Entropy flux vector

This relation can be split into two types of contributions The first term in (6.2.13) involves the divergence of the flux T 1(Jq - X(k)Mk k) In the context of Eq. (6.2.13) it therefore clearly makes sense to define an entropy flux vector by the following relation ... 

The starting point for the rigorous derivation of the diffusive fluxes in terms of the activity is the entropy equation as given by (1.170), wherein the entropy flux vector is defined by (1.171) and the rate of entropy production per unit volume is written (1.172) as discussed in sec. 1.2.4. 

Furthermore, it is noted that the first three terms in the brackets on the RHS of (2.452) are similar to the terms in the Gibbs-Duhem relation (2.451). However, two of these three terms do not contribute to the sum. The pressure term vanishes because js = 0. The two enthalpy terms obviously cancel because they are identical with opposite signs. Moreover, the last term in the brackets on the RHS of (2.452), involving the sum of external forces, is zero = 0- Oue of the two remaining terms, i.e., the one containing the enthalpy, we combine with the q term. Hence, the modified entropy flux vector (1.171) and production terms (1.172) become ... 

Here, S = 5 gj is the surface stress, f r the external body force per unit mass of material surface, Sa- the specific internal surface energy, q . = q a the surface heat flux vector, rjg. the surface entropy density, So- = i a the surface entropy flux vector, and h /d the surface entropy production. 

We reintroduce here the expression for the local rate of entropy production in an isotropic medium that involves the entropy flux vector, Eq. (6.1.27) ... 

In connection with the dissipation mechanism of the entropy inequality, the relations for the heat flux vector... 

While this expression looks much more complicated it is actually readily interpreted. Note that the quantity involving square brackets in the first term on the right-hand side represents the divergence of a set of flux vectors. Now, Eq. (6.1.3), specialized to the entropy balance, has the form... 

In order to solve the conservation or transport equations (mass, momentum, energy, and entropy) in terms of the dependent variables n, Vo,U, and , we must further resolve the expressions for the flux vectors— P, q, and s and entropy generation Sg. This resolution is the subject of closure, which will be treated in some detail in the next chapter. However, as a matter of illustration and for future reference, we can resolve the flux vector expression for what is called the local equilibrium approximation, i.e., we assume that the iV-molecule distribution function locally follows the equilibrium form developed in Chap. 4, i.e., we write [cf Eq. (4.34)]... 

The general equations of change given in the previous chapter show that the property flux vectors P, q, and s depend on the nonequi-lihrium behavior of the lower-order distribution functions g(r, R, t), f2(r, rf, p, p, t), and fi(r, P, t). These functions are, in turn, obtained from solutions to the reduced Liouville equation (RLE) given in Chap. 3. Unfortunately, this equation is difficult to solve without a significant number of approximations. On the other hand, these approximate solutions have led to the theoretical basis of the so-called phenomenological laws, such as Newton s law of viscosity, Fourier s law of heat conduction, and Boltzmann s entropy generation, and have consequently provided a firm molecular, theoretical basis for such well-known equations as the Navier-Stokes equation in fluid mechanics, Laplace s equation in heat transfer, and the second law of thermodynamics, respectively. Furthermore, theoretical expressions to quantitatively predict fluid transport properties, such as the coefficient of viscosity and thermal... 

Ji is the electron flux vector, C is the electrochemical potential acting on the electron flux, and Js represents the total entropy density flux vector. We choose this expression, rather than a version based on Eq. (6.1.29), because we wish to treat separately the effects of temperature and of electrochemical potential. The latter involves aU the contributions associated with temperature gradients, electron density gradients, and the externally imposed electrostatic field. It is expedient to introduce a current density vector as = e Ti- Then, along one dimension, we adopt i) = T 3s T + -V( /e) as our dissipation function. For this unidirectional flow pattern, the... 

Extended nonequilibrium thermodynamics is not based on the local equilibrium hypothesis, and uses the conserved variables and nonconserved dissipative fluxes as the independent variables to establish evolution equations for the dissipative fluxes satisfying the second law of thermodynamics. For conservation laws in hydrodynamic systems, the independent variables are the mass density, p, velocity, v, and specific internal energy, u, while the nonconserved variables are the heat flux, shear and bulk viscous pressure, diffusion flux, and electrical flux. For the generalized entropy with the properties of additivity and convex function considered, extended nonequilibrium thermodynamics formulations provide a more complete formulation of transport and rate processes beyond local equilibrium. The formulations can relate microscopic phenomena to a macroscopic thermodynamic interpretation by deriving the generalized transport laws expressed in terms of the generalized frequency and wave-vector-dependent transport coefficients. 

For each balance law, the values of -0, J and 4> defines the transported quantity, the diffusion flux and the source term, respectively, v denotes the velocity vector, T the total stress tensor, gc the net external body force per unit of mass, e the internal energy per unit of mass, q the heat flux, s the entropy per unit mass, h the enthalpy per unit mass, u>s the mass fraction of species s, and T the temperature. 

It should be stressed that because the vectors J and B lie in orthogonal subspaces, cf. (4.41), (4.174), the product B.J vanishes and B does not appear in (4.178). Consequently, theories of irreversible thermodynamics which try to find fiuxes and forces from the production of entropy overlook the dependence of reaction rate (flux) on force B, see also (4.179) below. 

As we noted below, the equation (4.489) the expressions (4.493) show that dependence of reaction rate on affinity is not so simple [158, 159] as it is assumed in classical non-equilibrium thermodynamics [1, 3, 4, 130] based on entropy production (by chemical reactions), i.e. as a product of fluxes and driving forces (4.178). Projection B of chemical potential vector p, to the subspace W also plays a role in expression for reaction rates J as (4.493) in our example the affinity A is projection of p into orthogonal reaction subspace V only, cf. (4.174). Cf. detailed discussion and criticism in review [108] and references [159, 160]. 

We shall study the system in Fig. 4. by means of the rininimitm principle of the global entropy production. Let the system be a so called current-tube, which is a tube where the boundary surface is constructed by those vector-lines of the thermodynamic fluxes, which fit on a closed line (for example circle, polygon). 

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