The Local Mass Transport

While it is possible to enhance mass transfer in the boundary layer by improving the flow conditions, this is impossible if a concentration profile exists within the porous support structure. 

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These effects do not involve the interfacial kinetics on bare portions of the metal which, for simplification, will be assumed to be inherently fast. The changes will affect only the local mass transport conditions towards the reaction sites. 

The main hypotheses for developing the EHD impedance theory are that the electrode interface is uniformly accessible and the electrode surface has uniform reactivity. However, in many cases, real interfaces deviate from this ideal picture due, for example, either to incomplete monolayer adsorption leading to the concept of partial blocking (2-D adsorption) or to the formation of layers of finite thickness (3-D phenomena). These effects do not involve the interfacial kinetics on bare portions of the metal, which, for simplification, will be assumed to be inherently fast. The changes will affect only the local mass transport toward the reaction sites. Before presenting an application of practical interest, the theoretical EHD impedance for partially blocked electrodes and for electrodes coated by a porous layer will be analyzed. 

In Eq. (13) km is the local mass transport coefficient D/S, that is, diffusion coefficient divided by the thickness of diffusion layer. High mass transport rates can be achieved by electrode movement,... 

Consider the net mass flow through the cylindrical differential element illustrated in Fig. 3.6. The following analysis makes no explicit reference to the scalar product of the flux vector and the outward normal, j ndA. Rather, it is based on a more direct observation of how mass diffuses into and out of the control volume. It is presumed that the spatial components of j are resolved into spatial components that are normal to the control-volume faces, jk,z, jk,r, and jk,e Further it is presumed that a positive value for a spatial component of jk means that the corresponding flux is in the direction of the positive coordinate. The components of the diffusive mass flux are presumed to be continuous and differentiable throughout the fluid. Therefore the flux components can be expanded in a first-order Taylor series to express the local variations in the flux. The net mass of species k that crosses the control surfaces diffusively is given by the incoming minus the outgoing mass transport. Consider, for example, transport in the radial direction ... 

Figure 13 shows the potential and concentration distributions for different values of dimensionless potential under conditions when internal pore diffusion (s = 0.1) and local mass transport (y = 10) are a factor. As expected the concentration and relative overpotential decrease further away from the free electrolyte (or membrane) due to the combined effect of diffusion mass transport and the poor penetration of current into the electrode due to ionic conductivity limitations. The major difference in the data is with respect to the variation in reactant concentrations. In the case when an internal mass transport resistance occurs (y = 10) the fall in concentration, at a fixed value of electrode overpotential, is not as great as the case when no internal mass transport resistance occurs. This is due to the resistance causing a reduction in the consumption of reactant locally, and thereby increasing available reactant concentration the effect of which is more significant at higher electrode overpotentials. 

Figure 14 shows the distributions removing the influence of the internal mass transport resistance on the current distribution, i.e., y is very high (model Eqs. 117 and 118 apply). In this case much higher local current densities are achieved, although the problem of a non-uniform distribution in current density prevails, with as before much of the electrode not very active. 

The curve shown in Fig. 3 cannot proceed indefinitely in either direction. In the cathodic direction, the deposition of copper ions proceeds from solution until the rate at which the ions are supplied to the electrode becomes limited by mass-transfer processes. In the anodic direction, copper atoms are oxidized to form soluble copper ions. While the supply of copper atoms from the surface is essentially unlimited, the solubility of product salts is finite. Local mass-transport conditions control the supply rate so a current is reached at which the solution supersaturates, and an insulating salt-film barrier is created. At that point the current drops to a low level further increase in the potential does not significantly increase the current density. A plot of the current density as a function of the potential is shown in Fig. 5 for the zinc electrode in alkaline electrolyte. The sharp drop in potential is clearly observed at -0.9 V versus the standard hydrogen electrode (SHE). At more positive potentials the current density remains at a low level, and the electrode is said to be passivated. 

It should be noted that the local mass transfer coefficient can only be obtained experimentally and is case specific. An analytical relationship for the local mass transfer rate coefficient can be obtained if a mathematical expression describing the gradient of the dissolved concentration at the NAPL-water interface is known. Unfortunately, the local mass transfer coefficient usually is not an easy parameter to determine with precision. Thus, in mathematical modeling of contaminant transport originating from NAPL pool dissolution, k(t, x,y) is often replaced by the average mass transfer coefficient, k(t), applicable to the entire pool, expressed as [41]... 

A second type of mass-balance approach is quantitative incorporation of mass balances within a reactive-transport model and could be applied to groundwaters, surface waters, and surface-water-groundwater interactions. Paces (1983, 1984) calls this the local mass-balance approach. There are numerous examples and explanations of this approach (e.g.. Freeze and Cherry, 1979 Domenico and Schwartz, 1990). 

The mass flux of a solute can be related to a mass transfer coefficient which gathers both mass transport properties and hydrodynamic conditions of the system (fluid flow and hydrodynamic characteristics of the membrane module). The total amount transferred of a given solute from the feed to the receiving phase can be assumed to be proportional to the concentration difference between both phases and to the interfacial area, defining the proportionality ratio by a mass transfer coefficient. Several types of mass transfer coefficients can be distinguished as a function of the definition of the concentration differences involved. When local concentration differences at a particular position of the membrane module are considered the local mass transfer coefficient is obtained, in contrast to the average mass transfer coefficient [37]. 

Brown C.J., Pletcher D., Walsh F.C., Hammond J.K. Robinson D., Local mass transport effects in the FM01 laboratory electrolyser. Jal of Applied Electrochemistry, 22, pp. 613-619, 1992. 

Siepmann J, Siepmann F, Florence AT. Local controlled drug delivery to the brain mathematical modeling of the underlying mass transport mechanisms. Int J Pharm 2006 314 101-119. 

Therefore, as basis we formulate the generalized form of the local instantaneous transport equation for the mixture mass, component mass, momentum, energy and entropy in a fixed control volume (CV) on microscopic scales, as illustrated in Fig. 1.1. 

So far SECM applications have been considered where enzymes immobilized at a surface catalyze redox reactions of low molecular weight compounds. The reaction products are detected at the ultramicroelectrode tip under diffusion-controlled conditions. This approach requires that the biochemically active layer continuously generate or consume redox active (for amperometric detection) or charged (for potentiometric detection) species. Since the tip signal depends on the diffusion coefficients and/or convective effects as well as the local concentration, it is possible to image localized mass transport phenomena instead of localized chemical fluxes (Chapter 9). In a general sense it is a process that is recorded with lateral resolution. 

The tip response in SECM is strongly dependent on local mass transport, and this may in fact be utilized to image local mass transport. Examples include the transport of oxygen and electroactive ions in cartilage (80), convective (81) and diffusional (34) transport in dentinal tubules, and ionic fluxes through skin (82), which are described in Chapters 9 and 12. In this section we discuss briefly experiments by Pohl, Antonenko and coworkers which, though not SECM, employ microelectrode techniques in a similar manner. 

Mass transport phenomena in biological systems can be investigated with SECM if the species of interest can be detected either by an potenti-ometric or amperometric microelectrode. Theory and selected studies of localized mass transport are covered in Chapter 9 of this volume. SECM is particularly appropriate in these studies in view of the intimate connection of the imaging mechanism to mass transport effects. The investigation of oxygen and ion transport in various tissues and under a variety of driving forces (concentration gradient, electric field, convection) has been demon-... 

As might be expected, the model leads to a great simplification over the calculations required for molecules with a continuous potential energy function, as it enables the analysis to be confined to binary collisions and permits the definition of a collision frequency. Because there is no molecular interaction between collisions, the velocity distributions of two colliding molecules may be assumed to be re-established by the time a second collision occurs between them. Thus a Maxwellian distribution around the local mass velocity may be postulated for the calculation of the mean frequency of collision and the average momentum and energy transported per collision in the nonuniform state of the liquid. 

Hence, the local mass transfer coefficient scales as the two-thirds power of a, mix for boundary layer theory adjacent to a solid-liquid interface, and the one-half power of A, mix for boundary layer theory adjacent to a gas-liquid interface, as well as unsteady state penetration theory without convective transport. By analogy, the local heat transfer coefficient follows the same scaling laws if one replaces a, mix in the previous equation by the thermal conductivity. 

In case of general rate models (Section 6.2.6) the local mass transfer inside the particles is considered additionally to the transport outside the particles. Therefore, the mass balances have to take into account the number of (spherical) particles per volume element (Figure 6.3) ... 

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