Diffusion-convection process steady-state

In the actual situation, both convective transfer of A to the film surface and diffusion of A through the film occur serially. At steady state these processes take place at equal rates, so that... 

For it to be useful, we need to couple Pick s law with mass balances. The first case considered is steady-state diffusion with no convection in the direction of diffusion. This is an inportant practical case for measuring diffusion coefficients, studying steady-state evaporation and steady-state permeation of gases and liquids in membranes, and in design of distillation and some other separation processes. The second case we consider is unsteady diffusion with no convection in the direction of diffusion, which is of practical significance in controlled-release drug delivery and in some batch reactors and separation processes. 

Steady-State Diffusion—Convection Process of Reactant 54... 

In a similar way for Eqns (2.55) and (2.56), the current density for the steady-state diffusion—convection process of oxidant in Reaction (2-II) (in.o) can be expressed as ... 

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 ... 

It is a typical feature of the diffusion processes at electrodes of small size, which are reached by converging diffusion fluxes, that a steady state can be attained even without convection (e.g., in gelled solutions). Such electrodes, which have dimensions comparable to typical values of 8, are called microelectrodes. 

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... 

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... 

Diffusion. The transport process may consist of two parts, diffusion and convection. When the liquid is stagnant and resting relative to the particle the transport is done by diffusion only. A steady state is quickly established in the solution around the particle (4 ). (Strictly it is a quasi-steady state since the particle is growing ( 5)). At the particle surface the concentration gradient becomes equal to (c-cs)/r, which leads to the growth rate... 

After the electrode reaction starts at a potential close to E°, the concentrations of both O and R in a thin layer of solution next to the electrode become different from those in the bulk, cQ and cR. This layer is known as the diffusion layer. Beyond the diffusion layer, the solution is maintained uniform by natural or forced convection. When the reaction continues, the diffusion layer s thickness, /, increases with time until it reaches a steady-state value. This behaviour is also known as the relaxation process and accounts for many features of a voltammogram. Besides the electrode potential, equations (A.3) and (A.4) show that the electrode current output is proportional to the concentration gradient dcourfa /dx or dcRrface/dx. If the concentration distribution in the diffusion layer is almost linear, which is true under a steady state, these gradients can be qualitatively approximated by equation (A.5). 

Next one needs an expression for (xA — xA"). The difference in concentration between the two streams results from two effects thermal diffusion, which tends to increase the concentration difference, and convection, which tends to decrease it. Each of these effects is considered separately by obtaining an approximate integrated form of the steady state equation of continuity as applied to that particular process. If the only effect tending to produce a concentration difference were thermal diffusion, then according to Eq. (131) dxA/dx = — (kT/T)(d,T/dx) this expression may be written in difference form over the distance from x = — ( 4)a to x — + (M)° thus ... 

The study of rotating disk electrode behavior provides a unique opportunity to develop a model that predicts the effect of diffusion and convection on the current. This is one of the few convective systems that have simple hydrodynamic equations that may be combined with the diffusion model developed herein to produce meaningful results. The effect of diffusion is modeled exactly as it has been done previously. The effect of convection is treated by integrating an approximate velocity equation to determine the extent of convective flow during a given At interval. Matter, then, is simply transferred from volume element to volume element in accord with this result to simulate convection. The whole process repeated results in a steady-state concentration profile and a steady-state representation of the current (the Levich equation). 

If the activity of the immobilised catalyst is sufficiently high, the reaction which it mediates occurs essentially at the interface between the catalyst and the substrate solution. In the case of the surface immobilised enzyme or a thin microbial film this will, of course, occur irrespective of the level of activity. Under these conditions the limiting process for transporting substrate from the bulk of the solution to the immobilised enzyme is molecular or convective diffusion through the layer of solution immediate to the carrier. Under steady-state conditions, the rate of reaction at the active sites is equal to the rate at which substrate arrives at the site. This... 

Let us consider the burning of an ideal spherical particle in static gas. The oxidant diffuses to the surface of the particle to react with the carbon C + CO2, while the latter diffuses out from the surface of the particle. The combustion heat is transferred to the surrounding gas partially by convection and partially by radiation. The following assumptions were made in the modeling (1) The process is at a pseudo steady state. (2) The temperature the highest at the surface, and continuously drops down outwards from the surface of the particle and the concentration of oxidant is highest in the bulk... 

Diffusion time (diffusion time constant) — This parameter appears in numerous problems of - diffusion, diffusion-migration, or convective diffusion (- diffusion, subentry -> convective diffusion) of an electroactive species inside solution or a solid phase and means a characteristic time interval for the process to approach an equilibrium or a steady state after a perturbation, e.g., a stepwise change of the electrode potential. For onedimensional transport across a uniform layer of thickness L the diffusion time constant, iq, is of the order of L2/D (D, -> diffusion coefficient of the rate-determining species). For spherical diffusion (inside a spherical volume or in the solution to the surface of a spherical electrode) r spherical diffusion). The same expression is valid for hemispherical diffusion in a half-space (occupied by a solution or another conducting medium) to the surface of a disk electrode, R being the disk radius (-> diffusion, subentry -> hemispherical diffusion). For the relaxation of the concentration profile after an electrical perturbation (e.g., a potential step) Tj = L /D LD being - diffusion layer thickness in steady-state conditions. All these expressions can be derived from the qualitative estimate of the thickness of the nonstationary layer... 

A reversible one-electron transfer process (19) is initially examined. For all forms of hydrodynamic electrode, material reaches the electrode via diffusion and convection. In the cases of the RDE and ChE under steady-state conditions, solutions to the mass transport equations are combined with the Nernst equation to obtain the reversible response shown in Fig. 26. A sigmoidal-shaped voltammogram is obtained, in contrast to the peak-shaped voltammetric response obtained in cyclic voltammetry. 

Catalyst deactivation in large-pore slab catalysts, where intrapaiticle convection, diffusion and first order reaction are the competing processes, is analyzed by uniform and shell-progressive models. Analytical solutions arc provid as well as plots of effectiveness factors as a function of model parameters as a basis for steady-state reactor design. 

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 ... 

The transport of surfactant on the interface is governed by the sequential process of diffusion to the interface, followed by adsorption-desorption, and then transport of the adsorbed surfactant on the interface by a combination of convection and diffusion. The latter process is described by Eq. (2-164), but now for steady-state and for a fixed shape. We express the flux to the interface in terms of the net diffusive flux ... 

It was already indicated in the preceding sections that this thermal boundary-layer structure does not occur when a particle (or body) is entirely surrounded by closed streamlines (or closed stream surfaces). In this case, the convection process near the body can no longer transfer heat directly to the streaming flow where it is carried into the wake, but instead circulates it only in a closed path around the body. Thus the heat transfer process is fundamentally altered, because heat can escape from the body only by diffusing slowly across the region of closed streamlines (or stream surfaces). Because the size of this region is independent of Pe, the steady-state temperature gradients will be 0(1), and we expect that... 

Transport. The mechanisms responsible for transport are considered to be both physical (convection or mass flow) and chemical (diffusion). When considered simultaneously, these processes have been summarized in the convective-dispersive, or miscible displacement, equation. For a non-interacting solute (such as chloride) under steady state water flow conditions in a homogeneous soil, this equation can be written as (10) ... 

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