Flux balance, diffusion

The material balance conditions (11.1) may be rewritten in terms of the total flux N and the diffusion fluxes J, when they take the form... 

At any point within the boundary layer, the convective flux of the macromolecule solute to the membrane surface is given by the volume flux,/ of the solution multipfled by the concentration of retained solute, c. At steady state, this convective flux within the laminar boundary layer is balanced by the diffusive flux of retained solute in the opposite direction. This balance can be expressed by equation 1 ... 

Material Balances Whenever mass-transfer applications involve equipment of specific dimensions, flux equations alone are inadequate to assess results. A material balance or continuity equation must also be used. When the geometiy is simple, macroscopic balances suffice. The following equation is an overall mass balance for such a unit having bulk-flow ports and ports or interfaces through which diffusive flux can occur ... 

A formal derivation of diffusion in a restricted, high diffusivity path which uses no atomic model of the grain boundary is that due to Fisher, who made a flux balance in unit width of a grain boundary having a drickness of <5. There is flux accumulation in the element according to Pick s second law given by... 

Under realistic conditions a balance is secured during current flow because of additional mechanisms of mass transport in the electrolyte diffusion and convection. The initial inbalance between the rates of migration and reaction brings about a change in component concentrations next to the electrode surfaces, and thus gives rise to concentration gradients. As a result, a diffusion flux develops for each component. Moreover, in liquid electrolytes, hydrodynamic flows bringing about convective fluxes Ji j of the dissolved reaction components will almost always arise. 

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 ... 

Having defined the balance regions, the next task is to identify all the relevant inputs and outputs to the system (Fig. 1.10). These may be well-defined physical flow rates (convective streams), diffusive fluxes, but may also include interphase transfer rates. 

The boundary conditions determine the form of balance equation for the inlet and outlet sections. These require special consideration as to whether diffusion fluxes can cross the boundaries in any particular physical situation. The physical situation of closed ends is considered here. This would be the case if a smaller pipe were used to transport the fluid in and out of the reactor, as shown in Figs. 4.13 and 4.14. 

Since no diffusive flux enters the closed entrance of the tube, the component balance for the first section, becomes... 

The end sections require special treatment to account for the fact that the no diffusive flux enters through the end wall of the column and are derived by omitting the diffusion term from the column end. In addition, the outlet concentrations X9 and Yq are extrapolated from the previous section. The component balances for section 8 can be derived with the aid of Figs. 5.199 and 5.200. 

Control Volume Mass Balance. We can now combine equations (2.1), (2.5), (2.6), (2.11), and (2.13) into a mass balance on our box for Cartesian coordinates. After dividing hyV = dx dy dz and moving the diffusive flux terms to the right-hand side, this mass balance is... 

The reactant A and the product B diffuse into and out of a cylindrical catalyst pore with length L and radius r. The material balance for reactant A at steady state for a differential length dr of the catalyst pore is written as diffusion flux in - diffusion flux out - disappearance by reaction = 0... 

Because the diffusive flux is enhanced by this drift of a charge under the influence of the coulomb potential [as represented in eqn. (142)], the partially reflecting boundary condition (127) has to be modified to balance the rate of reaction of encounter pairs with the rate of formation of encounter pairs [eqn. (46)]. However, the rate of reaction of ion-pairs at encounter is usually extremely fast and the Smoluchowski condition, eqn. (5), is adequate. The initial and outer boundary conditions are the same as before [eqns. (131) and (128), respectively], representing on ion-pair absent until it is formed at time t0 and a negligibly small probability of finding the ion-pair with a separation r - ... 

Let us start with the generalized balance equations (2.1.37). The stochastic differential equations arise due to a formal adding of the fluctuating particle sources. In a general case both the fluctuations of the diffusion flux... 

The balance over the ith species (equation IV. 5) consists of contributions from diffusion, convection, and loss or production of the species in ng gas-phase reactions. The diffusion flux combines ordinary (concentration) and thermal diffusions according to the multicomponent diffusion equation (IV. 6) for an isobaric, ideal gas. Variations in the pressure induced by fluid mechanical forces are negligible in most CVD reactors therefore, pressure diffusion effects need not be considered. Forced diffusion of ions in an electrical field is important in plasma-enhanced CVD, as discussed by Hess and Graves (Chapter 8). 

This kinetic current balances a diffusion flux in the polymer ... 

Writting at the metal-polymer interface that the current / balances a diffusion flux ... 

The profile of Mg2+ in Figure 8.25 indicates downward diffusion of this constituent into the sediments. Mass balance calculations show that sufficient Mg2+ can diffuse into the sediments to account for the mass of organogenic dolomite formed in DSDP sediments (Baker and Bums, 1985 Compton and Siever, 1986). In areas of slow sedimentation rates, the diffusive flux of Mg2+ is high, and the pore waters have long residence times. Dolomites form under these conditions in the zone of sulfate reduction, are depleted in 13c, and have low trace element contents. With more rapid sedimentation rates, shallowly-buried sediments have shorter residence times, and dolomites with depleted 13C formed in the sulfate-reduction zone pass quickly into the underlying zone of methanogenesis. In this zone the DIC is enriched in 13C because of the overall reaction... 

The generic equations of balance are statements of truth, which is a priori self-evident and which must apply to all continuum materials regardless of their individual characteristics. Constitutive relations relate diffusive flux vectors to concentration gradients through phenomenological parameters called transport coefficients. They describe the detailed response characteristics of specific materials. There are seven generic principles (1) conservation of mass, (2) balance of linear momentum, (3) balance of ro-... 

The generic balance relations and the derived relations presented in the preceding section contain various diffusion flux tensors. Although the equation of continuity as presented does not contain a diffusion flux vector, were it to have been written for a multicomponent mixture, there would have been such a diffusion flux vector. Before any of these equations can be solved for the various field quantities, the diffusion fluxes must be related to gradients in the field potentials . 

We note first that immediately following the injection of a sample at the head of the channel, the flow of carrier is stopped briefly to allow time for the sample particles to accumulate near the appropriate wall. As the particles concentrate near the wall, the growing concentration gradient leads to a diffusive flux which counteracts the influx of particles. Because channel thickness is small (approximately 0.25 mm), these two mass transport processes quickly balance one another, leading to an equilibrium distribution near the accumulation wall. This distribution assumes the exponential form... 

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