Diffusion layer steady transport through

The concentration at the boundary between the diffuse layer and diffusion layer, c(k , t), is unknown. To express this concentration in terms of the surfactant bulk concentration c the Eq. (7.28) describing the steady transport through the diffusion layer can be used. This leads to... 

In field-flow fractionation, a component undergoing flow transport through a thin channel is forced sideways against a wall by an applied field or gradient. The component is confined to a narrow region adjacent to the wall, by a combination of the wall s surface, which it cannot pass, and the driving force, which prevents its escape toward the center of the channel. The component molecules or particles soon establish a thin steady-state distribution in which outward diffusion balances the steady inward drift due to the field. The structure and dimensions of this layer determine its behavior in the separation process. 

Example Particles are transported through a thin layer of stationary gas to (he surface of a horizontal Hat plate. Derive an expression for the deposition rate in the steady state if diffusion and sedimentation are both operative. 

The Retardation of the Steady Transport of Adsorbing Ions Through the Diffuse Part of Double Layer... 

The model presented here is a comprehensive full three-dimensional, non-isothermal, singlephase, steady-state model that resolves coupled transport processes in the membrane, eatalyst layer, gas diffusion eleetrodes and reactant flow channels of a PEM fuel cell. This model accounts for a distributed over potential at the catalyst layer as well as in the membrane and gas diffusion electrodes. The model features an algorithm that allows for a more realistie representation of the loeal activation overpotentials which leads to improved prediction of the local current density distribution. This model also takes into aeeount convection and diffusion of different species in the channels as well as in the porous gas diffusion layer, heat transfer in the solids as well as in the gases, electrochemical reactions and the transport of water through the membrane. 

The external diffusion limitation (EDL) indicates that substrate transport through diffusion (stagnant) layer is a rate-limiting process [10]. At internal diffusion limitation (IDL), the substrate diffusion through the external diffusion layer is fast, and process is limited by the diffusion inside an enzyme membrane. The disadvantage of these approximate solutions is an error at the boundaries between the different approximate treatments. It is helpful to illustrate this approach by reference to a trivial problem of the substrate conversion in the biocatalytical membrane of the biosensor and at the concentration of the substrate less than the Michaelis-Menten constant Km)- The calculated profile of substrate concentration at steady-state or stationary conditions is shown in Fig. 1. 

As it has been discussed above, MEs display common features in the response of a potential step experiment, regardless of the geometry. At short times, where the diffusion layer is thin compared to the critical dimension of the ME, the current can be predicted by the Cottrell equation and planar diffusion is prevailing. At long times, where the diffusion layer is large compared to the critical dimension, the current is steady state or quasi-steady state. Under the latter conditions, the current at the MEs is related to the mass transport coefficient, Mq, through... 

A situation which is frequently encountered in the production of microelectronic devices is when vapour deposition must be made into a re-entrant cavity in an otherwise planar surface. Clearly, the gas velocity of the major transporting gas must be reduced in the gas phase entering the cavity, and transport down the cavity will be mainly by diffusion. If the mainstream gas velocity is high, there exists the possibility of turbulent flow at the mouth of the cavity, but since this is rare in vapour deposition processes, the assumption that the gas within the cavity is stagnant is a good approximation. The appropriate solution of the diffusion equation for the steady-state transport of material through the stagnant layer in the cavity is... 

In the model, the internal structure of the root is described as three concentric cylinders corresponding to the central stele, the cortex and the wall layers. Diffu-sivities and respiration rates differ in the different tissues. The model allows for the axial diffusion of O2 through the cortical gas spaces, radial diffusion into the root tissues, and simultaneous consumption in respiration and loss to the soil. A steady state is assumed, in which the flux of O2 across the root base equals the net consumption in root respiration and loss to the soil. This is realistic because root elongation is in general slow compared with gas transport. The basic equation is... 

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... 

Several definitive experiments have shown that during the thermal oxidation of silicon the oxidizing species (some form of oxygen) diffuses through the oxide layer and reacts with the silicon to produce more oxide at the Si-Si02 interface (70, 71). For oxidation to occur, three consecutive oxidant fluxes must exist (1) transport from the furnace ambient to the outer oxide surface, (2) diffusion through the oxide layer of thickness x0, and (3) reaction with silicon at the interface. At steady state, all three fluxes are equal. 

The salt flux through the membrane is given by the product of the permeate volume flux. /,. and the permeate salt concentration c,p. For dilute liquids the permeate volume flux is within 1 or 2% of the volume flux on the feed side of the membrane because the densities of the two solutions are almost equal. This means that, at steady state, the net salt flux at any point within the boundary layer must also be equal to the permeate salt flux Jvcip. In the boundary layer this net salt flux is also equal to the convective salt flux towards the membrane Jvc, minus the diffusive salt flux away from the membrane expressed by Fick s law (Didcildx). So, from simple mass balance, transport of salt at any point within the boundary layer can be described by the equation... 

In the fourth subtechnique, flow FFF (F/FFF), an external field, as such, is not used. Its place is taken by a slow transverse flow of the carrier liquid. In the usual case carrier permeates into the channel through the top wall (a layer of porous frit), moves slowly across the thin channel space, and seeps out of a membrane-frit bilayer constituting the bottom (accumulation) wall. This slow transverse flow is superimposed on the much faster down-channel flow. We emphasized in Section 7.4 that flow provides a transport mechanism much like that of an external field hence the substitution of transverse flow for a transverse (perpendicular) field is feasible. However this transverse flow—crossflow as we call it—is not by itself selective (see Section 7.4) different particle types are all transported toward the accumulation wall at the same rate. Nonetheless the thickness of the steady-state layer of particles formed at the accumulation wall is variable, determined by a combination of the crossflow transport which forms the layer and by diffusion which breaks it down. Since diffusion coefficients vary from species to species, exponential distributions of different thicknesses are formed, leading to normal FFF separation. 

Diffusion through a barrier layer is a special case of diffusion through a laminate film composed of several layers with different thicknesses and diffusion coefficients. The mathematical treatment of the non-steady state case is complicated. The steady state permeation case allows the overall transport to be simply treated. Let n films with thicknesses dpi, dp2,..., dpn with corresponding diffusion coefficients DP1, Dp2,..., Dpn be bound together in a laminate. Because in steady state, the flux J of the diffusing substance i is the same through every individual component of the laminate, one obtains an expression for the concentration gradient ... 

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