Entropy production density

The final expression for the local entropy production density now becomes... 

Equation (3.200) is the expression for a nonconservative change in local entropy density, and allows the determination of the entropy production from the total change in entropy and the evaluation of the dependence of on flows and forces. 

The total entropy production as needed in Equation (A.22) is related to the entropy density as follows ... 

If the system is nonuniform in temperature and pressure, the derived equations pertain to processes in an elementary volume—that is, to the density of distribution of corresponding quantities. Therefore, determining the system total rates of entropy production and energy dissipation must be done with the integration over the volume. 

The expression for the irreversible entropy production of a flowing nematic liquid crystal given in Section 4 did not include the contribution from the streaming angular velocity and its conjugate torque density. Therefore we present a more general expression that includes this contribution [26],... 

It follows trivially that dS/dt = 0 for all time in the equilibrium state. Thus, there is no purely microscopic mechanism that will give rise to entropy production, dS/dt > 0. The reason for this is that the phase space density / q accounts for all of the microstructure of phase space. In reality, we can never know this full microstructure, as was recognized by Ehrenfest, who suggested that one should work with a coarse-grained density, /eq, obtained by averaging over suitably small cells in phase space. Then one can define an entropy in analogy with Eq. [50], but in terms of /eq. In this way, entropy production can be realized microscopically, as discussed in detail in Ref. 23. Thus, the fine-grained entropy as defined in Eq. [50] always has a zero time derivative. 

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. 

In order to obtain an additional balance equation for the microstructural parameter K, the principle of dissipation is utilized. The starting point is the entropy balance [Eq. (9)] with entropy density p t], entropy flux entropy supply a, and entropy production f >0. 

The Soret coefficient is not a constant it may vary due to changes in concentration or temperature. A positive Soret coefficient means a thermophobic particle. The Soret coefficient can change sign as a function of concentration [1] or temperature [9] (see Fig. 2). The inversion temperature for latex particles, which only have electrostatic interactions with water, lies close to the maximum density of the solvent at 4 °C. lacopini et al. [9] conclude based on empirical findings that the thermal expansivity is the major factor in thermophoresis. They claim that this effect can be added to the Soret coefficient caused by the entropy production of particle solvent interactions. The similarity between the... 

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. 

The entropy production rate at macroscopic level, (equation 8.5) can be obtained after integration of the local entropy production rate density over the volume element in phase (a). For the sake of simplification the following assumptions are adopted (i) entropy contributions of all involving phases in the GLRDVE are combined in a single variable (E = /(E )) (m) entropy is produced due to mass and heat diffusion in... 

In continuous systems, the local increase in entropy can be defined by using the entropy density s(x,t), which is the entropy per unit volume. The total entropy change is ds = dgS + diS and results from the flow of entropy due to exchanges with surroundings (dgs) and from the changes inside the system (dis). Therefore, the local entropy production can be defined by... 

The entropy production is obtained in terms of the chemical potential expressed in probability density ... 

Each thermodynamic system strives to equalise gradients in temperature, pressure and chemical potentials which induces heat flow, liquid or gas flow or diffusive mass flows. The entropy production associated with a diffusive mass flow is proportional to the product of mass flow density and gradient in chemical potential. Gradients in the chemical potentials are therefore considered within the scope of thermodynamics of irreversible processes as the generalised driving force for diffusive mass flow (Prigogine 1961). In a thermodynamic equilibrium state not only temperature and pressure are equal, but the chemical potentials are also. 

So, the density of total entropy change contains, as was expected, two summands the divergence of density of entropy flux (describing the redistribution of already available entropy between the sites), and the density of entropy production including the sum of products of fluxes and corresponding forces . Apparently, for the... 

In the more generalized case, taking into account the convective motion at a certain velocity V and the corresponding viscosity, external forces Fj acting on each component (for example, electric field, chemical reactions), the density of entropy flux, and the density of entropy production become (1) ... 

Determine the thermodynamic reaction forces for a membrane having multiple transports of chemical components forced by a current density from outside source. Determine the OM-function of the process. What is the entropy production resulted by the reaction forces in this case ... 

Prescribe the coordinate of the current density in direction of outer normal vector on Qj part of the boundary Q = Qj u Q(o and the electric potential on Q.

vector-scalar ordered pair is the unique solution of the equations (126), which are identical(126), and uniform with the (127) transport equation. The entropy production of the system could be calculated by the Joule-law, [19],... 

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