Diffusion-convection transport

In the film-penetration model (H19), it is assumed that the reactant A penetrates through the surface element by one-dimensional unsteady-state molecular diffusion. Convective transport is assumed to be insignificant. The diffusing stream of the reactant A is depleted along the path of diffusion by its reversible reaction with the reactant B, which is an existing component of the liquid surface element. If such a reaction can be represented as... 

If the integration reaction is very fast or the rate corrstant oo, the growth rate is controlled by the diffusive/convective transport of elementary units. With (c-Cj) = (c- c ) = Ac the molar flux density vahd for small rates is given by... 

Uitto OD, White HS, Aoki K (2002) Diffusive-convective transport into a porous membrane. A comparison of theory and experiment using scanning electrochemical microscopy operated in reverse imaging mode. Anal Chem 74(17) 4577-4582. doi 10.1021/ac0256538... 

We examine the problem of diffusion in a porous medium using a homogenization analysis (HA). Diffusion problems have important applications in environmental geosciences. We clarily the mechanism of diffusion, convective transport and adsorption in porous media at both the microscale and macroscale levels. Attention is particularly focused on diffusion processes in bentonite, which is an engineered geological barrier to be used to buffer the transport of radionuclides from deep geologic repositories. 

Macpherson et al. have used SECM to study molecular transport in laryngeal cartilage. Cartilage acts as a load-bearing material in the human body. Understanding transport mechanisms in this material is of interest for several reasons. First, the rate at which water is redistributed under applied loads determines the viscoelastic properties of the material. Second, the physiological functioning of the material is determined by the diffusive-convective transport of nutrients and metabolites. ... 

The distribution of the reactant concentrations along the electrodes is needed to calculate the transfer currents in electrochemical cells. The concentrations along the gas channels/gas diffusers/catalyst layers vary because of diffusion-convection transport and electrokinetics in the catalyst layers. These distributions depend therefore on the gas and medium... 

The concentration boundary layer forms because of the convective transport of solutes toward the membrane due to the viscous drag exerted by the flux. A diffusive back-transport is produced by the concentration gradient between the membranes surface and the bulk. At equiUbrium the two transport mechanisms are equal to each other. Solving the equations leads to an expression of the flux ... 

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 ... 

Exceptions to these rules of thumb abound, however. For example, although diffusion is faster at elevated temperatures, dissolved oxygen concentration may be lower. Convective transport is stimulated by high temperatures, but increased... 

Eventually oxygen becomes depleted in the crevice. Replenishment of oxygen by convective transport cannot occur since the crevice is too tight to allow water to move in and out freely. Also, oxygen diffusion... 

Combining hindered diffusion theory with the diffusion/convection problem in the model pore, Trinh et al. [399] showed how the effective transport coefficients depend upon the ratio of the solute to pore size. Figure 28 shows that as the ratio of solute to pore size approaches unity, the effective mobility function becomes very steep, thus indicating that the resolution in the separation will be enhanced for molecules with size close to the size of the pore. Similar results were found for the effective dispersion, and the implications for the separation of various sizes of molecules were discussed by Trinh et al. [399]. 

Trinh, S Locke, BR Arce, P, Diffusive-Convective and Diffusive-Elechoconvective Transport in Non-Uniform Channels with Application to Macromolecular Separations, Separation and Purification Technology 15, 255, 1999. 

Similar convection-diffusion equations to the Navier-Stokes equation can be formulated for enthalpy or species concentration. In all of these formulations there is always a superposition of diffusive and convective transport of a field quantity, supplemented by source terms describing creation or destruction of the transported quantity. There are two fundamental assumptions on which the Navier-Stokes and other convection-diffusion equations are based. The first and most fundamental is the continuum hypothesis it is assumed that the fluid can be described by a scalar or vector field, such as density or velocity. In fact, the field quantities have to be regarded as local averages over a large number of particles contained in a volume element embracing the point of interest. The second hypothesis relates to the local statistical distribution of the particles in phase space the standard convection-diffusion equations rely on the assumption of local thermal equilibrium. For gas flow, this means that a Maxwell-Boltzmann distribution is assumed for the velocity of the particles in the frame-of-reference co-moving with the fluid. Especially the second assumption may break dovm when gas flow at high temperature or low pressure in micro channels is considered, as will be discussed below. 

The governing equation for the transport of heat contains convective and diffusive contributions. Inside the fluid phase, convective transport often dominates. Within... 

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 ... 

The Peclet number Pe = v /Dc, where Dc is the diffusion coefficient of a solute particle in the fluid, measures the ratio of convective transport to diffusive transport. The diffusion time Tp = 2/D< is the time it takes a particle with characteristic length to diffuse a distance comparable to its size. We may then write the Peclet number as Pe = xD/xs, where xv is again the Stokes time. For Pe > 1 the particle will move convectively over distances greater than its size. The Peclet number can also be written Pe = Re(v/Dc), so in MPC simulations the extent to which this number can be tuned depends on the Reynolds number and the ratio of the kinematic viscosity and the particle diffusion coefficient. 

Two examples will now be given of solution of the convective diffusion problem, transport to a rotating disk as a stationary case and transport to a growing sphere as a transient case. Finally, an engineering approach will be mentioned in which the solution is expressed as a function of dimensionless quantities characterizing the properties of the system. 

The mass-transfer correlation obtained by Bohm et al. (B9), Eq. (33), in Table VII, is conspicuous for its remarkably high exponent (0.85) on the GrSc product. Since the current is almost independent of diffusivity, this must mean that the reacting ion is depleted at the downstream end of the narrow slit between the cathode and diaphragm. The total current then is determined largely by the convective transport of reactant into the slit, which, in turn, depends on the density difference but not on diffusivity. 

In this equation the entire exterior surface of the catalyst is assumed to be uniformly accessible. Because equimolar counterdiffusion takes place for stoichiometry of the form of equation 12.4.18, there is no net molar transport normal to the surface. Hence there is no convective transport contribution to equation 12.4.21. Let us now consider two limiting conditions for steady-state operation. First, suppose that the intrinsic reaction as modified by intraparticle diffusion effects is extremely rapid. In this case PA ES will approach zero, and equation 12.4.21 indicates that the observed rate per unit mass of catalyst becomes... 

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