Non-equilibrium Thermodynamics of Heterogeneous Systems. Signe KjelstrupЧитать онлайн книгу.
is that the system is in mechanical equilibrium, a condition we shall use throughout this book. Contrary to the situation in vacuum, the electric field in a medium is not a conservative field.
The total heat flux across the volume element is given by
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(4.8) |
It consists of the measurable heat flux J′q(x, t) and the partial molar enthalpies Hj(x, t) carried by the neutral component fluxes Jj(x, t). The measurable heat flux is independent of the frame of reference. This is not the case for the total heat flux. We discuss the frame of reference further in Secs. 4.2, 4.4, and also in Chapter 12, which deals with multi-component diffusion. The first law is illustrated in Fig. 4.2. De Groot and Mazur [23] used charged and uncharged components, which are appropriate for systems that are not electroneutral, and defined a reduced heat flux in analogy with Eq. (4.8). In the electroneutral systems, which we consider, it is more appropriate to use the smaller number of uncharged components. The measurable heat flux differs from the reduced heat flux by a contribution proportional to the electric current density, see Appendix 4.A for a discussion.
Figure 4.2The energy change of a volume element with fluxes of heat and charge across the boundaries.
4.2The entropy production
The Gibbs equation in its local form, Eq. (3.26) together with Eq. (3.7),
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(4.9) |
is needed to calculate the entropy production. The time derivative of the entropy density becomes, using Eq. (4.9):
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(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:
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(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
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(4.12) |
and the entropy production in the system
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(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
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(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
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(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)
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(4.16) |
This gives for the entropy production
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(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