Diffusive fluxes

It is assumed that irreversible aggregation occurs on contact. The rate of coagulation is expressed as the aggregation flux J of particles towards a central particle. Using a steady-state approximation, the diffusive flux is derived to be... 

C3.6.13 where large diffusion fluxes are indicated by —> and smaller diffusion fluxes by —+. For tire part of tire B front tliat protmdes into tire A region, fast diffusion of B leads to dispersal of B and suppresses tire autocatalytic reaction tliat requires two molecules of B. The front will have difficulty advancing here. In tire region where A protmdes into B, A will react leading to advancement of tire front. The net effect is to remove any initial nonplanarity and give rise to a planar front. 

Now consider Che cross-sectional average N of the total molar flux and the cross-sectional average diffusion fluxes J, defined by... 

But from the definition of the diffusion fluxes given above it follows immediately that... 

In Chapter 4 we introduced the total flux N and the diffusion fluxes... 

This determines the diffusion fluxes at the limit of bulk, diffusion control. 

This is an explicit solution of the Stefan-Maxwell equations for the diffusion fluxes. The species flux vectors are then given by... 

The incorporation of surface diffusion into a model of transport in a porous medium is quite straightforward, since the surface diffusion fluxes simply combine additively with the diffusion fluxes in the gaseous phase. 

Since -D c is the diffusion flux which would be carried by the cnicro-... 

The diffusion flux for substance 2 then follows from equation (9.14), using equation (9.20) ... 

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... 

The tme driving force for any diffusive transport process is the gradient of chemical potential rather than the gradient of concentration. This distinction is not important in dilute systems where thermodynamically ideal behavior is approached. However, it becomes important at higher concentration levels and in micropore and surface diffusion. To a first approximation the expression for the diffusive flux may be written... 

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 ... 

In these cases, the diffusion flux may be written in terms of the adsorbed solute concentration as... 

Diffusion in the bulk crystals may sometimes be short circuited by diffusion down grain boundaries or dislocation cores. The boundary acts as a planar channel, about two atoms wide, with a local diffusion rate which can be as much as 10 times greater than in the bulk (Figs. 18.8 and 10.4). The dislocation core, too, can act as a high conductivity wire of cross-section about (2b), where b is the atom size (Fig. 18.9). Of course, their contribution to the total diffusive flux depends also on how many grain boundaries or dislocations there are when grains are small or dislocations numerous, their contribution becomes important. 

The flux is the sum of the diffusive flux with respect to the center of mass plus the contribution to the flux of A caused hy the hulk mass movement. 

The units on A are mol/(m s). This is the convective flux. The student of mass transfer will recognize that a diffusion term like —3>Adaldz is usually included in the flux. This term is the diffusive flux and is zero for piston flow. The design equation for the variable-density, variable-cross-section PFR can be written as... 

Laminar flow reactors have concentration and temperature gradients in both the radial and axial directions. The radial gradient normally has a much greater effect on reactor performance. The diffusive flux is a vector that depends on concentration gradients. The flux in the axial direction is... 

This section derives a simple version of the convective diffusion equation, applicable to tubular reactors with a one-dimensional velocity profile V (r). The starting point is Equation (1.4) applied to the differential volume element shown in Figure 8.9. The volume element is located at point (r, z) and is in the shape of a ring. Note that 0-dependence is ignored so that the results will not be applicable to systems with significant natural convection. Also, convection due to is neglected. Component A is transported by radial and axial diffusion and by axial convection. The diffusive flux is governed by Pick s law. 

Hint Use a version of Equation (11.49) but correct for the spherical geometry and replace the convective flux with a diffusive flux. Example 11.14 assumed piston flow when treating the moving-front phenomenon in an ion-exchange column. Expand the solution to include an axial dispersion term. How should breakthrough be defined in this case The transition from Equation (11.50) to Equation (11.51) seems to require the step that dVsIAi =d Vs/Ai] = dzs- This is not correct in general. Is the validity of Equation (11.51) hmited to situations where Ai is actually constant ... 

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