Entropy flux

The entropy flux from the control volume associated with the heat transfer is... 

The magnitude of the entropy flux is the entropy transported through unit area per unit time, which is the divergence V Js- It is convenient to define all extensive variables per unit volume (denoted here by lower case symbols)... 

The entropy is given in terms of the specific entropy shy S = J pscPr so that the entropy flux out of the system is related to the entropy current density by... 

Unlike mass, which is conserved, leading to dp/dt= —V ], entropy is not conserved. That is, in addition to entropy flux that would lead to variation in entropy in a given volume, entropy is also produced from nowhere. Hence, entropy balance is written as... 

In a hypothetical system for modeling kinetics, the microscopic cells must be open systems. It is useful to consider entropy as a fluxlike quantity capable of flowing from one part of a system to another, just like energy, mass, and charge. Entropy flux, denoted by Js, is related to the heat flux. An expression that relates Js to measurable fluxes is derived below. Mass, charge, and energy are conserved quantities and additional restrictions on the flux of conserved quantities apply. However, entropy is not conserved—it can be created or destroyed locally. The consequences of entropy production are developed below. 

Comparison with terms in Eq. 1.20 identifies the entropy flux and entropy production ... 

The terms in Eq. 2.10 for the entropy flux can be interpreted using Eq. 2.4. The entropy flux is related to the sum of all potentials multiplying their conjugate fluxes. Each extensive quantity in Eq. 2.4 is replaced by its flux in Eq. 2.10. 

Available energy Mass flow rate Mass flux Pressure Gas Constant Entropy per unit mass Entropy flux Entropy production Time... 

Here, q/7 is the entropy flux and the first term on the right is the entropy production. The simplest constitutive equations satisfying the requirement of the positive entropy production are... 

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 phenomenological equations (6.9.1) have thus been reexpressed in (6.9.9) solely in terms of the measurable transport coefficients a, k, and o. The Seebeck coefficient may be interpreted as the entropy carried per electronic charge. Equation (6.9.9a) represents a further generalization of Ohm s Law, showing how the current density behaves in the presence of a temperature gradient see also Exercise 6.9.3. Equation (6.9.9b) specifies the entropy flux under the joint action of a gradient in electrochemical potential and in temperature this represents a generalization of Fourier s Law. 

FIGURE 6.10.1 Parallelepiped geometry for current and/or for entropy flux along the x— and or y—directions in a magnetic field oriented along the z-direction. 

It is easy to apply the Ostrogradsky theorem for deriving the entropy fluxes and the thermodynamic forces that initiate these fluxes. In fact, the balance of entropy (it also is an extensive parameter, S = ps) is... 

The comparison of this equation with equation (1.63) gives the entropy flux density... 

Hence, the density of the entropy flux for the case of the matter diffu sion is... 

Note first that even in the absence of a temperature gradient a flux of entropy and matter can occur. For, when Xq = 0, /i = L/i Zi -b L/2X2, where ( = 0, 1, 2. For purposes of identification we first consider the constant temperature case, where we define (5 ) and (5 )2 as the entropy intrinsic to (i.e., exclusive of entropy transport) one mole of species 1 and 2. Then at constant T the entropy flux is given by the postulated form... 

Entropy flux in the absence of a net particle flow is equivalent to Jq/T where Jq is the heat flux. Thus, Eq. (6.9.4) is a formulation of Fourier s Law for heat conduction, Jq — —kVT, thereby identifying the thermal conductivity associated with the transport of charge carriers as... 

Fig. 6.10.1. Parallelepiped geometry for current/heat/entropy flux along the x and y directions in the presence of a magnetic field aligned with the z direction.

Non-equilibrium thermodynamics (NET) offers a systematic way to derive the local entropy production rate, c, of a system. The total entropy production rate is the integral of the local entropy production rate over the volume, V, of the system, but, in a stationary state, it is also equal to the entropy flux out, J, minus the entropy flux into the system,... 

The entropy flux difference and the integral over a can be calculated independently, and they must give the same answer. The entropy production rate governs the transport processes that take place in the system. We have... 

The first term on the right hand side of (3.78) represents the entropy fed in with the heat. This is known as the entropy flux. The two remaining terms represent the production of entropy. The second term stems from the finite temperature differences in thermal conduction, the third from the mechanical energy. We call... 

The entropy flux, J, and the entropy production, totab are given by... 

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 ... 

This is the local equation for the entropy density. It is of fundamental importance in what follows. The quantity a, the local entropy production, is the entropy produced irreversibly per unit time per unit volume and is analogous to the property a a defined prior to Eq. (10.3.4). The quantity Js is the entropy flux and V Js represents the rate of change of the local entropy due to an inflow of entropy from neighboring regions of the fluid. 

The calculation of entropy fluxes for each unit leads to quantitative evaluation of the potential working performance of a system. For this purpose in technical thermodynamics a variable known as exergy is introduced. 

Step 7. Manipulate the equation of change for specific entropy, via definitions of convective and molecular entropy fluxes, to identify all terms that correspond to entropy generation. These terms appear as products of fluxes and forces. 

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