Transport kinetics, coupled diffusion processes

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]... 

It was seen that in the experiments performed hydrogen ion was the trace species. Therefore, for coupled Al -H transport, the interdiffusion coefficient approached the self-diffusion coefficient of the faster-diffusing ions. This conclusion appears counterintuitive, but explains the good kinetics of the process. 

Appropriate initial and boundary conditions should also be added to complete the mathematical formulation. In Equation (1) C(r,t) is the local concentration vector, F(C 5) a vector function representing the reaction kinetics, B stands for a set of control parameters and D is the matrix of transport coefficients. In most chemical systems involving small molecules in aqueous solutions, the diffusion processes are well described by a diagonal matrix with constant positive diffusion coefficients. However, in some systems it is the coupling between the transport processes that provides the engine of the instability. For instance, stratification occurs in electron-hole plasmas in semiconductors subjected to electromagnetic radiations because of the effect of the temperature field on the carrier density distribution (thermodiffusion)... 

A fundamental fuel cell model consists of five principles of conservation mass, momentum, species, charge, and thermal energy. These transport equations are then coupled with electrochemical processes through source terms to describe reaction kinetics and electro-osmotic drag in the polymer electrolyte. Such convection—diffusion—source equations can be summarized in the following general form... 

Because of its cyclic nature, this process presents analogies with molecular catalysis it may be considered as physical catalysis operating a change in location, a translocation, on the substrate, like chemical catalysis operates a transformation into products. The carrier is the transport catalyst which strongly increases the rate of passage of the substrate with respect to free diffusion and shows enzyme-like features (saturation kinetics, competition and inhibition phenomena, etc.). The active species is the carrier-substrate supermolecule. The transport of substrate Sj may be coupled to the flow of a second species S2 in the same (symport) or opposite antiport) direction. 

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. 

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. 

Dimensional analysis of the coupled kinetic-transport equations shows that a Thiele modulus (4> ) and a Peclet number (Peo) completely characterize diffusion and convection effects, respectively, on reactive processes of a-olefins [Eqs. (8)-(14)]. The Thiele modulus [Eq. (15)] contains a term ( // ) that depends only on the properties of the diffusing molecule and a term ( -) that includes all relevant structural catalyst parameters. The first term introduces carbon number effects on selectivity, whereas the second introduces the effects of pellet size and pore structure and of metal dispersion and site density. The Peclet number accounts for the effects of bed residence time effects on secondary reactions of a-olefins and relates it to the corresponding contribution of pore residence time. 

The formalism introduced in the previous subsections is able to incorporate the effect of these influences in the crystallization kinetics, thus providing a more realistic modeling of the process, which is mandatoiy for practical and industrial purposes. Due to the strong foundations of our mesoscopic formalism in the roots of standard non-equilibrium thermodynamics, it is easy to incorporate the influence of other transport processes (like heat conduction or diffusion) into the description of crystallization. In addition, our framework naturally accounts for the couplings between all these different influences. 

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