Multi-Component Reaction-Diffusion Systems

The reaction-diffusion equations for a system of n species in one-dimensional space read, see Sect. 2.1.2, 

and D is the n X n diffusion matrix. We focus on front solutions and illustrate how to determine the front velocity for KPP kinetics. Let us write, for simplicity. 

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As a pronounced contrast, in the field of high performance non-oxide ceramics, currently only binaries are in use. Thus, the manifold opportunities for creating new and ever more capable nitride or carbide ceramics offered by the use of multi-component systems seem to have remained essentially unexplored. The main chemical reason for this lagging behind of non-oxide ceramics are clearly the extremely low self-diffusion coefficients of silicon or boron in their nitrides or carbides [6, 7]. Although the experimental data available are rather limited, the numbers presented in Table 1 [8] suggest that the temperatures needed to complete a solid state reaction between SiC and Si3N4 in an acceptable length of time reach, or even exceed, the decomposition temperature of at least one of the reactants. 

As a preliminary test, the governing equation (i.e., convection-diffusion-reaction PDE) is solved using an ODE time integrator for the binary system Including only A and B components. Fig. 2 shows effects of the three kernels on liquid concentrations. Abnormal concentration profiles are found in inactive zones for the conventional rate model (Fig. 2a) and that with the sum kernel (Fig. 2b). The product kernel will play an effective role for multi-component systems. 

In this book we considered mass transfer and elemental migration between the atmosphere, hydrosphere, soils, rocks, biosphere and humans in earth s surface environment on the basis of earth system sciences. In Chaps. 2, 3, and 4, fundamental theories (thermodynamics, kinetics, coupling model such as dissolution kinetics-fluid flow modeling, etc.) of mass transfer mechanisms (dissolution, precipitation, diffusion, fluid flow) in water-rock interaction of elements in chemical weathering, formation of hydrothermal ore deposits, hydrothermal alteration, formation of ground water quality, seawater chemistry. However, more complicated geochemical models (multi-components, multi-phases coupled reaction-fluid flow-diffusion model) and phenomenon (autocatalysis, chemical oscillation, etc.) are not considered. 

The reaction under consideration may involve a solvent and/or one or more liquid-phase products, thus making it a multi-component diffusion system. In such cases, Z>b represents the solute diffusivity in the liquid mixture including the products. To simplify the effort to make a reasonable estimate of Z>b, the relatively insignificant components may be neglected. For example, the following expression can then be used to calculate the diffusivity of a solute, gas or liquid, in 2-solvent liquid system (39) ... 

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