Local entropy production

As we noted in the previous section, the Second Law of thermodynamics must be a local law. If we divide a system into r parts, then not only is 

The local increase of entropy in continuous systems can be defined by using the entropy density s x, t). As was the case for the total entropy, ds = d s + df,s, with diS 0. We define local entropy production as 

Nonequilibrium thermodynamics is founded on the explicit expression for a in terms of the irreversible processes that we can identify and study experimentally. Before we begin deriving this expression, however, we shall write the explicit local forms of balance equations for energy and concentrations. 

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The partial -derivative of this expression therefore gives the rate of local entropy production... 

Here Dt is a positive proportionality constant ( diffusion constant for Et), Jfz is z-ward flow induced by the gradient, and superscript e denotes eigenmodt character of the associated force or flow. The proportionality (13.25) corresponds to Fick s first law of diffusion when Et is dominated by mass transport or to Fourier s heat theorem when Et is dominated by heat transport, but it applies here more deeply to the metric eigenvalues that control all transport phenomena. In the near-equilibrium limit (13.25), the local entropy production rate (13.24) is evaluated as... 

The local entropy production rate (13.24) is then expressed in terms of eigenforces and eigenfiows as... 

The eigenmode expansion (13.30) for the local entropy production rate can be expressed in terms of usual laboratory variables Rh Rt (13.14a, b) using transformation equations analogous to those of Section 11.6. In the present Abased framework, the expansion of dEt in intensities [cf. (11.89)] becomes... 

NEAR-EQUILIBRIUM IRREVERSIBLE THERMODYNAMICS DIFFUSIONAL GEOMETRY 435 This leads to the final expression for the local entropy production rate... 

The basic postulate of irreversible thermodynamics is that, near equilibrium, the local entropy production is nonnegative ... 

From Eqs. (3.118), (3.137), and (3.138), the entropy source strength or the rate of local entropy production per unit volume < > is defined by... 

If the local entropy production, <, is integrated over the volume, it is called the volumetric rate of entropy production... 

We relate the dissipation function to the rate of local entropy production using Eqs. (3.151)—(3.153)... 

Using Eq. (3.220) in the local entropy production equation, we have... 

Using the linear relation between the thermodynamic forces, 0 = yX, + X2, the local entropy production becomes... 

The local entropy production for diffusion of several substances per unit volume is... 

For an elementary chemical reaction, the local entropy production and the linear phenomenological equation are... 

The local entropy production of a reacting mixture in a system with gradients in temperature T, and chemical potentials /u, is given by... 

If the phenomenological coefficients are constant, then < > is constant through the reactor, and local entropy productions... 

By setting Eqs. (A.27) and (A.29) equal, we obtain an expression for the local entropy production term ... 

Entropy Balance Equation and Rate of Local Entropy Production... 

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,... 

To reduce the lost work in industrial process plants, the minimization of entropy production rates in process equipment is suggested as a strategy for future process design and optimization [81]. The method is based on the hypothesis that the state of operation that has a minimum total entropy production is characterized by equipartition of the local entropy production. In this context we need to quantify the entropy sources of the various irreversible unit operations that occur in the industrial system. 

The last term is characteristic of the thermodynamics of irreversible processes. Its magnitude becomes positive if the system s processes are irreversible. Typical irreversible processes are the adsorption or desorption of surfactants at liquid interfaces. The derivative of the second term of Eq. (2C.2), as local entropy production is... 

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 an irreversible process the total entropy production as well as the local entropy production must be positive (a > 0). 

The equation of change of entropy can be obtained by a balance [1, p. 372]. In a volume element, the accumulation of entropy with time arises from convective transport of entropy attached to matter, from inflow and outflow by molecular transport, and from local entropy production in the volume element itself. Thus, the equation of... 

The first term on the right-hand side of Eq. (19.3) refers to convective transport, the second term with the entropy flow vector s refers to the entropy flow by molecular transport, and term gs is the rate of local entropy production. Equation (19.3) goes back to Jaumann ... 

Ross, J., Corlan, A.D., MiiUer, S.C. Proposed principles of maximum local entropy production. J. Phys. Chem. B 116(27), 7858-7865 (2012)... 

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 time derivative of entropy production is called the rate of entropy production, and can be calculated from the laws of the conservation of mass, energy, and momentum, and the second law of thermodynamics expressed as equality. If the local entropy production, a, is integrated over the volume, it is called the volumetric rate of entropy production. ... 

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