Irreversible thermodynamics, development

Another theoretical basis of the superheated liquid-film concept lies on the irreversible thermodynamics developed by Prigogine [43]. According to this theory, irreversible chemical processes would be described (Equation 13.17) by extending the equation of De Donder, provided that simultaneous reactions were coupled in a certain thermodynamic model, as follows ... 

R D - research and development AG, AH, AS, q, w - classical thermodynamic significance J, X, L - fluxes, forces and phenomenological coefficients of irreversible thermodynamics ... 

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 1977. Professor Ilya Prigogine of the Free University of Brussels. Belgium, was awarded Ihe Nobel Prize in chemistry for his central role in the advances made in irreversible thermodynamics over the last ihrec decades. Prigogine and his associates investigated Ihe properties of systems far from equilibrium where a variety of phenomena exist that are not possible near or al equilibrium. These include chemical systems with multiple stationary states, chemical hysteresis, nucleation processes which give rise to transitions between multiple stationary states, oscillatory systems, the formation of stable and oscillatory macroscopic spatial structures, chemical waves, and Lhe critical behavior of fluctuations. As pointed out by I. Procaccia and J. Ross (Science. 198, 716—717, 1977). the central question concerns Ihe conditions of instability of the thermodynamic branch. The theory of stability of ordinary differential equations is well established. The problem that confronted Prigogine and his collaborators was to develop a thermodynamic theory of stability that spans the whole range of equilibrium and nonequilibrium phenomena. 

The experimental material concerning direct application of irreversible thermodynamics is very scarce. This is partly due to the fact that it is a recent development. 

Cells in which at least two electrolytic solutions are in contact are known as cells with liquid junction or with transference. Such cells are inherently irreversible and a complete thermodynamic development of them is beyond the scope of this book. However, cells with liquid junction are of sufficient importance that we discuss here the type that approximates a reversible cell most closely. 

Abstract A permeameter was developed for measurement of coupled flow phenomena in clayey materials. Results are presented on streaming potentials in a Na-bentonite induced by hydraulic flow of electrolyte solutions. Transport coefficients are derived from the experiments, assuming the theory of irreversible thermodynamics to be applicable. Hydraulic and electro-osmotic conductivities are consistent with data reported elsewhere. However the electrical conductivity of the clay is substantially lower. This is ascribed to the high compaction of the clay resulting in overlap of double layers... 

Multidisciplinary analytical and numerical models require development. These models should involve considerations of equilibrium and irreversible thermodynamics and kinetics of carbonate mineral-organic matter-water interactions within a sound hydrodynamic and basin evolution framework. 

While the formalism of irreversible thermodynamics provides an elegant framework for describing molecular displacements, it provides too little substance and too much conceptual difficulty to justify its development here. For instance, it provides no values, not even estimates, for various transport coefficients such as the diffusion coefficient. Cussler has noted the disappointment of scientists in several disciplines with the subject [7]. It is the author s opinion that a clearer understanding of the transport processes and interrelationships that underlie separations can be obtained from a mechanical-statistical approach. This is developed in the subsequent sections. 

Phenomenological transport relationships can be developed even in the absence of any knowledge of the mechanisms of transport through the membrane or any information about the membrane structure.10 The basis of irreversible thermodynamics assumes that if the system is divided into small enough subsystems in which local equilibrium exists, thermodynamic equations can be written for the subsystems. 

In this chapter we will derive the Boyle-Van t Hoff relation using the chemical potential of water, and in Chapter 3 (Section 3.6B) we will extend the treatment to penetrating solutes by using irreversible thermodynamics. Although the Boyle-Van t Hoff expression will be used to interpret the osmotic responses only of chloroplasts, the equations that will be developed are general and can be applied equally well to mitochondria, whole cells, or other membrane-surrounded bodies. 

In the next section we will use Equation 3.29 as the starting point for developing the expression from irreversible thermodynamics that describes the volume flux density. Because the development is lengthy and the details may obscure the final objective, namely, the derivation of Equation 3.40 for the volume flux density in terms of convenient parameters, the steps and the equations involved are summarized first ... 

We may stress at this point that the conception of the role played by irreversible processes developed here is quite different from that in classical thermodynamics. In the latter, irreversible changes appear only as undesirable effects which reduce the efficiency of heat engines and which one must attempt to eliminate. On the other hand thermodynamic coupling, enables us to predict results such as separations and syntheses, which would be quite impossible to derive in the absence of a consideration of irreversible changes. 

Equation 3.3.7 expresses the Onsager reciprocal relations (ORR), named after Lars Onsager who first established the principles of irreversible thermodynamics (Onsager, 1931). The ORR have been the subject of many journal papers receiving support as well as criticism, the latter from, in particular, Coleman and Truesdell (1960) and Truesdell (1969). We shall assume the validity of the ORR in the development that follows. 

Chapter 1 serves to remind readers of the basic continuity relations for mass, momentum, and energy. Mass transfer fluxes and reference velocity frames are discussed here. Chapter 2 introduces the Maxwell-Stefan relations and, in many ways, is the cornerstone of the theoretical developments in this book. Chapter 2 includes (in Section 2.4) an introductory treatment of diffusion in electrolyte systems. The reader is referred to a dedicated text (e.g., Newman, 1991) for further reading. Chapter 3 introduces the familiar Fick s law for binary mixtures and generalizes it for multicomponent systems. The short section on transformations between fluxes in Section 1.2.1 is needed only to accompany the material in Section 3.2.2. Chapter 2 (The Maxwell-Stefan relations) and Chapter 3 (Fick s laws) can be presented in reverse order if this suits the tastes of the instructor. The material on irreversible thermodynamics in Section 2.3 could be omitted from a short introductory course or postponed until it is required for the treatment of diffusion in electrolyte systems (Section 2.4) and for the development of constitutive relations for simultaneous heat and mass transfer (Section 11.2). The section on irreversible thermodynamics in Chapter 3 should be studied in conjunction with the application of multicomponent diffusion theory in Section 5.6. 

Finely Porous Model. In this model, solute and solvent permeate the membrane via pores which connect the high pressure and low pressure faces of the membrane. The finely porous model, which combines a viscous flow model eind a friction model (7, ), has been developed in detail and applied to RO data by Jonsson (9-12). The most recent work of Jonsson (12) treated several organic solutes including phenol and octanol, both of which exhibit solute preferential sorption. In his paper, Jonsson compared several models including that developed by Spiegler eind Kedem (13) (which is essentially an irreversible thermodynamics treatment), the finely porous model, the solution-diffusion Imperfection model (14), and a model developed by Pusch (15). Jonsson illustrated that the finely porous model is similar in form to the Spiegler-Kedem relationship. Both models fit the data equally well, although not with total accuracy. The Pusch model has a similar form and proves to be less accurate, while the solution-diffusion imperfection model is even less accurate. 

The methods of irreversible thermodynamics are useful in providing a quantitative approach to the phenomenon of electro-osmotic dewatering and its connection to other electrokinetic effects. The main ideas were developed by Overbeeki" and reviewed by DeGroot these ideas were applied by many workers to a number of problems, following the earlier papers of Overbeek and co-workers - on the treatment of electrokinetic phenomena in terms of irreversible thermodynamics. Recently we have shown that this approach can also be applied to EOD, as follows. 

Considerable effort has been expended in the attempt to develop a general theory of reaction rates through some extension of thermodynamics or statistical mechanics. Since neither of these sciences can, by themselves, yield any information about rates of reactions, some additional assumptions or postulates must be introduced. An important method of treating systems that are not in equilibrium has acquired the title of irreversible thermodynamics. Irreversible thermodynamics can be applied to those systems that are not too far from equilibrium. The theory is based on the thermodynamic principle that in every irreversible process, that is, in every process proceeding at a finite rate, entropy is created. This principle is used together with the fact that the entropy of an isolated system is a maximum at equilibrium, and with the principle of microscopic reversibility. The additional assumption involved is that systems that are slightly removed from equilibrium may be described statistically in much the same way as systems in equilibrium. 

The theory of absolute reaction rates, which i s based on statistical mechanics, was developed in full generality by H. Eyring in 1935, although it was foreshadowed in kinetic theory investigations as early as 1915. A simplified development of the equations will be given here. In this theory, we have a postulate of equilibrium away from equilibrium, applied more broadly here than in the irreversible thermodynamics. 

In the introduction of this section, we showed that the spacial derivative of the energy is a force and we illustrated associated changes, which we addressed as a flux. In irreversible thermodynamics the concept of force and flux is generalized. Actually, the development of the thermodynamics of irreversible processes was launched in the 1930s by Onsager [4, 5],... 

It should be emphasized that the above development is a lowest-order development in that only the first corrections to the local equilibrium values of the internal energy and heat capacity have been used, This is consistent with current practice in fluid dynamics and nonequilibiium thermodynamics. Further refinements can be developed if it is desired to make contact with some of the recent work in extended irreversible thermodynamics [36, 37]. 

I pay my tribute to Prof. B.N. Srivastava, who was primarily responsible for the development of my interest in Irreversible Thermodynamics. I am equally grateful to my former Ph.D. students and other collaborators who had been involved in theoretical and experimental studies in the areas covered in the book, whose references appear in... 

The Maxtwell-Stefan theory can be derived from continuum mechanics fPatta and Vilekar. 2010). irreversible thermodynamics fBird et al.. 2002). or the kinetic theory of gases (Hirschfelder et al.. 19541. Although the continuum mechanics approach is probably most powerful, for an introductory development, a simplified kinetic theory is easier to follow. The presentations of Taylor and Krishna (1993) and Wesselingh and Krishna (2Q00) are paraphrased in a somewhat loose manner here. 

All classic developments based on irreversible thermodynamics assume implicitly that the process does not deviate significantly from thermodynamic equilibrium In consequence, despite the fact the system is in evolution therefore in non-equilibrium, the state equation expressing the condition of thermodynamic equilibrium can still be used to reduce the number of independent state parameters by one in complex problems (for example, the density, pressure and temperature of the pore fluid transiting a porous solid is related by a state equation). This is strictly speaking an approximation. Its efficiency can only be assessed a posteriori by the results. 

The thermodynamics of irreversible processes are very useful for understanding and quantifying coupling phenomena. However, structure-related membrane models are more useful than the irreversible thermodynamic approach for developing specific membranes. A number of such transport models have been developed, partly based on the principles of the thermodynamics of irreversible processes, both for porous and nonporous membranes. Again, two types of structure will be considered here porous membranes, as found in microfUtration/uItrafiltration, and nonporous membranes of the type used in pervaporation/gas separation. 

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