Mass transport concentration profile

To ensure that the detector electrode used in MEMED is a noninvasive probe of the concentration boundary layer that develops adjacent to the droplet, it is usually necessary to employ a small-sized UME (less than 2 /rm diameter). This is essential for amperometric detection protocols, although larger electrodes, up to 50/rm across, can be employed in potentiometric detection mode [73]. A key strength of the technique is that the electrode measures directly the concentration profile of a target species involved in the reaction at the interface, i.e., the spatial distribution of a product or reactant, on the receptor phase side. The shape of this concentration profile is sensitive to the mass transport characteristics for the growing drop, and to the interfacial reaction kinetics. A schematic of the apparatus for MEMED is shown in Fig. 14. 

The nature of mass transport in MEMED has been confirmed with both ampero-metric and potentiometric studies of bromine transfer from an aqueous phase to DCE [79]. Figure 17 shows typical amperometric data for this case, in which the DCE phase acts as a sink for Br2, and a depleted region of Br2 is measured adjacent to the droplet in the aqueous phase. Video images are also provided, which correspond to particular times during the amperometric transient at position (3) the edge of the developing concentration boundary layer, around the drop, reaches the electrode the concentration profile is then mapped out between points (3) and (4). The measured current, i, can be related to the local concentration, c, via... 

Ordinary or bulk diffusion is primarily responsible for molecular transport when the mean free path of a molecule is small compared with the diameter of the pore. At 1 atm the mean free path of typical gaseous species is of the order of 10 5 cm or 103 A. In pores larger than 1CT4 cm the mean free path is much smaller than the pore dimension, and collisions with other gas phase molecules will occur much more often than collisions with the pore walls. Under these circumstances the effective diffusivity will be independent of the pore diameter and, within a given catalyst pore, ordinary bulk diffusion coefficients may be used in Fick s first law to evaluate the rate of mass transfer and the concentration profile in the pore. In industrial practice there are three general classes of reaction conditions for which the bulk value of the diffusion coefficient is appropriate. For all catalysts these include liquid phase reactions... 

When the two liquid phases are in relative motion, the mass transfer coefficients in either phase must be related to the dynamical properties of the liquids. The boundary layer thicknesses are related to the Reynolds number, and the diffusive transfer to the Schmidt number. Another complication is that such a boundary cannot in many circumstances be regarded as a simple planar interface, but eddies of material are transported to the interface from the bulk of each liquid which change the concentration profile normal to the interface. In the simple isothermal model there is no need to take account of this fact, but in most industrial circumstances the two liquids are not in an isothermal system, but in one in which there is a temperature gradient. The simple stationary mass transfer model must therefore be replaced by an eddy mass transfer which takes account of this surface replenishment. 

As suggested before, the role of the interphasial double layer is insignificant in many transport processes that are involved with the supply of components from the bulk of the medium towards the biosurface. The thickness of the electric double layer is so small compared with that of the diffusion layer 8 that the very local deformation of the concentration profiles does not really alter the flux. Hence, in most analyses of diffusive mass transport one does not find any electric double layer terms. For the kinetics of the interphasial processes, this is completely different. Rate constants for chemical reactions or permeation steps are usually heavily dependent on the local conditions. Like in electrochemical processes, two elements are of great importance the local electric field which affects rates of transfer of charged species (the actual potential comes into play in the case of redox reactions), and the local activities... 

Figure 2. Development of elemental concentration-distance profiles as a function of time for a mass-transport-limiting situation, (a) Diffusion to a large (r0 > <5) organism, (b) Diffusion to a small (ro

The microstructure of a catalyst layer is mainly determined by its composition and the fabrication method. Many attempts have been made to optimize pore size, pore distribution, and pore structure for better mass transport. Liu and Wang [141] found that a CL structure with a higher porosity near the GDL was beneficial for O2 transport and water removal. A CL with a stepwise porosity distribution, a higher porosity near the GDL, and a lower porosity near the membrane could perform better than one with a uniform porosity distribution. This pore structure led to better O2 distribution in the GL and extended the reaction zone toward the GDL side. The position of macropores also played an important role in proton conduction and oxygen transport within the CL, due to favorable proton and oxygen concentration conduction profiles. 

Important hints on the reaction site can be gained by the Hatta numbers (Ha) of mass transport at the G/L- and L/L-phase boundaries. These numbers are also essential in order to estimate mass transport rates and concentration profiles within the boundary layer. Since the main resistance of mass transport is in the aqueous phase, mass transport coefficients and Ha numbers mentioned in the text are related to the aqueous phase. 

We will try our hand at applying the diffusion equation to a couple of mass transport problems. The first is the diffusive transport of oxygen into lake sediments and the use of oxygen by the bacteria to result in a steady-state oxygen concentration profile. The second is an unsteady solution of a spill into the groundwater table. 

There will also be a temporal mean concentration. If there is a source or sink in the flow, or transport across the boundaries as in Figure 5.1, then the temporal mean concentration profile will eventually reach a value such as that given in Figure 5.1. This flux of compound seems to be from the bottom toward the top of the flow. Superimposed on this temporal mean concentration profile will be short-term variations in concentration caused by turbulent transport. The concentration profile is flatter in the middle of the flow because the large turbulent eddies that transport mass quickly are not as constrained by the flow boundaries in this region. Now, if we put a concentration-velocity probe into the flow at one location, the two traces of velocity and concentration versus time would look something like that shown in Figure 5.2. 

In the ideal plug flow reactor, the flow traverses through the reactor hke a plug, with a uniform velocity profile and no diffusion in the longitudinal direction, as illustrated in Figure 6.2. A nonreactive tracer would travel through the reactor and leave with the same concentration versus time curve, except later. The mass transport equation is... 

As noted previously, most environmental flows are turbulent. The diffusive sublayer, where only diffusion acts to transport mass and the concentration profile is linear, is typically between 10 /xm and 1 mm thick. Measurements within this sublayer are not usually feasible. Thus, the interfacial flux is typically expresses as a bulk transfer... 

C. Mass Transport and Reactant Concentration Profiles through the Rod. 182... 

B. Derivation of Equations for Reactant Concentration Profile THROUGH Carbon Rods Depending upon Type of Mass Transport... 

When fluid is pumped through a cell such as that shown in Fig. 12, transport of dissolved molecules from the cell inlet to the IRE by convection and diffusion is an important issue. The ATR method probes only the volume just above the IRE, which is well within the stagnant boundary layer where diffusion prevails. Figure 13 shows this situation schematically for a diffusion model and a convection-diffusion model (65). The former model assumes that a stagnant boundary layer exists above the IRE, within which mass transport occurs solely by diffusion and that there are no concentration gradients in the convection flow. A more realistic model of the flow-through cell accounts for both convection and diffusion. As a consequence of the relatively narrow gap between the cell walls, the convection leads to a laminar flow profile and consequently to concentration gradients between the cell walls. 

Figure 24 depicts schematically the concentration profile of dissolved hydrogen in a porous catalyst particle brought about by interaction of diffusive mass transport of hydrogen into the porous catalyst particles and the consumption of hydrogen by anodic oxidation at the inner surface of the catalyst. 

When the substrate is liquid, or a solid dissolved in a liquid, the reaction condition becomes more complex. The amount of hydrogen that can be dissolved in a liquid is generally much lower than the stoichiometric need for the reaction. The hydrogenation is completely based on mass-transport from the gas phase through the liquid and to the catalyst. This transport is described with the concentration profiles in Fig. 9.3-2. 

Note that D is defined by the Einstein equation using finite differences as D = (Ax)2/2 At. Equation 2.8c is known as Fick s first law and dC/dx is, of course, the concentration gradient. It should be clear that the net transport of solute mass per unit time across a plane intersecting a concentration profile will be proportional to the steepness of the profile and in the downhill direction. This fact must be intuitively obvious to anyone who wishes to use electrochemical techniques effectively. 

The vast majority of small-amplitude methods are based on small-amplitude potential excitations with potential control of the surface concentrations. In earlier chapters, the relationship between surface concentration and electrode potential was explored and the concept of concentration profiles was presented. Whenever there is a flux of electrons at the electrode surface, the concentration profiles of at least two species will exhibit nonzero slopes at the electrode surface, as the electrochemical conversion of one member of a couple into another takes place and mass transport processes act to reestablish a uniform concentration distribution. These processes occur irrespective of whether the current flux arises from a potential or current excitation of the cell. In either case, they result in a perturbation from the previously existing concentration profile. The initial surface concentrations (which existed prior to the application of the new perturbation) are often termed the dc surface concentrations. It is useful to note that at any time, the distance integral of the concentration excess or defect is directly proportional to the charge passed due to that per-... 

Given the inherently competitive nature of emission and photocurrent, it should not be surprising that photocurrent was generally observed to increase with temperature. Figure 2 is a photocurrent-temperature profile obtained at +0.7 V vs. Ag (PRE) in sulfide electrolyte the light intensity and sulfide concentration employed ensured that photocurrents were limited by excitation rate and not by mass transport, i.e., photocurrents... 

An example of the concentration profiles of the oxidized species O, calculated for different times and corresponding to the application of a constant potential under linear diffusion conditions, is shown in Fig. 1.20. The electrode reaction at the interface leads to the depletion of species O at the solution region adjacent to the electrode surface. As the time increases, the layer in the solution affected by the diffusive mass transport becomes thicker, which indicates that linear diffusion is unable to restore the initial situation (for a more detailed discussion on concentration profiles and their relation with the current, see Sects. 2.2.1 and 2.2.2). 

Mass transport gives rise to the appearance of concentration profiles of an electroactive species O like those shown in Fig. 1.20, obtained for the application of a constant potential to a macroelectrode. From this figure it can be seen that there is a region adjacent to the electrode surface where the concentration of species O is different from its bulk value, Cq, and, therefore, mass transport takes place. In the following discussion, diffusion will be the only transport mode considered. The thickness of this diffusion layer, <5real, can be accurately calculated from the concentration profile as the distance from the electrode surface to a point in solution at which the following condition holds ... 

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