diffusion of oxygen in La2CuC>4. Writing down the one-dimensional transport equations (Eq. 103) for ionic and electronic carriers and considering flux coupling and electroneutrality, we immediately obtain for this case (see, e.g., Ref.173)... 

Generally, the concentration profile of analytes in FFF can be obtained from the solution of the general transport equation. For the sake of simplicity, the concentration profile of the steady-state zone of the analyte along the axis of the applied field is calculated from the one-dimensional transport equation ... 

The multiple-trapping model can also be solved analytically for the transit time by the method of Laplace transforms. The one-dimensional transport equations for the fiee-electron density n(x, t) in a semiconductor with a distribution of discrete trapping levels are... 

In order to solve the transport problem we have to complete the set of necessary equations and, therefore, boundary conditions must be formulated. Depending on the boundary conditions we impose, quite different transport situations will arise. Let us analyze the one-dimensional transport in a binary electrolyte as an illustration. Two different boundary conditions will be introduced. 1) AX is brought between different chemical potentials relative to one of its component (open electrical circuit). 2) AX is brought between two inert electrodes to which a voltage A U is applied. Figures 4-3a and 4-3b show the experimental schemes. Let us examine them separately. 

When viewed from a reference frame that is stationary with respect to the solidification interface, the melt moves uniaxially toward the interface and the crystal moves away. Then solute transport in the melt is governed by the one-dimensional balance equation... 

This equation can be used to describe one-dimensional transport of radionuclides through porous media (e.g. radionuclide elution curves from laboratory columns packed with interbed solids) assuming instantaneous sorption and desorption. Van Genuchten and coworkers have demonstrated the importance of using both sorption and desorption isotherms in this equation when hysteresis is significant. Isotherm data for sorption and desorption reactions of radionuclides with interbed materials are presented in this paper which can be used to predict radionuclide transport. 

A complete derivation of the mass transport equation is presented in detail elsewhere, and an abbreviated derivation is outlined here for one-dimensional transport of a chemical species in porous media saturated with water (e.g.. Ref. P °l). As with the flow equation, the transport equation begins with the mass conservation principle the change in mass within a control... 

The isotherms are incorporated into the transport models through the retardation factor R. The one-dimensional ADR equation for saturated media becomes... 

A realistic geologic model would be the one-dimensional transport of carbon isotopes by CO2 infiltration through a quartzite with constant porosity. Note that oxygen isotope transport is described by reactive transport equations, since oxygen isotopes will at least partially exchange with the quartzite (see below). The initial carbon isotope com-... 

The effluent concentration can be predicted by solving the mass conservation equation. The conservation equations of particulate matter consider the change in concentration of particulate and change of porosity with time. The amount of fines retained in the porous medium is represented by a, while u signifies the superficial velocity of the incompressible transport fluid. For constant volumetric, incompressible flow, neglecting dispersion and gravitational effects, the one dimensional conservation equation follows. 

This table gives values of the diffusion coefficient, D, for diffusion of several common gases in water at various temperatures. For simple one-dimensional transport, the diffusion coefficient describes the time-rate of change of concentration, dddt, through the equation... 

Assuming one dimensional transport the heat balance equation for the interconnect can be summarized as... 

Below, a reactive fiow system will be discussed which can be described by the one-dimensional governing equations using the geometry of the problem. So, the resulting independent variables are the time and the distance normal to the catalytic surface. Detailed models for the chemical reactions as well as for the molecular transport are used. In order to include surface chemistry, the gas-phase problem is closely coupled with the transport to the gas-surface interface and the reaction thereon. The elementary-reactions concept is extended to heterogeneous reactions. Therefore, the boundary conditions for the governing equations at the catalyst become more complex compared to the pure gas-phase problem. 

If there is no transformation of the species inside the polymer phase, the balance equation for electronic or ionic species i in the case of a one-dimensional transport takes the universal form... 

Samples were cut in rectangular shape (9 cm x 1 cm) using a warm scalpel. All of the samples had an aspect ratio (length over thickness) greater than 10, which ensured application of a one-dimensional diffusion equation for analysis of the transport data. 

The one-dimensional continuity equation for the ambipolar transport of charge carriers will be posed as [378]... 

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