Applied Non-Equilibrium Thermodynamics

Non-equilibrium thermodynamics describes all kinds of transport processes. This chapter must focus on a few, namely transport of heat and mass in homogeneous and heterogeneous systems, in the absence or presence of chemical reactions. This introduction gives a brief history of the field, a list of good reasons for why the field is important, and a discussion of a basic assumption (See section 14.1.2). We then proceed to examples of applications in the three sections that follow. 

The field resulted from the work of many scientists with the objective to find a more useful formulation of the second law of thermodynamics than the familiar inequality AS 0. The effort started in 1856 with Thomson s studies of thermoelectricity. Onsager is, however, considered as the founder of the field with his papers from 1931, see also his collected works, because he put earlier research by Thomson, Boltzmann, Nernst, Duhem, Jauman and Einstein into the proper perspective. Onsager was given the Nobel prize in chemistry in 1968 for this work. 

Edited by A. R. H. Goodwin, J. V. Sengers and C. J. Peters International Union of Pure and Applied Chemistry 2010 Published by the Royal Society of Chemistry, www.rsc.org 

In non-equilibrium thermodynamics, the second law is reformulated using the local entropy production in the system, cr, which is given by the product sum of the so-called conjugate fluxes, and forces, A, in the system. Using the assumption of local equilibrium, the second law becomes 

In non-equilibrium thermodynamics, we first need to choose a complete set of independent extensive variables, a,-. The fluxes and forces are then determined from 

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Applied Non-Equilibrium Thermodynamics The entropy production reduces further to... 

Applied Non-Equilibrium Thermodynamics Substitution of eq 14.63 into eq 14.62 gives ... 

In addition, cure time is increased five minutes for every 0.25 inches of thickness of a molding [6, 7]. In general, these rules do not apply to most polymeric systems because the phenomena of heat transfer and cure kinetics have been over-simplified. The cure rate depends on the basic polymers, curatives, cure temperature, and filler loading. The prediction of cure rate will be discussed from a new model of cure kinetics which is developed from the concept of a non-equilibrium thermodynamic fluctuation theory of chemical relaxation. 

Modern opinion views the Nernst-Plank theory as a special case of applying the thermodynamics of irreversible processes to ion exchange. It may also be argued theoretically and experimentally that the observed characteristics of ion exchange rate behaviour can only be fully explained by including chemical reaction as a flux-coupling mechanism as well as the diffusion potential. From a research standpoint it is most probable that future theoretical advances in ion exchange kinetics will result from the further application of non-equilibrium thermodynamics. 

Non-equilibrium thermodynamics was founded by Onsager. The theory was further elaborated by de Groot and Mazur and Prigogine. The theory is based on the hypothesis of local equilibrium a volume element in a non-equilibrium system is in local equilibrium when the normal thermodynamic relations apply to the element. Evidence is emerging that show that many systems of interest in the process industry are in local equilibrium by this criterion. " Onsager prescribed that each flux be connected to its conjugate force via the extensive variable that defines the flux. - ... 

Mesoscopic non-equilibrium thermodynamics provides a description of activated processes. In the case considered here, when crystallization proceeds by the formation of spherical clusters, the process can be characterized by a coordinate y, which may represent for instance the number of monomers in a cluster, its radius or even a global-order parameter indicating the degree of crystallinity. Polymer crystallization can be viewed as a diffusion process through the free energy barrier that separates the melted phase from the crystalline phase. From mesoscopic non-equilibrium thermodynamics we can analyze the kinetic of the process. Before proceeding to discuss this point, we will illustrate how the theory applies to the study of general activated processes. 

Steady states are among the phenomena that non-equilibrium thermodynamics studies. When one is not too far from equilibrium it can be shown that the steady states are stable. On the other hand, when far from equilibrium, certain systems can make transitions to states exhibiting "dissipative structures." The theory of non-equilibrium developed by I. Prigogine, is quite general and has been applied to a wide range of phenomena. It is the aim of this lecture to introduce this field with a few examples. 

While Belousov was describing his e)q)eriments into oscillatory chemical reactions, Ilya Prigogine in Brussels was developing theoretical models of nonequilibrium thermodynamics and ended with the notion of "structure dissipative" for which he was awarded the 1977 Nobel Prize in Chemistry. The concept of "Dissipative Structure" is ejq)licitly mentioned in the Nobel quotation "The 1977 Nobel Prize in Chemistry has been awarded to Professor Ilya Prigogine, Brussels, for his contributions to non-equilibrium thermodynamics, particularly the theory of dissipative structures". In the first half of the 1950s, Glansdorff and Balescu defined with Prigogine the thermodynamic criteria necessary for oscillatory behavior in dissipative systems [7]. Nicohs and Lefever then applied these to models of autocatalytic reactions [8]. 

V.l. Concepts of Non-Equilibrium Thermodynamics as Applied to Transfer Processes in Disperse Systems. General Principles of the Theory of Percolations... 

It was the proof of dispersion [4b], the conductivity jump based on it due to flocculation [17a] and the optimisation thereof [17b,d], that first forced us to develop the new non-equilibrium thermodynamics theory of heterogeneous systems [19,17c,37] that applies to carbon-black compounds and ICP blends. The presumed necessity to achieve good dispersions compelled us to polymerise ICPs of increasingly high purity and to develop, to this end, the analytical techniques that yielded contributions to the understanding of structural aspects (see Sections 3,7.1 and 3.6). 

If the entropy export is supercritical, as shown by modem non-equilibrium thermodynamics, a sudden self-organisation of dissipative structures is observed. If the results of non-equilibrium thermodynamics are being applied to them, many systems in nature and technology are becoming better understood [77c], The most important consequence for the colloidal scientists is that the phenomenon of sudden occurrence of... 

We have seen that the basic issues of coupled water transport can be formulated in quite a general way. This general framework has been applied to the several elementary models discussed in this chapter but may equally well be applied to comprehensive computer simulations of epithelial function. For any model, comparison with experimental data necessitates computing the fluxes at equal bathing media, the derivatives of these fluxes with respect to the bath conditions, the bath conditions required for mucosal and serosal transport equilibrium, and the strength of transport. Via the formulation of non-equilibrium thermodynamics the flux derivatives can be related to the experimentally determined epithelial permeabilities. For analytical work, the fluxes and flux derivatives at equal bathing solutions can be used to assess transport isotonicity, and, with less certainty, estimate the strength of transport. 

In order to simplify these equafimis, the basic principle of microscopic reversibility introduced by Onsager in the 1930s can be evoked [70]. According to this principle, used to derive the reciprocal relations in non-equilibrium thermodynamics, there is a local reversibility of all (sub-)processes even though the system is out of equilibrium. Hence, applying this principle, which is strictly speaking only likely to be valid close to equilibrium, we can write ... 

The data reduction of vapor-pressure osmometry (VPO) follows to some extent the same relations as outlined above. However, from its basic principles, it is not an equilibrium method, since one measures the (very) small difference between the boiling point temperatures of the pure solvent drop and the polymer solution drop in a dynamic regime. This temperature difference is the starting point for determining solvent activities. There is an analogy to the boiling point elevation in thermodynamic equilibrium. Therefore, in the steady state period of the experiment, the following relation can be applied if one assumes that the steady state is sufficiently near the vapor-liquid equilibrium and linear non-equilibrium thermodynamics is valid ... 

Theories based on non-equilibrium thermodynamics [3-8] have been applied extensively to elucidate the phenomenon of thermo-osmosis. The methodology of nonequilibrium thermodynamics essentially involves the evaluation of entropy production by the application of the laws of conservation of mass and energy and Gibbs equation. Appropriate fluxes and forces are chosen by suitably splitting the expression for entropy production and subsequently, thermodynamic transport equations are written. The theory of thermo-osmosis based on non-equilibrium thermodynamics is discussed below. 

Kedem ° attempted to circumvent the paradox implicit in the driving of vectorial transport processes with supposedly scalar chemical forces by introducing a vectorial cross coefficient in a non-equilibrium thermodynamic definition of active transport. Jardetzky , however, stated that the direct coupling between a metabolic reaction and a transport process, implied by Kedem s vectorial cross coefficient, was impossible because it would contravene the Curie principle (see also ref. 11). Katchalsky and Kedem, later supported by Moszynski et al, answered the criticism of Jardetzky by saying that Langeland had shown that the principle of Curie applies... 

B. Baranowski, Non-equilibrium thermodynamics as applied to membrane transport, J. Membrane Sci, 57 (1991) 119-159... 

Non-equilibrium thermodynamics have been applied to all kinds of membrane processes, as well as to dilute solutions consisting of a solvent (usually water) and a solute [5,6]. The characteristics of a membrane in such systems may be described in terms of three coefficients or transport parameters the solvent permeability L, the solute permeability co and the reflection coefficient o. Using water as the solvent (index w) and with a given solute (index s), the dissipation function (entropy production) in a dilute solution is the sum of the solvent flow and solute flow multiplied by their conjugated driving forces ... 

The history and fundamentals of continuous thermodynamics will be briefly presented here and has been discussed in detail elsewhere. Before the 1980 s many authors applied continuous distribution functions to specific cases of non-equilibrium thermodynamics, statistical thermodynamics, the VLE of petroleum fractions and the LLE of polydisperse polymer systems. Starting in 1980 a consistent version of chemical thermodynamics directly based on continuous distribution functions was developed and called continuous thermodynamics. The work of Kehlen and Ratzsch," " Gualtieri et al., Salacuse and Stell, Briano and Glandt," are to be mentioned as sources of information. In the following years several groups applied continuous thermodynamics to nearly all important types of polydisperse systems." Cotterman and Prausnitz reviewed the literature up until about 1990. In the 1980 s continuous modelling of phase equilibria was mostly focused on polymer systems, petroleum fractions and natural gases. In the last ten years, this has been expanded to also include problems with asphaltene precipitation from crude oils and wax precipitation from hydrocarbon mixtures. In section 9.4 the more recent papers are discussed. 

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