Chemical waves and stationary patterns

In the earlier chapter, we have discussed the emergence of time-order in chemical reactions in continuously stirred tank reactor (CSTR) and have discussed the concept of negative and positive feedback for occurrence of oscillatory reactions. In this respect, experimental studies of oscillatory reactions in batch reactors have been investigated in great depth which has provided convincing evidence for the important role of auto-catalytic and inhibitory reactions in oscillatory reactions. Rate of internal production is controlled by the influx of reactants from external source. 

Increase in entropy is associated with (i) decrease in order, (ii) increase in disorder and (iii) loss of information as predicted by Statistical Mechanics. Everything tends towards disorder. Any process that converts from one energy to another must lose heat. The universe is one-way street. Entropy must always increase in the universe and in any hypothetical system, within it. Although d S is greater than zero d S can be less than zero under certain circumstances, thereby generating a particular type of ordered structure which is called dissipative structure. 

Nature forms patterns. Some are orderly in space. Some patterns are fractals, exhibiting self-similar in scale. Others give rise to steady states or oscillating ones. We will discuss fractals in Chapter 13. 

In case of space order and spatio-temporal oscillations, diffusion of the reactants control the internal production of entropy so that dissipative structure occurs. Diffusion contributes to the dissipation of entropy. 

Reacting systems involving non-linear kinetics (and systems) involving interplay of reaction and diffusion (R-D systems) exhibit a number of exotic phenomena. Several reviews (and symposium volumes) have been addressed to these issues [1-9]. In general, we come across the following types of systems  

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Example 4.8 Chemical reactions and reacting flows The extension of the theory of linear nonequilibrium thermodynamics to nonlinear systems can describe systems far from equilibrium, such as open chemical reactions. Some chemical reactions may include multiple stationary states, periodic and nonperiodic oscillations, chemical waves, and spatial patterns. The determination of entropy of stationary states in a continuously stirred tank reactor may provide insight into the thermodynamics of open nonlinear systems and the optimum operating conditions of multiphase combustion. These conditions may be achieved by minimizing entropy production and the lost available work, which may lead to the maximum net energy output per unit mass of the flow at the reactor exit. 

The extension of the theory of linear nonequilibrium thermodynamics to nonlinear systems can describe systems far from equilibrium, such as open chemical reactions. Some chemical reactions may include multiple stationary states, periodic and nonperiodic oscillations, chemical waves, and spatial patterns. Determine the optimum operating conditions. 

Under non-equilibrium conditions, some nonlinear phenomena such as oscillation, chaos and stationary pattern occur are a result of the loss of stability by the steady states, caused by the feedback loops in the processes determining the dynamics of such systems. Such self-organization can be obvious itself as a function of either only time coordinate including simultaneous oscillations of the entire system s state or only spatial coordinate including Turing stmctures or both coordinates including both traveling and chemical waves. The universal fact discovered of such phenomenon in different systems is remarkable in the context of mathematical description. 

Waves of chemical reaction may travel through a reaction medium, but the ideas of important stationary spatial patterns are due to Turing (1952). They were at first invoked to explain the slowly developing stripes that can be exhibited by reactions like the Belousov-Zhabotinskii reaction. This (rather mathematical) chapter sets out an analysis of the physically simplest circumstances but for a system (P - A - B + heat) with thermal feedback in which the internal transport of heat and matter are wholly controlled by molecular collision processes of thermal conductivity and diffusion. After a careful study the reader should be able to ... 

A typical feature of a non-potential systems is the non-stationary oscillatory behavior that usually manifests itself in the propagation of waves. We have shown that the nonlinear evolution of waves near the instability threshold is described by the complex Ginzburg-Landau (CGL) equation. This equation is capable of describing various kinds of instabilities of wave patterns, like the Benjamin-Feir instability. In two dimensions, the CGL equation describes the formation of spiral waves that are observed in many biological and chemical systems characterized by the interplay of diffusion and chemical reactions at nano-scales. 

The above classical detonation theory was proposed by Zebdovich [529, 530, 534] (see also [334]), Doring [113] and Grib [169] on the basis of a unidimensional model of a stationary detonation wave. Further studies showed (for references see Strehlow s review [458, 535]) that the real gas-kinetic and the chemical-kinetic pattern of a detonation wave are much more complicated than the idealized plane shock wave. Moreover, the flat chemical reaction fronts which follow from the classical theory are non-stationary, thus leading to distortions and discontinuities in the flame front resulting in the violation of the idealized detonation wave pattern. 

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