NON-LINEAR STEADY STATES

In Part One, we have discussed theory and related experiments concerning steady states in the range of validity of linear phenomenological equations. We shall call such states as linear steady states. We come across stable steady states even beyond the domain of validity of linear non-equilibrium thermodynamics (LNT) where the flux equations are non-linear. 

We will first discuss the form of non-linear flux equations for electro-osmosis and streaming current, mass and heat flux in thermal diffusion and chemical reactions from experimental and theoretical angle. 

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Yablonsky, G. S., and Lazman, M. Z., Non-Linear Steady-State Kinetics of Complex Catalytic Reactions Theory and Experiment, Dynamics of Surfaces and Reaction Kinetics in Heterogeneous Catalysis, Proceedings of the International Symposium, Antwerpen, September 371-378 (1997). 

Dhakar, S.P., and Burdige, D.J. (1996) A coupled, non-linear, steady state model for early diagenetic processes in pelagic sediments. Am. J. Sci. 296, 296-330. 

For CBD operation, the problem presented above results in a non-linear dynamic optimisation problem, which is solved using the technique in Mujtaba and Macchietto (1993, 1996) as outlined in earlier chapters. For continuous column operation the problem OP results in a non-linear steady state optimisation problem which is solved using the computer software SRQPDV1.1 due to Chen (1988). 

Non-Linear Steady-State Kinetics of Complex Catalytic Reactions Theory and Applications... 

The analysis of non-linear mechanisms and corresponding kinetic models are much more difficult than that of linear ones. The obvious difficulty in this case is the follows an explicit solution for steady-state reaction rate R can be obtained only for special non-linear algebraic systems of steady-state (or pseudo-steady-state) equations. In general case it is impossible to solve explicitly a system of non-linear steady-state (or pseudo-steady-state) equations. However, in the case of mass-action-law-model it is always possible to apply to this system a method of elimination of variables and reduce it to a polynomial in one variable [4], i.e., a polynomial in terms of the steady-state reaction rate. We refer a polynomial in the steady-state reaction as a kinetic polynomial. The idea of this polynomial was firstly emphasized in [5]. 

Steady-state thermodynamics in the linear range provides a good glimpse of the non-equilibrium region close to equilibrium. The utility of steady-state thermodynamics is illustrated in the case of electro-kinetic phenomena in Parts Two and Three in the regions more and more distant from equilibrium (non-linear steady state, bistability, oscillations, pattern formation) including complexity and complex phenomena. 

NON-LINEAR STEADY STATES - DISSIPATIVE STRUCTURE (TIME ORDER AND SPACE ORDER)... 

Non-linear steady states 7.3.1. Electro-osmotic pressure... 

Chapter 7. Non-linear Steady States where Ai/ is equal to ATred. Accordingly... 

Non-linear flux equations and non-linear steady states in chemical reactions... 

Steady states, as we have seen in Part One, are obtained when fluxes in opposite directions are involved. In electro-osmosis, hydrodynamic flow is opposed by electro-osmotic flux. In thermo-osmosis, hydrodynamic flow is opposed by thermo-osmotic flux. In case of chemical reactions, such situations can arise when positive feedback is opposed by negative feedback. For example, when autocatalysis is opposed by inhibitory reaction, steady state can be attained. However, the reaction rates are non-linear and have only non-linear steady states in practice. We illustrate this point by the following example. 

On account of severe limitations, experimental studies on linear steady state could not be undertaken so far. However, the non-linear range experimental studies on bistability in reacting systems (Chapter 8) and chemical oscillations in CSTR do provide convincing examples of stable non-linear steady states under specific circumstances. 

Specific features of non-linear steady states (beyond linear steady state) can be summarized as follows ... 

Non-linear steady state bistability in Economics and Physiology Oscillations spatio-temporal oscillatory features in population Chaos and highly complex time series in Sociology and Economics Fractal growth of cities... 

Fluxes can be non-linear as pointed out in Chapter 7. Beyond non-linear steady-state range, non-linear dynamics as well as non-linear kinetics can be applied profitably for investigating the far from equilibrium phenomena, using the concepts based on the corresponding studies of physico-chemical phenomena as discussed in Parts Two to Four. 

Equilibrium state —Linear steady state close to equilibrium —Steady state —> Non-linear steady state — Bifurcation phenomena —> Multi-stability —> Temporal and spatio-temporal oscillations —> More complex situations (chaos, turbulence, pattern formation, fractal growth). All these stages have been discussed in different chapters of the book. 

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