Nonlinear irreversible processes

GEN.107. P. Glansdorff et I. Prigogine, Thermodynamique Processus irreversibles non lineaires (Thermodynamics Nonlinear irreversible processes). Encyclopaedia Universahs, 1177-1178. 

Lohmander U (1964) Non-Newtonian flow of dilute macromolecular solutions studied by capillary viscosimetry. Macromol Chem 72(1) 159-173 Maugin GA (1999) Thermomechanics of nonlinear irreversible processes. World Scientific, Singapore... 

Rather lately, an attempt to generalize this principle to a nonlinear irreversible process has been made by Klein, based on a statistical method for a simplified model. He demonstrated that the minimum production properties are also a useful approximation criterion for the stationary state even when the latter is very far from equilibrium. According to a simple example of an irreversible process given in his calculation, it has been found that the entropy production does not decrease monotonically, but passes through its minimum en route to the stationary state. We have already mentioned these results in connection with the variational principle in nonlinear irreversible processes. 

The fundamental question in transport theory is Can one describe processes in nonequilibrium systems with the help of (local) thermodynamic functions of state (thermodynamic variables) This question can only be checked experimentally. On an atomic level, statistical mechanics is the appropriate theory. Since the entropy, 5, is the characteristic function for the formulation of equilibria (in a closed system), the deviation, SS, from the equilibrium value, S0, is the function which we need to use for the description of non-equilibria. Since we are interested in processes (i.e., changes in a system over time), the entropy production rate a = SS is the relevant function in irreversible thermodynamics. Irreversible processes involve linear reactions (rates 55) as well as nonlinear ones. We will be mainly concerned with processes that occur near equilibrium and so we can linearize the kinetic equations. The early development of this theory was mainly due to the Norwegian Lars Onsager. Let us regard the entropy S(a,/3,. ..) as a function of the (extensive) state variables a,/ ,. .. .which are either constant (fi,.. .) or can be controlled and measured (a). In terms of the entropy production rate, we have (9a/0f=a)... 

In Fig. 3.14a, the dimensionless limiting current 7j ne(t)/7j ne(tp) (where lp is the total duration of the potential step) at a planar electrode is plotted versus 1 / ft under the Butler-Volmer (solid line) and Marcus-Hush (dashed lines) treatments for a fully irreversible process with k° = 10 4 cm s 1, where the differences between both models are more apparent according to the above discussion. Regarding the BV model, a unique curve is predicted independently of the electrode kinetics with a slope unity and a null intercept. With respect to the MH model, for typical values of the reorganization energy (X = 0.5 — 1 eV, A 20 — 40 [4]), the variation of the limiting current with time compares well with that predicted by Butler-Volmer kinetics. On the other hand, for small X values (A < 20) and short times, differences between the BV and MH results are observed such that the current expected with the MH model is smaller. In addition, a nonlinear dependence of 7 1 e(fp) with 1 / /l i s predicted, and any attempt at linearization would result in poor correlation coefficient and a slope smaller than unity and non-null intercept. 

Irreversible processes may promote disorder at near equilibrium, and promote order at far from equilibrium known as the nonlinear region. For systems at far from global equilibrium, flows are no longer linear functions of the forces, and there are no general extremum principles to predict the final state. Chemical reactions may reach the nonlinear region easily, since the affinities of such systems are in the range of 10-100 kJ/mol. However, transport processes mainly take place in the linear region of the thermodynamic branch. 

A fundamental corollary of the Glansdorf Prigogine criterion (3.2) is a potentiality of the formation of ordered structures at the occurrence of irreversible processes in the region of nonlinear thermodynamics in open systems that are far from their equilibrium. Prigogine created the term dissipative structures to describe the structures that arise when some controlling parameters exceed certain critical values and are classified as spatial, temporal, or spatial temporal. Some typical dissipative structures are discussed in Sections 3.5 and 4.6. 

A second procedure, using the methods of thermodynamics applied to Irreversible processes, offers another new approach for understanding the failure of materials. For example, the equilibrium thermodynamics of closed systems predicts that a system will evolve In a manner that minimizes Its energy (or maximizes Its entropy). The thermodynamics of Irreversible processes In open systems predicts that the system will evolve In a manner that minimizes the dissipation of energy under the constraint that a balance of power Is maintained between the system and Its environment. Application of these principles of nonlinear Irreversible thermodynamics has made possible a formal relationship between thermodynamics, molecular and morphological structural parameters. 

Before introducing the notion of nonequilibrium thermodynamics we shall first summarize briefly the linear and nonlinear laws between thermodynamic fluxes and forces. A key concept when describing an irreversible process is the macroscopic state parameter of an adiabatically isolated system These parameters are denoted by. At equilibrium the state parameters have values A , while an arbitrary state which is near or far from the equilibrium may be specified by the deviations from the equilibrium state ... 

Samohyl, I., Thermodynamics of Irreversible Processes in Fluid Mixtures Approached by Rational Thermodynamics. 1987, Leipzig B. G. Teubner. Stratonovich, R. L., Nonlinear Nonequilibrium Thermodynamics. 1992, New York Springer-Verlag. 

If the steady state concentrations of the components are shifted, but not too far from their equilibrium values, the interconnection between the fluxes and chemical forces (chemical affinities, in our case) should satisfy the well-known linear relationships that are usually postulated in the linear thermodynamics of irreversible processes [15-18]. We do not consider here the phenomenological equations of nonequilibrium thermodynamics. For details the reader can refer to numerous excellent monographs and review articles devoted to the applications of nonequilibrium thermodynamics in the description of chemical reactions and biological processes (see, for instance, [22-30]). In many cases, the conventional phenomenological approaches of linear and nonlinear nonequilibrium thermodynamics appear to be useful tools for the... 

There are several control problems in chemical reactors. One of the most commonly studied is the temperature stabilization in exothermic monomolec-ular irreversible reaction A B in a cooled continuous-stirred tank reactor, CSTR. Main theoretical questions in control of chemical reactors address the design of control functions such that, for instance (i) feedback compensates the nonlinear nature of the chemical process to induce linear stable behavior (ii) stabilization is attained in spite of constrains in input control (e.g., bounded control or anti-reset windup) (iii) temperature is regulated in spite of uncertain kinetic model (parametric or kinetics type) or (iv) stabilization is achieved in presence of recycle streams. In addition, reactor stabilization should be achieved for set of physically realizable initial conditions, (i.e., global... 

Two experimental systems have been used to illustrate the theory for two-step surface electrode mechanism. O Dea et al. [90] studied the reduction of Dimethyl Yellow (4-(dimethylamino)azobenzene) adsorbed on a mercury electrode using the theory for two-step surface process in which the second redox step is totally irreversible. The thermodynamic and kinetic parameters have been derived from a pool of 11 experimental voltammograms with the aid of COOL algorithm for nonlinear least-squares analysis. In Britton-Robinson buffer at pH 6.0 and for a surface concentration of 1.73 X 10 molcm, the parameters of the two-step reduction of Dimethyl Yellow are iff = —0.397 0.001 V vs. SCE, Oc,i = 0.43 0.02, A sur,i =... 

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