Convection steady state

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... 

A simple example is the rotating disc electrode described in detail in chapter 2. The horizontal spinning disc draws liquid up and then flings it out sideways, creating a continuous but steady-state convection pattern. If the distance down from the disc is denoted by z and the distance across the disc surface by the radial distance, r, then it is not difficult to show that ... 

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 ... 

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... 

At time th the heat flow by conduction matches the steady-state convective heat transfer from the body. If conditions are met to initiate convection before t, (this will depend on Pr and Ra), the heat flow falls monotonically in the transition regime, as along path A in Fig. 4.37b. Otherwise convection will not be initiated until to > t so the heat flow will have fallen below the steady-state value and must therefore recover from the undershoot in the transition regime as shown by path B. 

The equations governing the fluid motion and heat transfer in these quasi-steady regimes are (if the property values and boundary conditions are the same) identical to those for steady-state convection in the same geometry with a uniform internal generation of energy [73]. The heat transfer equations from one situation can therefore be readily transferred to the other by replacing the constant pcp dT/dt by the internal generation rate q " (in W/m3 or Btu/h-ft3). 

At the turn of the century, Henri Benard, a young French physicist, published the first truly systematic study of natural convection in a horizontal fluid layer (B4, B5, B6). In a horizontal liquid layer heated from below B6nard, sought to measure and to define the most stable steady-state convection currents prevailing under given conditions. He utilized liquid layers only a few millimeters in thickness, initially in an apparatus giving a free upper surface, and of considerable horizontal extent (about 20 cm) so that edge effects could not influence the form of the convection pattern. For these studies. 

---