Groot and Mazur

Having clarified these details, we can now introduce the momentum as defined by de Groot and Mazur as being equal to  

It is evident, as noted above, that only the difference V P - F is perfectly defined once the forces are known, as we can control that fact using equation [3.50] in the case of the electrical field. Equation [3.57] was written in relation to the barycentric motion. Equation [3.54] can also be transformed to reveal the material derivative. We obtain  

In reality, the Maxwell pressure tensor in relation to the Galilean frame of reference which, at time t and at the point in question, has the velocity v of the fluid, is expressed not by equation [3.60], but rather  

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The momentum equation (the Navier-Stokes equation) for fluid flow (De Groot and Mazur, 1962) is complicated and difficult to solve. It is the subject of fluid mechanics and dynamics and is not covered in this book. When fluid flow is discussed in this book, the focus is on the effect of the flow (such as a flow of constant velocity, or boundary flow) on mass transfer, not the dynamics of the flow itself. 

The full treatment of multicomponent diffusion requires a diffusion matrix because the diffusive flux of one component is affected by the concentration gradient of all other components. For an N-component system, there are N-1 independent components (because the concentrations of all components add up to 100% if mass fraction or molar fraction is used). Choose the Nth component as the dependent component and let n = N 1. The diffusive flux of the components can hence be written as (De Groot and Mazur, 1962)... 

Uphill diffusion occurs in binary systems because, strictly speaking, diffusion brings mass from high chemical potential to low chemical potential (De Groot and Mazur, 1962), or from high activity to low activity. Hence, in a binary system, a more rigorous flux law is (Zhang, 1993) ... 

A more rigorous way to generalize Pick s law is to use phenomenological equations based on linear irreversible thermodynamics. In this treatment of an N-component system, the diffusive flux of component i is (De Groot and Mazur,... 

In irreversible thermod3mamics, the second law of thermodynamics dictates that entropy of an isolated system can only increase. From the second law of thermodynamics, entropy production in a system must be positive. When this is applied to diffusion, it means that binary diffusivities as well as eigenvalues of diffusion matrix are real and positive if the phase is stable. This section shows the derivation (De Groot and Mazur, 1962). 

We start from the fundamental thermodynamic relation for an irreversible process (de Groot and Mazur, 1962),... 

Fundamental results in substantiating and extending the principle of detailed equilibrium to a wide range of chemical processes were obtained in 1931 by Onsager, though chemists had also applied this principle (see Chap. 2). A derivation of this principle from that of microscopic reversibility was reported by Tolman [19] and Boyd [20], In the presence of an external magnetic field it is possible that equilibrium is not detailed. Respective modifications of this principle were reported by de Groot and Mazur [21]. 

At thermodynamic equilibrium, Ttj must vanish for every reaction otherwise, the equilibrium could be shifted by adding catalysts or inhibitors to alter the nonzero rates TZj. Formal proofs of this "detailed balance principle are presented by de Groot and Mazur (1962). Therefore, setting 7 = 0 at equilibrium and using Eq. (2.4-3), we get... 

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. - ... 

This first chapter has been somewhat in the nature of a housekeeping exercise with regard to the definitions of mass and molar fluxes, reference velocities, and transformations from one reference frame to another. We shall not have occasion to use all of these definitions, but they have been included for reasons of completeness. Any reader who is interested in furthering their knowledge on this topic must refer to De Groot and Mazur (1962). 

Until now, we have considered that the ditfusion process took place under essentially isobaric conditions, in nonelectrolyte systems and in the absence of external force fields, such as centrifugal or electric fields. In this section we shall generalize our analysis to include the influence of external force fields. The best starting point for a generalized treatment is the theory of irreversible thermodynamics. The treatment below is similar to that given by Lightfoot (1974) but readers will also find the books by de Groot and Mazur (1962) and Haase (1969) very useful. 

The term, — 57 i Pci c gc)> reduces to the term applied for one component fluids, —p(v g), when gc is the same for all species. The interested reader is referred to de Groot and Mazur [32] for further discussion of these terms. 

In this section, an analysis based on the Boltzmann equation will be given. Before we proceed it is essential to recall that the translational terms on the LHS of the Boltzmann equation can be derived adopting two slightly different frameworks, i.e., considering either a fixed control volume (i.e., in which r and c are fixed and independent of time t) or a control volume that is allowed to move following a trajectory in phase space (i.e., in which r(t) and c t) are dependent of time t) both, of course, in accordance with the Liouville theorem. The pertinent moment equations can be derived based on any of these two frameworks, but we adopt the fixed control volume approach since it is normally simplest mathematically and most commonly used. The alternative derivation based on the moving control volume framework is described by de Groot and Mazur [22] (pp. 167-170). 

That is, as described by de Groot and Mazur [22] (chap. XI), Taylor and Krishna [96] and Cussler [20], it can be shown using the linearity postulate and the Onsager relations from irreversible thermodynamics that not all of... 

The derivation of such an expression is tedious and would require us to develop detailed microscopic equations of change for a mixture. Rather than doing this, we will merely write down the final result, referring the reader to the book by de Groot and Mazur for the details." Their result, with the slight modification of writing the chemical reaction term on a molar rather than a mass basis, is... 

Section 15.6 describes the deficiencies in the Fickian model and points out why an alternative model (the fourth) is needed for some situations. The alternative Maxwell-Stefan model of mass transfer and diffusivity is explored in Section 15.7. The Maxwell-Stefan model has advantages for nonideal systems and multicomponent mass transfer but is more difficult to couple to the mass balances when designing separators. The fifth model of mass transfer, the irreversible thermodynamics model fde Groot and Mazur. 1984 Ghorayeb and Firoozabadi. 2QQQ Haase. 1990T is useful in regions where phases are unstable and can split into two phases, but it is beyond the scope of this introductory treatment. The... 

Irreversible thermodynamics model. This model is useful in regions where phases are unstable and can split into two phases fde Groot and Mazur. 1984 Ghorayeb and Firoozabadi. 2000 Haase. 19901. However, this model is beyond the scope of this introductory treatment. 

Non-equilibrium thermodynamics (de Groot and Mazur, 1984 Prigogine, 1977) affords an abstract, and therefore very general, foundation for understanding pattern formation. The abstract nature of this approach makes direct application to geochemical problems difficult, but non-equilibrium thermodynamics does provide a powerful conceptual basis for thinking about pattern formation. 

Onsager s approach, by definition, is valid in the vicinity of equilibrium, and deviations of Cj from c, eq are assumed to be small. The symmetry of the Onsager matrix, = Lji, follows from the principle of microreversibility (see the classical monograph by de Groot and Mazur, 1962). 

Tolman (1938) and Boyd (1974) have described a derivation of the principle of detailed balance from the principle of microscopic reversibility. In the presence of an external magnetic field, it is possible that the equilibrium is not a detailed balance. De Groot and Mazur (1962) have formulated a modification of the principle of detailed balance for this case. 

As far as equilibrium thermodynamics is regarded, we shall closely follow the lines of CALLEN s (1960) book which is particularly recommended for further reading An application of thermodynamics especially to chemical processes is presented in detail in the book of PRIGOGINE and DEFAY (1954). For an introduction into thermo-dynanjics of irreversible processes the interested reader is referred to the books of de GROOT (1951), de GROOT and MAZUR (1962) and KATCHALSKY and CURRAN (1967). 

For a detailed proof, the reader is referred to the monograph by de GROOT and MAZUR (1962). 

If the parallel plates that bind the Uquid layer deviate fi om the horizontal plane by angle

convective heat loss, Qc, can then be calculated from the de Groot theory (de Groot and Mazur, 1962 Michels et al, 1962) as... 

Classical thermodynamics provides valuable information that can assist in the solution of practical problems in many fields of science and engineering. In particular, when applied to coupled problems that include mechano-thermal and chemo-thermal phenomena, it gives extremely valuable results (see e.g., de Groot and Mazur 1962 Kestin 1979 Kondepudi and Prigogine 1998). An informative account of thermodynamics of solids is given by Ericksen (1991). 

Note 3.6 (Notation of pressure and volume change for fluid and thermodynamic functions). Most textbooks on thermodynamics (de Groot and Mazur 1962 Kestin 1979 Kondepudi and Prigogine 1998) contain treatments of the perfect gas, therefore we have [Pg.100]

First, some fundamental concepts of thermodynamics are introduced and these will be discussed in greater detail in subsequent sections. In this Appendix we treat phenomena relevant to mechanics and temperature other effects including chemical fields (cf. Appendix E) and electromagnetic effects are excluded. Further expositions are given by de Groot and Mazur (1962), Kestin (1979) and Kondepudi and Prigogine (1998). 

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