Simultaneous transport equations

In gridpoint models, transport processes such as speed and direction of wind and ocean currents, and turbulent diffusivities (see Section 4.8.1) normally have to be prescribed. Information on these physical quantities may come from observations or from other (dynamic) models, which calculate the flow patterns from basic hydrodynamic equations. Tracer transport models, in which the transport processes are prescribed in this way, are often referred to as off-line models. An on-line model, on the other hand, is one where the tracers have been incorporated directly into a d3mamic model such that the tracer concentrations and the motions are calculated simultaneously. A major advantage of an on-line model is that feedbacks of the tracer on the energy balance can be described... 

Here one makes an effort to describe simultaneously transport-controlled and chemical kinetics processes (Skopp, 1986). Thus, an attempt is made to describe both the chemistry and physics accurately. For example, outflow curves from miscible displacement experiments on soil columns are matched to solutions of the conservation of mass equation. The matching process introduces a potential ambiquity such that experimental uncertainties are translated into model uncertainties. Often, an error in the description of the physical process is compensated for by an error in the chemical process and vice-versa (i.e., Nkedi-Kizza etal, 1984). 

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

Example 10.7 Energy conversion in the electrokinetic effect Electrokinetic effects are the consequence of the interaction between the flow of matter and flow of electricity through a porous membrane. The linear phenomenological equations for the simultaneous transport of matter and electricity are (Eqs. (10.89) and (10.90))... 

We shall, as before, use Newton s method to solve all the independent equations simultaneously. The independent variables that are to be determined by iteration are the fluxes and the interface compositions and temperature. However, the use of the turbulent eddy diffusion model for the vapor-phase mass transport means that the mass fluxes and n 2, and the molar fluxes Aj and A2, appear in the set of model equations. These fluxes are related by... 

Nonequilibrium treatment " of EOD under these conditions yields the following rate equations for the simultaneous transport of matter (i.e., water) and electricity (i.e., current), assuming that the diaphragm is uniform ... 

Here c is the solute concentration, [ML-3 ], Ds is the solute dispersion coefficient, [L2T], and v is the average pore-water velocity, [LHM ]. This equation describes movement of particles participating in Fickian diffusion-like transport and simultaneously transported with the mean pore velocity. 

Recognizing the above facts, the following equations based on Equation 7 and using Equation 2-4 to represent can be developed to describe the simultaneous transport and transformation of A, A-SO and A-SO2 ... 

We know that the reaction rate depends on temperature and concentration. If the temperature and concentration differences between the interior of the catalyst particles and the bulk fluid are significant, then these differences must be taken into account in solving the design equation. In essence, this would require simultaneously solving the design equation and equations that describe heat transport, mass transport, and reaction kinetics in the interior of the catalyst particle, using the equations for transport through the boundary layer as boundary conditions. 

Overall, the RDE provides an efficient and reproducible mass transport and hence the analytical measurement can be made with high sensitivity and precision. Such well-defined behavior greatly simplifies the interpretation of the measurement. The convective nature of the electrode results also in very short response tunes. The detection limits can be lowered via periodic changes in the rotation speed and isolation of small mass transport-dependent currents from simultaneously flowing surface-controlled background currents. Sinusoidal or square-wave modulations of the rotation speed are particularly attractive for this task. The rotation-speed dependence of the limiting current (equation 4-5) can also be used for calculating the diffusion coefficient or the surface area. Further details on the RDE can be found in Adam s book (17). 

Especially for the electrons, the fluid model has the advantage of a lower computational effort than the PIC/MC method. Their low mass (high values of the transport coefficients) and consequent high velocities give rise to small time steps in the numerical simulation (uAf < Aa) if a so-called explicit method is used. This restriction is easily eliminated within the fluid model by use of an implicit method. Also, the electron density is strongly coupled with the electric field, which results in numerical Instabilities. This requires a simultaneous implicit solution of the Poisson equation for the electric field and the transport equation for the electron density. This solution can be deployed within the fluid model and gives a considerable reduction of computational effort as compared to a nonsi-multaneous solution procedure [179]. Within the PIC method, only fully explicit methods can be applied. 

Equation (A9) accounts for the change in the amount of daughter within the solid while Equation (A 10) accounts for that in the fluid. The transfer of the nuclide from the solid to the fluid phase is governed by the first term on the right side of both equations. Note that in this formulation, as given in Spiegelman and Elliott (1993), the time of melt transport is accounted for and depends on the physically based transport velocity. Like Equation (A8), both equations can be solved simultaneously by numerical integration or more sophisticated numerical method. 

This approach is possible only if density gradients are not formed in the solution as a result of transport processes (e.g. in dilute solutions). Otherwise, both differential equations must be solved simultaneously—a very difficult task. 

This equation describes the cathodic current-potential curve (polarization curve or voltammogram) at steady state when the rate of the process is simultaneously controlled by the rate of the transport and of the electrode reaction. This equation leads to the following conclusions ... 

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