Convective flux

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 ... 

For the region near the attachment point, Mullis found a strong effect of axial position on flux, but no satisfactory general correlation for this effect. In addition, he found no quantitative relation for the heat-transfer characteristics of jets directed toward the propellant surface. Under most conditions studied by Mullis, the radiation contribution is approximately 10% of the convective flux. The effects of solid-particle impingement were not investigated. 

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... 

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 ... 

It can easily be shown that for the upwind scheme all coefficients a appearing in Eq. (37) are positive [81]. Thus, no unphysical oscillatory solutions are foimd and stability problems with iterative equation solvers are usually avoided. The disadvantage of the upwind scheme is its low approximation order. The convective fluxes at the cell faces are only approximated up to corrections of order h, which leaves room for large errors on course grids. 

Besides the convective fluxes, the diffusive fluxes on the control volume faces have to be determined. As apparent from Eq. (33), an expression for dO/dsc containing the nodal values of O is needed. In the case of an orthogonal grid aligned with the axes of a Cartesian coordinate frame, the expression... 

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. 

It was shown in Section 1.8 that in addition to ion migration, diffusion and convection fluxes are a substantial part of mass transport during current flow through electrolyte solutions, securing a mass balance in the system. In the present chapter these processes are discussed in more detail. 

Convective transport is the transport of substances with a moving medium (e.g the transport of a solute in a liquid flow). The convective flux is given by... 

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 ... 

Let us estimate the ratios of diffusion and maximum convective fluxes, =... 

Solute flux within a pore can be modeled as the sum of hindered convection and hindered diffusion [Deen, AIChE33,1409 (1987)]. Diffusive transport is seen in dialysis and system start-up but is negligible for commercially practical operation. The steady-state solute convective flux in the pore is J, = KJc = where c is the radially... 

According to Faraday s law, the current passing through the electrode is equivalent to the material flux of electroactive substances. The disappearance of electroactive substances in the electrode reaction is considered as their transport through the electrode surface. Consequently, only diffusion and migration but not convection flux need be considered at the electrode surface, as the electrode is impenetrable to the solution components. 

The corresponding flux of x-momentum in the x direction is Fx/Ax = pVx. This x-momentum is also the driving force for convective transport of x-momentum in the — y direction (toward the wall), i.e., Tyx=Fx/Ay. Therefore, the convective flux of x-momentum from the fluid to the wall (or the stress exerted by the fluid on the wall) can be expressed as... 

The left-hand sideofEq. (40)isthe accumulationofparticlesofagivensize. The terms on the right-hand side are, in turn, the bulk flow into and out of the control volume, the convective flux along the size axis due to layering and attrition, the birth of new particles due to nucleation, and birth and death of granules due to coalescence. 

Fig. 25. Schematic representation of the Barton scheme for the convective flux of a quantity D by velocity Vi+i/2 in the v-direction.

PeL can be used in place of Dh as the single parameter of the axial dispersion model. The physical interpretation of PeL is that it represents the ratio of the convective flux to fee diffusive (disposed) flux ... 

One conclusion from these results is that the axial diffusion model begins to fail as Pe, - small, when an open boundary condition is used at the outlet. The case Pe, - small means increasing backmixing, or that the diffusive flux becomes increasingly significant compared with the convective flux. For an open boundary condition, it is also questionable whether the actual response C(e) can be identified with E(B). Furthermore, regardless of the boundary conditions chosen, it is difficult to envisage that cA... 

The general case of mass transfer includes both diffusion and convection. Hence, there are both diffusive flux and convective flux for a component. Therefore, the total flux is the sum of the two fluxes. For a given component in a binary and isotropic system, the total flux is... 

The diffusion equation will be developed by considering each term in equation (2.1) separately. In addition, the flux terms wiU be divided into diffusive and convective flux rates. 

The convective flux rate into our control volume is simply the chemical mass carried in by convection. If we consider the same box of Figure 2.3, except with a velocity component u in the x-direction, the convective flux rate into the box from the left-hand side is... 

Convective Flux Rates. We will deal with the convective and diffusive flux rates separately. They will eventually be separated in the final diffusion equation, and it is convenient to make that break now. The x-component of the convective flux rate is equal to the x-component of velocity, u, times the concentration, C, times the area of our box normal to the x-axis. Therefore, in terms of convective flux rates, equation (2.9a) becomes... 

Because the normal area. Ax = dy dz, of our box does not change with x it can be pulled out of the partial with respect to x. This is done in the second part of equation (2.10a). The same can be done with the y- and z-components of convective flux rate... 

Finally, adding equations (2.10a) to (2.10c) results in the total net convective flux rate ... 

Our solution is similar to Example 2.3, except that we have a convective flux, a porosity, and... 

Upwind differences are typical for convective flux, where the upstream concentration is important to determine the convective flux at the upstream interface. Upwind differences have a lower numerical diffusion than central differences... 

We will again consider the flux at the / -I-1/2 interface between the (/, j, k) control volume and the (/ -i-1, , k) control volume given in Figure 7.1. The convective flux, however, is given with equation (2.4) ... 

When convection delivers ions or molecules to react at an electrode, there is (in addition to any diffusion) a convective flux, and this quantity is given by Fick s first law. 

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