Diffusion steady-state, convection

H. Steady-State Convective Diffusion with Simultaneous First-Order Irreversible... 

As an alternative to film models, McNamara and Amidon [6] included convection, or mass transfer via fluid flow, into the general solid dissolution and reaction modeling scheme. The idea was to recognize that diffusion was not the only process by which mass could be transferred from the solid surface through the boundary layer [7], McNamara and Amidon constructed a set of steady-state convective diffusion continuity equations such as... 

Valdes models the electro-deposition of Brownian particles on a RDE, by solving the steady-state convective diffusion equation ... 

In many interfacial conversion processes, certainly those at biological interphases, the diffusion situation is complicated by the fact that the concentration at the organism surface is not constant with time (c°(f) not constant). However, in most cases of steady-state convective diffusion, the changes in the surface... 

The steady-state convective dissolution rate calculated above applies only when the unperturbed diffusion distance is greater than the boundary... 

Fo being the Fourier number and d the diameter of the disk. The mass transfer coefficient k can be considered as interpolating between the steady-state convective diffusion at large times (t - oo) and unsteady-state diffusion at short times (t — 0 and v = 0). The constants A and B of Eq. (147) follow from the solutions for these two limiting cases. For these two limiting cases... 

What is the quantitative relationship between the steady state, convection-with-diffusion current density and the potential difference across the interface How is the steady-state potential difference at a steady current density related to the zero-current, or equilibrium, potential difference These questions are the relevant ones for steady passage of current in convection-aided situations. 

As is thoroughly discussed in Chap. 2 of this volume, the convective diffusion conditions can be controlled under steady state conditions by use of hydrodynamic electrodes such as the rotating disc electrode (RDE), the wall-jet electrode, etc. In these cases, steady state convective diffusion is attained, becomes independent of time, and solution of the convective-diffusion differential equation for the particular electrochemical problem permits separation of transport and kinetics from the experimental data. 

The concentration polarization model, which is based on the stagnant fihn theory, was developed to describe the back-diffusion phenomenon during filtration of macromolecules. In this model, the rejection of particles gives rise to a thin fouling layer on the membrane surface, overlaid by a concentration polarization layer in which particles diffuse away from the membrane surface, where solute concentration is high, to the bulk phase, where the solute concentration is low [158]. At steady state, convection of particles toward the membrane surface is balanced by diffusion away from the membrane. Thus, integrating the onedimensional convective-diffusion equation across the concentration polarization layer gives... 

Theoretical developments show that it is possible to deduce hydrodynamic information from the limiting current measiuement, either in quasi-steady state where /(f) cx py t) or, at higher frequency, in terms of spectral analysis. In the latter case, it is possible to obtain the velocity spectra from the mass-transfer spectra, where the transfer function between the mass-transfer rate and the velocity perturbation is known. However, in most cases, charge transfer is not infinitely fast, and the analysis also requires knowledge of the convective-diffusion impedance, i.e., the transfer function between a concentration modulation at the interface and the resulting flux of meiss under steady-state convection. 

In order to study properties of the schemes, we principally consider the steady state convection and diffusion of a property with a source term in a one dimensional domain as sketched Fig 12.3 using a staggered grid for the velocity components so that the rr-velocity components are located at the w and e GCV faces. Preliminary, we assume that the velocity is constant and constant fluid properties. The convective and diffusive processes are then governed by a balance equation of the form ... 

Step 1 It is easy to add the convective diffusion equation by starting with the fluid flow model of the T-sensor. Choose Multiphysics/Model Navigator. A window appears with a hst of possible equations. Scroll down and select Chemical Engineering Module/Mass Balance/Convection and Diffusion/Steady-state Analysis. Click Add. Now FEMLAB will solve both equations. 

Let us investigate steady-state convective diffusion on the surface of a flat plate in a longitudinal translational flow of a viscous incompressible fluid at high Reynolds numbers (the Blasius flow). We assume that mass transfer is accompanied by a volume reaction. In the diffusion boundary layer approximation, the concentration distribution is described by the equation... 

In view of this, let us consider steady-state convective mass transfer to a solid particle, drop, or bubble for an arbitrary dependence D = D(C) of the diffusion coefficient on concentration. We assume that the concentrations at the particle surface and remote from it are uniform and are equal to Cs and C, respectively (Cs Ci). We also assume that the concentration nonuniformity does not affect the flow parameters. In dimensionless variables, this nonlinear problem is described by the equation... 

We now turn to the second criterion, in particular bearing in mind the criticism, alluded to above, about the difficulty associated with the theoretical description of processes at non-uniformly accessible electrodes. Again, we will compare and contrast the channel electrode and the RDE. Now the theoretical description of electrode reactions involves, typically, the solution of perhaps several coupled steady-state convective-diffusion equations of the form... 

We next consider the behaviour for DISP1 reactions at channel electrodes. The normalised steady-state convective-diffusion equations for this case, under the Levich approximation, are... 

The dimensionless steady-state convective-diffusion equation requiring solution is... 

This could be, for example, the RRDE illustrated in Figure 9.4.1 with the disk electrode at open circuit. When this electrode is rotated at an angular velocity, co, the solution flow velocity profile is that discussed in Section 9.3.1. The steady-state convective-diffusion equation that must be solved is... 

However, most transport processes take place under steady-state convective diffusion, driven by a (linear) concentration gradient. This results in... 

The computational model assumes the mixing process of one species dissolved in water without chemical reactions occurring during the mixing and thus without the reaction term included in the species equation. Therefore, only the steady-state convection-diffusion equation for species A has to be considered as... 

The scan rate, u = EIAt, plays a very important role in sweep voltannnetry as it defines the time scale of the experiment and is typically in the range 5 mV s to 100 V s for nonnal macroelectrodes, although sweep rates of 10 V s are possible with microelectrodes (see later). The short time scales in which the experiments are carried out are the cause for the prevalence of non-steady-state diflfiision and the peak-shaped response. Wlien the scan rate is slow enough to maintain steady-state diflfiision, the concentration profiles with time are linear within the Nemst diflfiision layer which is fixed by natural convection, and the current-potential response reaches a plateau steady-state current. On reducing the time scale, the diflfiision layer caimot relax to its equilibrium state, the diffusion layer is thiimer and hence the currents in the non-steady-state will be higher. 

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 ... 

The net transport of component A in the +2 direction in the centrifuge is equal to the sum of the convective transport and the axial diffusive transport. At the steady state the net transport of component A toward the product withdrawal point must be equal to the rate at which component A is being withdrawn from the top of the centrifuge. Thus, the transport of component is given by equation 72 ... 

Note Equation (4.241) characterizes diffusion when the mixture element is in steady state with no turbulence. Diffusion in a pipe can be represented by Eq. (4.241) in convective mass transfer the flow and turbulence are important. 

Transport of a species in solution to and from an electrode/solution interface may occur by migration, diffusion and convection although in any specific system they will not necessarily be of equal importance. However, at the steady state all steps involved in the electrode reaction must proceed at the same rate, irrespective of whether the rate is controlled by a slow step in the charge transfer process or by the rate of transport to or from the electrode surface. It follows that the rate of transport must equal the rate of charge transfer ... 

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