Transport kinetics, 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. 

Equation (2.19), which concerns a situation without processes in the biofilm, can be extended to include transformation of a substrate, an electron donor (organic matter) or an electron acceptor, e.g., dissolved oxygen. If the reaction rate is limited by j ust one substrate and under steady state conditions, i.e., a fixed concentration profile, the differential equation for the combined transport and substrate utilization following Monod kinetics is shown in Equation (2.20) and is illustrated in Figure 2.8. Equation (2.20) expresses that under steady state conditions, the molecular diffusion determined by Fick s second law is equal to the bacterial uptake of the substrate. 

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... 

If electron transport is fast, the system passes from zone R to zone S+R and then to zone SR. In the latter case there is a mutual compensation of diffusion and chemical reaction, making the substrate concentration profile decrease within a thin reaction layer adjacent to the film-solution interface. This situation is similar to what we have termed pure kinetic conditions in the analysis of an EC reaction scheme adjacent to the electrode solution interface developed in Section 2.2.1. From there, if electron transport starts to interfere, one passes from zone SR to zone SR+E and ultimately to zone E, where the response is controlled entirely by electron transport. 

In Fig. 1, various elements involved with the development of detailed chemical kinetic mechanisms are illustrated. Generally, the objective of this effort is to predict macroscopic phenomena, e.g., species concentration profiles and heat release in a chemical reactor, from the knowledge of fundamental chemical and physical parameters, together with a mathematical model of the process. Some of the fundamental chemical parameters of interest are the thermochemistry of species, i.e., standard state heats of formation (A//f(To)), and absolute entropies (S(Tq)), and temperature-dependent specific heats (Cp(7)), and the rate parameter constants A, n, and E, for the associated elementary reactions (see Eq. (1)). As noted above, evaluated compilations exist for the determination of these parameters. Fundamental physical parameters of interest may be the Lennard-Jones parameters (e/ic, c), dipole moments (fi), polarizabilities (a), and rotational relaxation numbers (z ,) that are necessary for the calculation of transport parameters such as the viscosity (fx) and the thermal conductivity (k) of the mixture and species diffusion coefficients (Dij). These data, together with their associated uncertainties, are then used in modeling the macroscopic behavior of the chemically reacting system. The model is then subjected to sensitivity analysis to identify its elements that are most important in influencing predictions. 

Solution of the coupled mass-transport and reaction problem for arbitrary chemical kinetic rate laws is possible only by numerical methods. The problem is greatly simplified by decoupling the time dependence of mass-transport from that of chemical kinetics the mass-transport solutions rapidly relax to a pseudo steady state in view of the small dimensions of the system (19). The gas-phase diffusion problem may be solved parametrically in terms of the net flux into the drop. In the case of first-order or pseudo-first-order chemical kinetics an analytical solution to the problem of coupled aqueous-phase diffusion and reaction is available (19). These solutions, together with the interfacial boundary condition, specify the concentration profile of the reagent gas. In turn the extent of departure of the reaction rate from that corresponding to saturation may be determined. Finally criteria have been developed (17,19) by which it may be ascertained whether or not there is appreciable (e.g., 10%) limitation to the rate of reaction as a consequence of the finite rate of mass transport. These criteria are listed in Table 1. 

For transporters, relatively low protein expression level and limited transport capacity makes for nonlinear, enzyme-like transport kinetics that is, the transport rate saturates with increasing substrate concentration. This phenomenon is the basis for the competitive interactions generally found for chemicals that are handled by one or more common transporters this is usually manifest as inhibition of the transport of one chemical by a structural analog. The extent to which these competitive interactions are important depends on the concentrations of the chemicals involved, their relative affinities for the common transporter, and their phar-macological/toxicological profiles (effects, effective concentrations, therapeutic index). Competition for transport is discussed below in the context of drug-drug interactions. 

The evaluation of catalyst effectiveness requires a knowledge of the intrinsic chemical reaction rates at various reaction conditions and compositions. These data have to be used for catalyst improvement and for the design and operation of many reactors. The determination of the real reaction rates presents many problems because of the speed, complexity and high exo- or endothermicity of the reactions involved. The measured conversion rate may not represent the true reaction kinetics due to interface and intraparticle heat and mass transfer resistances and nonuniformities in the temperature and concentration profiles in the fluid and catalyst phases in the experimental reactor. Therefore, for the interpretation of experimental data the experiments should preferably be done under reaction conditions, where transport effects can be either eliminated or easily taken into account. In particular, the concentration and temperature distributions in the experimental reactor should preferably be described by plug flow or ideal mixing models. 

The diffusion-layer imaging technique which was developed by McCreery is another method for studying intermediates in the diffusion layer [71-75]. A laser beam is directed in a parallel direction through the diffusion layer of the electrode and the light is then magnified and focused on a diode-array detector. With this method, spatial resolution of the diffusion layer of 1.25 pm is achieved, and concentration profiles in the diffusion layer are mapped. A detailed description of mass transport processes as well as the kinetics and spectra of intermediates can be obtained. Diffusion coefficients and extinction coefficients for, for example, the benzophenone radical anion were measured with this technique [74, 75]. 

In Section 12.2.4 we showed that after time z,ja = RjJit2Daq the concentration profile inside a cloud droplet becomes uniform. The characteristic time for aqueous-phase reaction was found to be equal to xra = [A]// aq. If the characteristic aqueous-phase diffusion time is much less than the characteristic reaction time, then aqueous-phase diffusion will be able to maintain a uniform concentration profile inside the droplet. In this case, there will be no concentration gradients inside the drop and therefore no aqueous-phase mass transport limitations on the aqueous-phase kinetics. This criterion can be written as... 

To describe the concentration profiles and release kinetics of fluorescein from single NP-shelled capsules, Munoz Tavera et al. solved a mathematical model that describes unsteady-state transport from multilayered spheres using the Sturm-Liouville approach [92], Several aspects of dye release, such as the asymptotic plateau effect of diffusive release and the effects of capsule diameter and shell thickness distribution on dye release, were captured by the model and confirmed by experimental data. 

The dimensionless substrate concentration profile in a porous membrane, where the biocatalyst is entrapped, depends on the Thiele modulus heterogeneous catalyst, since it allows for the estimation of the penetration depth within the support and can also be used to identify the mechanism that controls the process rate. The Thiele modulus can be interpreted as a ratio between a diffusion time (SVD if.s) and a kinetic time (A a/ max) or, equivalently, as a ratio between a characteristic kinetic rate Vmax/ m AS = AS and a merely diffusive transport mechanism, characterized by (D s s/S) x (A5/5). When (/) 1 kinetics is the limiting step, the process rate coincides with the reaction rate and the concentration profile is uniform, completely penetrating the support. 

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