Spatially distributed systems and reaction-diffusion modeling

Spatially distributed systems and reaction-diffusion modeling 

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Photoinduced electron transfer is a subject characterised, particularly at the present time, by papers with a strongly theoretical content. Solvent relaxation and electron back transfer following photoinduced electron transfer in an ensemble of randomly distributed donors and acceptors, germinate recombination and spatial diffusion a comparison of theoretical models for forward and back electron transfer, rate of translational modes on dynamic solvent effects, forward and reverse transfer in nonadiabatic systems, and a theory of photoinduced twisting dynamics in polar solvents has been applied to the archetypal dimethylaminobenzonitrile in propanol at low temperatures have all been subjects of very detailed study. The last system cited provides an extended model for dual fluorescence in which the effect of the time dependence of the solvent response is taken into account. The mechanism photochemical initiation of reactions involving electron transfer, with particular reference to biological systems, has been discussed by Cusanovich. ... 

Development-controlling prepattern mechanisms have been modelled in reaction-diffusion context. In the celebrated paper of Turing (1952) a model was presented in terms of reaction-diffusion equations to show how spatially inhomogeneous arrangements of material might be generated and maintained in a system in which the initial state is a homogeneous distribution. Two components were involved in the model, and the reactions were described by linear differential equations. The model in one spatial dimension s is ... 

Great efforts are needed even in a laboratory to achieve a homogeneous spatial distribution of the concentrations, temperature and pressure of a system, even in a small volume (a few mm or cm ). Outside the confines of the laboratory, chemical processes always occur under spatially inhomogeneous conditions, where the spatial distribution of the concentrations and temperature is not uniform, and transport processes also have to be taken into account. Therefore, reaction kinetic simulations frequently include the solution of partial differential equations that describe the effect of chemical reactions, material diffusion, thermal diffusion, convection and possibly turbulence. In these partial differential equations, the term f defined on the right-hand side of Eq. (2.9) is the so-called chemical source term. In the remainder of the book, we deal mainly with the analysis of this chemical source term rather than the full system of model equations. 

A similar procedure can be applied to the device we call a one-dimensional (1-D) pipe, shown in Figure 2.1b. In this configuration, the properties of the system are time invariant, but vary with distance in the direction of flow. The system is said to be at steady state and distributed in one spatial coordinate. This is the exact reverse of the conditions that prevailed in a compartment. The physical phenomena that take place in the two cases, however, are similar. Mass is again transported by bulk flow, enters and leaves the device by exchange with the surroundings, and is generated or consumed by chemical reaction. The only difference here is that mass can also enter and leave by diffusion, which was not the case in the compartmental model. 

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