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The Gibbs equation in its local form, Eq. (3.26) together with Eq. (3.7),

(4.9)

is needed to calculate the entropy production. The time derivative of the entropy density becomes, using Eq. (4.9):

(4.10)

Partial derivatives are used since the variables are position and time dependent.

Exercise 4.2.1.Derive Eq. (4.1) by considering changes in a volume element fixed with respect to the walls.

•Solution: The change of entropy is due to the flux in and out of the volume element plus an increase due to the entropy production:

where σ(x, t) is the entropy production per unit of volume. The cross section is equal to the volume divided by dx. We obtain in the limit of small dx

By dividing this equation left and right by the volume, one obtains Eq. (4.1).

By introducing Eqs. (4.3) and (4.7) into (4.10), using the rule for derivation of products, and solving for ∂s(x, t)/∂t, we obtain as balance equation for the entropy density:

(4.11)

To simplify notation, we usually suppress the (x, t) for all variables. By comparing the above equation with Eq. (4.1), we identify the entropy flux

(4.12)

and the entropy production in the system

(4.13)

Here, Sj is the partial molar entropy of component j. By replacing the total heat flux Jq by the entropy flux Js, we obtain an alternative expression

(4.14)

We furthermore replace the total heat flux Jq by the often more practical measurable heat flux J′q using Eq. (4.12). The result is

(4.15)

where ∂μj, T/∂x = ∂μj/∂x + Sj∂T/∂x is the derivative of the chemical potential keeping the temperature constant, see Appendix 3.A. Finally, when one describes heat and charge transport, it is sometimes convenient to replace the total heat flux by the energy flux Ju, which is defined by (see Chapters 9, 15 and 19)

(4.16)

This gives for the entropy production

(4.17)

The results for σ were derived using only the assumption of local equilibrium, see Sec. 3.5. Local equilibrium does not imply local chemical equilibrium. Local chemical equilibrium is a special case of local equilibrium [23, 71].

The entropy production contains pairs of fluxes and forces. These are the conjugate fluxes and forces defined in Chapter 1. The conjugate flux–force pairs in Eqs. (4.13)–(4.15) and (4.17) are different. The different sets of pairs are, however, equivalent and describe the same physical situation. The problem one wants to describe dictates the form that is most convenient. Convenient is a form that describes the system in the most direct way. If, for instance, the system is such that the chemical potentials of all components are constant, Eq. (4.14) is convenient because all terms containing their gradients are zero. The four alternative expressions have been given, as a help to find the appropriate final form. All forces and fluxes have a direction, except the last pair, and are thus vectors. The chemical reaction has a scalar flux and force. We shall discuss the consequences of this for isotropic systems in Sec. 7.6.

Remark 4.1.De Groot and Mazur [23] use fluxes in the barycentric frame of reference. We always use fluxes in the laboratory frame of reference. The entropy production is invariant for this transformation. This can be verified for instance in Eq. (4.15). In this expression, the only fluxes that depend on the frame of reference are the component fluxes. Subtracting from these component fluxes, the molar densities times the barycentric velocity, gives a total change which is zero according to Gibbs–Duhem’s equation. It follows that one may replace the component fluxes in the laboratory frame of reference by those in the barycentric frame of reference. See also Sec.4.4.2.

The separate products do not only give pure losses of work (cf. Secs. 2.4 and 7.5). Their sum does. For instance, the electric power per unit of volume −(j∂ϕ/∂x) does not necessarily give only an ohmic contribution to the entropy production, there may also be electric work terms included in the product, as we shall see in detail in Chapters 9 and 10. Each product contains normally work terms as well as energy storage terms. It is their combination which gives the entropy production rate, and the work that is lost per unit of time, dW_lost/dt = T₀ ∫σdV (see Sec. 2.4). The expressions (4.13)–(4.15) and (4.17) can thus be used to find the second law efficiency of a process [32].

De Groot and Mazur [23] used the affinity A of the chemical reaction rather than the reaction Gibbs energy in Eq. (4.15). According to Kondepudi and Prigogine [59, p. 111], the reaction Gibbs energy is primarily used in connection with equilibrium states and reversible processes, while De Donder’s affinity concept is more general, it relates chemical reactions to entropy. We dispute that there is a principle difference between the two concepts. The affinity is simply equal to minus the reaction Gibbs energy. We use the reaction Gibbs energy, because chemist are more familiar with this concept.

Ross and Mazur [118] showed that the contribution to σ from the chemical reaction was equal to the product of r and the driving force −Δ_rG/T also when the reaction rate is a nonlinear function of the driving force, provided that the ensemble of particles is nearly Maxwellian. Prigogine showed that the Gibbs relation was valid for such conditions [113].

Haase [24] defined and used the dissipation function in his monograph on Thermodynamics of Irreversible Processes. The dissipation function is still in focus in many books, see, e.g., [119]. In homogeneous systems, Haase defined Ψ = TdS_irr/dt, where dS_irr/dt is the rate of increase of the entropy due to processes which occur inside the system. For continuous systems, in which the temperature can vary from point to point, he used the definition Ψ = Tσ [25, p. 83]. This last definition is analogous to Raleigh’s dissipation function for hydrodynamic flow. For heterogeneous systems, which are discussed in this book, neither of the above definitions can be used. On pp. 161–164, Haase considered the case of two phases in thermal contact with each other. After calculating the entropy production, he multiplied it with the temperature of one of the phases. In the resulting expression, he then needs to linearize in the temperature difference of the phases in order to obtain an expression for Ψ which is linear in the temperature difference. He contributed thereby to the erroneous idea that a non-equilibrium thermodynamic description should be fully linearized. The entropy production he first derived was perfectly linear in the difference of the inverse temperature. When one uses the entropy production rather than Ψ, as is systematically done in this book, there is no need to further linearize. Instead of multiplying with the temperature of one of the phases, it is appropriate to multiply with the temperature of the environment as is done in exergy analysis to find the lost work. We discussed in Sec. 2.4 that this gives the lost work. We strongly advise against using Ψ in heterogeneous systems, see Sec. 4.2.1 for further arguments.

Changing the frame of reference does not change the value or the physical interpretation of σ. The entropy production is an absolute quantity. It is invariant for transformations to other frames of reference, i.e. it is “Galilean invariant”. Possible choices for the frame of reference will be discussed in Sec. 4.4.

As mentioned above, we are dealing only with one-dimensional transport problems in this book. That is, we consider only transport normal to the surface. All densities and fluxes are assumed to be independent of the coordinates along the surface and the vectorial fluxes are assumed to be directed normal to the surface. These assumptions can be made because the systems we consider are isotropic along the surface. A system is isotropic if it is invariant for translation along the surface, for rotation around a normal on the surface, and for reflection with respect to planes normal to the surface.

Compression is a scalar phenomenon that couples to the reactions in the bulk phases. This coupling is small, however, since density variations relax on a timescale of the size of the system divided by the velocity of sound. The assumption of mechanical equilibrium, grad p = 0, used to derive Eq. (4.7), is therefore often valid, see [23] for a discussion of the more general case. In the absence of compression, the compressional (bulk) viscosity does not play a role.

Convection and shear flow will disturb chemical reactions and transport of heat and mass, but do not couple to them directly. Convection and/or shear flow can be described by non-equilibrium thermodynamics, see, e.g., [33, 34], but will not be considered in this book.