Reaction-diffusion—transport system

Sedimentary denitrification rates have been estimated from measured pore-water solute profiles using diagenetic models, determined direcdy via sediment incubation both on deck and in situ, and determined from N-incubation techniques. Sedimentary diagenetic process can be thought of as a simple reaction—diffusion-transport system (Berner, 1980 Boudreau, 1997). In a simple fine-grained sediment system, transport is via molecular diffusion and the diagenetic equation describing this system can be expressed as ... 

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. 

Pollard and Newman" have also studied CVD near an infinite rotating disk, and the equations we solve are essentially the ones stated in their paper. Since predicting details of the chemical kinetic behavior is a main objective here, the system now includes a species conservation equation for each species that occurs in the gas phase. These equations account for convective and diffusive transport of species as well as their production and consumption by chemical reaction. The equations stated below are given in dimensional form since there is little generalization that can be achieved once large chemical reaction mechanisms are incorporated. 

The form of the effective mobility tensor remains unchanged as in Eq. (125), which imphes that the fluid flow does not affect the mobility terms. This is reasonable for an uncharged medium, where there is no interaction between the electric field and the convective flow field. However, the hydrodynamic term, Eq. (128), is affected by the electric field, since electroconvective flux at the boundary between the two phases causes solute to transport from one phase to the other, which can change the mean effective velocity through the system. One can also note that even if no electric field is applied, the mean velocity is affected by the diffusive transport into the stationary phase. Paine et al. [285] developed expressions to show that reversible adsorption and heterogeneous reaction affected the effective dispersion terms for flow in a capillary tube the present problem shows how partitioning, driven both by electrophoresis and diffusion, into the second phase will affect the overall dispersion and mean velocity terms. 

Attaching the catalyst molecules to the electrode surface presents an obvious advantage for synthetic and sensor applications. Catalysis can then be viewed as a supported molecular catalysis. It is the object of the next section. A distinction is made between monolayer and multilayer coatings. In the former, only chemical catalysis may take place, whereas both types of catalysis are possible with multilayer coatings, thanks to their three-dimensional structure. Besides substrate transport in the bathing solution, the catalytic responses are then under the control of three main phenomena electron hopping conduction, substrate diffusion, and catalytic reaction. While several systems have been described in which electron transport and catalysis are carried out by the same redox centers, particularly interesting systems are those in which these two functions are completed by two different molecular systems. 

In this section we studied the phenomenon of enhanced (hydrodynamic) transport, induced by population growth in reaction-diffusion systems. Based on our Fisher theorem approach, we have shown that the expressions for the emerging hydrodynamic speeds have a simple physical interpretation They are proportional to space specific fitness functions, which express the ability of a population to fill out space. Based on our approach, we came up with simple rules for solving inverse problems in geographical population genetics. 

Carrier-mediated passage of a molecular entity across a membrane (or other barrier). Facilitated transport follows saturation kinetics ie, the rate of transport at elevated concentrations of the transportable substrate reaches a maximum that reflects the concentration of carriers/transporters. In this respect, the kinetics resemble the Michaelis-Menten behavior of enzyme-catalyzed reactions. Facilitated diffusion systems are often stereo-specific, and they are subject to competitive inhibition. Facilitated transport systems are also distinguished from active transport systems which work against a concentration barrier and require a source of free energy. Simple diffusion often occurs in parallel to facilitated diffusion, and one must correct facilitated transport for the basal rate. This is usually evident when a plot of transport rate versus substrate concentration reaches a limiting nonzero rate at saturating substrate While the term passive transport has been used synonymously with facilitated transport, others have suggested that this term may be confused with or mistaken for simple diffusion. See Membrane Transport Kinetics... 

Chemical reactions with autocatalytic or thermal feedback can combine with the diffusive transport of molecules to create a striking set of spatial or temporal patterns. A reactor with permeable wall across which fresh reactants can diffuse in and products diffuse out is an open system and so can support multiple stationary states and sustained oscillations. The diffusion processes mean that the stationary-state concentrations will vary with position in the reactor, giving a profile , which may show distinct banding (Fig. 1.16). Similar patterns are also predicted in some circumstances in closed vessels if stirring ceases. Then the spatial dependence can develop spontaneously from an initially uniform state, but uniformity must always return eventually as the system approaches equilibrium. 

In this section we mov6 on to examine the behaviour of an open system in which the transport of reactants and products relies on molecular diffusion processes. The situation we envisage is that of a reaction zone in which various species diffuse and react. Outside this zone there exists an external reservoir in which the reactants have fixed concentrations. The reservoir provides a source of reactants which can diffuse across the boundary into the reaction zone, and either a source or a sink for the intermediates and products. The reaction-diffusion cell is sketched in Fig. 9.2. 

The objective is to derive a system of equations in general vector form that describes the overall gas-phase mass continuity and the species continuity equations for A and all other species k in the mixture. Assume that there is convective and diffusive transport of the species, but no chemical reaction. 

In summary, the steady state and transient performance of the poly(acrylamide) hydrogel with immobilized glucose oxidase and phenol red dye (pAAm/GO/PR) demonstrates phenomena common to all polymer-based sensors and drag delivery systems. The role of the polymer in these systems is to act as a barrier to control the transport of substrates/products and this in turn controls the ultimate signal and the response time. For systems which rely upon the reaction of a substrate for example via an immobilized enzyme, the polymer controls the relative importance of the rate of substrate/analyte delivery and the rate of the reaction. In membrane systems, the thicker the polymer membrane the longer the response time due to substrate diffusion limitations as demonstrated with our pAAm/GO/PR system. However a membrane must not be so thin as to allow convective removal of the substrates before undergoing reaction, or removal of the products before detection. The steady state as well as the transient response of the pAAm/GO/ PR system was used to demonstrate these considerations with the more complicated case in which two substrates are required for the reaction. 

In a spatially extended system, fluctuations that are always present cause the variables to differ somewhat in space, inducing transport processes, the most common one being diffusion. In the case of constant diffusion coefficients /), the system s dynamics is then governed by reaction-diffusion equations ... 

In this chapter, we present most of the equations that apply to the systems and processes to be dealt with later. Most of these are expressed as equations of concentration dynamics, that is, concentration of one or more solution species as a function of time, as well as other variables, in the form of differential equations. Fundamentally, these are transport (diffusion-, convection-and migration-) equations but may be complicated by chemical processes occurring heterogeneously (i.e. at the electrode surface - electrochemical reaction) or homogeneously (in the solution bulk chemical reaction). The transport components are all included in the general Nernst-Planck equation (see also Bard and Faulkner 2001) for the flux Jj of species j... 

This chapter has developed the basic concepts of modeling diffusive transport and coupled diffusion, advection, and reaction in physiological systems. The emphasis... 

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