Читать книгу Non-equilibrium Thermodynamics of Heterogeneous Systems - Signe Kjelstrup - Страница 27

We derive the entropy production in a volume element in a homogeneous phase for transport of heat, mass, charge and chemical reactions. The entropy production determines the conjugate fluxes and forces in the phase. Equivalent forms of the entropy production are given.

The second law of thermodynamics, Eq. (2.14), says that the entropy change of the system plus its surroundings is positive for irreversible processes and zero for reversible processes. The law gives the direction of a process; it does not give its rate. Non-equilibrium thermodynamics assumes that the second law remains valid locally, cf. Eq. (1.1). In this chapter, we shall derive the entropy production for a volume element in a homogeneous phase. In the following chapters, we shall find the corresponding expressions for a surface area element and a three-phase contact line element. In total, we will then be able to describe the rate of changes in heterogeneous systems.

The change of the entropy in a volume element in a homogeneous phase is given by the flow of entropy in and out of the volume element and by the entropy production inside the volume element:

(4.1)

Here, s(x, t) is the entropy density, Js(x, t) is the entropy flux in the laboratory frame of reference and σ(x, t) is the entropy production. As the entropy density and flux depend on both position and time, we use partial derivatives. We consider only one-dimensional transport problems.

Below a more explicit expression for σ will be found by combining

•mass balances;

•the first law of thermodynamics;

•the local form of the Gibbs equation.

We shall do this and compare the resulting expression for the time rate of change of the entropy density with Eq. (4.1). This will make it possible to identify the entropy flux as well as the entropy production. We shall find that σ can be written as the product sum of the conjugate fluxes and forces in the system, see Sec. 1.1. In the derivation, we follow Refs. [23, 30]. Electroneutral, polarizable, non-equilibrium systems, with and without chemical reactions are of interest. These are the systems that we encounter most often in nature, and also in industry. We shall model transport of heat, mass and charge in systems where reactions occur. We recommend to use the symbol list for check of dimensions in the equations. An introduction to non-equilibrium thermodynamics for homogeneous systems was given by Kjelstrup, Bedeaux, Johannessen and Gross [2, 3].

Consider a volume element V which is in local equilibrium in a polarizable, electroneutral bulk phase. The volume element does not move with respect to the walls (the laboratory frame of reference), see Fig. 4.1. All fluxes are with respect to the laboratory frame of reference.

The volume element has a sufficient number of particles to give a statistical basis for thermodynamic calculations. Its state is given by the temperature T(x, t), chemical potentials μj(x, t) for all the n neutral components, pressure p(x, t) and the equilibrium electric field E_eq(x, t) = D_eq(t)/ε(x, ^t). We shall take ε constant in the homogeneous phases. Both equilibrium electric fields are then independent of the position:

Figure 4.1A volume element in a bulk phase with transport where j′ indicates one of the fluxes.

(4.2)

The number of positive particles equals the number of negative particles, but their distribution within the volume element can lead to a polarization density.