Coupled transport processes diffusion

The occurrence of kinetic instabilities as well as oscillatory and even chaotic temporal behavior of a catalytic reaction under steady-state flow conditions can be traced back to the nonlinear character of the differential equations describing the kinetics coupled to transport processes (diffusion and heat conductance). Studies with single crystal surfaces revealed the formation of a large wealth of concentration patterns of the adsorbates on mesoscopic (say pm) length scales which can be studied experimentally by suitable tools and theoretically within the framework of nonlinear dynamics. [31]... 

An alternative approach is to make the simplification that the rate of chemical reaction is fast compared to the rate of diffusion that is, the membrane diffusion is rate controlling. This approximation is a good one for most coupled transport processes and can be easily verified by showing that flux is inversely proportional to membrane thickness. If interfacial reaction rates were rate controlling, the flux would be constant and independent of membrane thickness. Making the assumption that chemical equilibrium is reached at the membrane interfaces allows the coupled transport process to be modeled easily [9], The process is... 

Facilitated or carrier-mediated transport is a coupled transport process that combines a (chemical) coupling reaction with a diffusion process. The solute has first to react with the carrier to fonn a solute-carrier complex, which then diffuses through the membrane to finally release the solute at the permeate side. The overall process can be considered as a passive transport since the solute molecule is transported from a high to a low chemical potential. In the case of polymeric membranes the carrier can be chemically or physically bound to the solid matrix (Jixed carrier system), whereby the solute hops from one site to the other. Mobile carrier molecules have been incorporated in liquid membranes, which consist of a solid polymer matrix (support) and a liquid phase containing the carrier [2, 8], see Fig. 7.1. The state of the art of supported liquid membranes for gas separations will be discussed in detail in this chapter. 

Theoretical studies of catalytic conversion in a flow reactor reveal that a compensation effect will be observed under certain restrictive conditions. It appears that the compensation effect is observed when two or more coupled transport processes are involved and consequently may be a general law. Compensation effects have been observed in electronic conductivity in semiconductors, diffusion of atoms in solids, etc however, more work is needed to establish its generality. 

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. 

In conclusion to this introduction it may be remarked that a new branch of thermodynamics has developed during the past few decades which is not limited in its applications to systems at equilibrium. This is based on the use of the principle of microscopic reversibility as an auxiliary to the information contained in the laws of classical thermodynamics. It gives useful and interesting results when applied to non-equilibrium systems in which there are coupled transport processes, as in the thermo-electric effect and in thermal diffusion. It does not have significant applications in the study of chemical reaction or phase change and for this reason is not included in the present volume.f... 

A discussion of the charge transfer reaction on the polymer-modified electrode should consider not only the interaction of the mediator with the electrode and a solution species (as with chemically modified electrodes), but also the transport processes across the film. Let us assume that a solution species S reacts with the mediator Red/Ox couple as depicted in Fig. 5.32. Besides the simple charge transfer reaction with the mediator at the interface film/solution, we have also to include diffusion of species S in the polymer film (the diffusion coefficient DSp, which is usually much lower than in solution), and also charge propagation via immobilized redox centres in the film. This can formally be described by a diffusion coefficient Dp which is dependent on the concentration of the redox sites and their mutual distance (cf. Eq. (2.6.33). 

Transport is a three-phase process, whereas homogeneous chemical and phase-transfer [2.87, 2.88] catalyses are single phase and two-phase respectively. Carrier design is the major feature of the organic chemistry of membrane transport since the carrier determines the nature of the substrate, the physico-chemical features (rate, selectivity) and the type of process (facilitated diffusion, coupling to gradients and flows of other species, active transport). Since they may in principle be modified at will, synthetic carriers offer the possibility to monitor the transport process via the structure of the ligand and to analyse the effect of various structural units on the thermodynamic and kinetic parameters that determine transport rates and selectivity. 

In PEMFC systems, water is transported in both transversal and lateral direction in the cells. A polymer electrolyte membrane (PEM) separates the anode and the cathode compartments, however water is inherently transported between these two electrodes by absorption, desorption and diffusion of water in the membrane.5,6 In operational fuel cells, water is also transported by an electro-osmotic effect and thus transversal water content distribution in the membrane is determined as a result of coupled water transport processes including diffusion, electro-osmosis, pressure-driven convection and interfacial mass transfer. To establish water management method in PEMFCs, it is strongly needed to obtain fundamental understandings on water transport in the cells. 

As with coupled transport, two assumptions are made to simplify the treatment first, that the rate of chemical reaction is fast compared to the rate of diffusion across the membrane, and second, that the amount of material transported by carrier facilitated transport is much larger than that transported by normal passive diffusion, which is ignored. The facilitated transport process can then be represented schematically as shown in Figure 11.17. 

The general solutions of the fundamental systems of nonlinear equations [Eq. (2)] will be of the type wherein the state variables are dependent both on time and space, which will manifest in the form of wave propagation. Coupling between several parts of the system will be transmitted through the generalized diffusion coefficient D. If the associated transport process proceeds on a time scale comparable to or slower than the period of the temporal oscillation, macroscopic wave propagation phenomena are to be expected, as, for example, realized with the Belousov-Zhabotinsky... 

Spatiotemporal pattern formation at the electrode electrolyte interface is described by equations that belong in a wider sense to the class of reaction-diffusion (RD) systems. In this type of coupled partial differential equations, any sustained spatial structure comes about owing to the interplay of the homogeneous dynamics or reaction dynamics and spatial transport processes. Therefore, the evolution of each variable, such as the concentration of a reacting species, can be separated into two parts the reaction part , which depends only on the values of the other variables at one particular location, and another part accounting for transport processes that are induced by spatial variations in the variables. These latter processes constitute a spatial coupling among different locations. 

In this book we are concerned only with mass transport, or diffusion, in solids. Self-diffusion refers to atoms diffusing among others of the same type (e.g., in pure metals). Interdiffusion is the diffusion of two dissimilar substances (a diffusion couple) into one another. Impurity diffusion refers to the transport of dilute solute atoms in a host solvent. In solids, diffusion is several orders of magnitude slower than in liquids or gases. Nonetheless, diffusional processes are important to study because they are basic to our understanding of how solid-liquid, solid-vapor, and solid-solid reactions proceed, as well as [solid-solid] phase transformations in single-phase materials. 

Other processes that lead to nonlinear compartmental models are processes dealing with transport of materials across cell membranes that represent the transfers between compartments. The amounts of various metabolites in the extracellular and intracellular spaces separated by membranes may be sufficiently distinct kinetically to act like compartments. It should be mentioned here that Michaelis-Menten kinetics also apply to the transfer of many solutes across cell membranes. This transfer is called facilitated diffusion or in some cases active transport (cf. Chapter 2). In facilitated diffusion, the substrate combines with a membrane component called a carrier to form a carrier-substrate complex. The carrier-substrate complex undergoes a change in conformation that allows dissociation and release of the unchanged substrate on the opposite side of the membrane. In active transport processes not only is there a carrier to facilitate crossing of the membrane, but the carrier mechanism is somehow coupled to energy dissipation so as to move the transported material up its concentration gradient. 

Transport in membranes is mostly a complex and coupled process coupling between the solute and the membrane, and coupling between diffusion and the chemical reaction may play an important role in efficiency. It is important to understand and quantify the coupling to describe the transport in membranes. Kinetic studies may also be helpful. However, thermodynamics might offer a new and rigorous approach toward understanding the coupled transport in composite membranes without the need for detailed examination of the mechanism of diffusion through the solid structure. Table 10.4 shows some of the applications of facilitated transport. 

These equations display the spatial order with the thermodynamically coupled heat and mass flows. Here, the coupling between chemical reactions and transport processes of heat and mass is excluded. The analysis of reaction-diffusion systems would be more complete if we consider heat effects and coupling among fluxes of mass and heat. The nonequi-librium thermodynamics approach is useful for incorporating the coupling phenomena into reaction-diffusion systems. 

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