Internal entropy production

The internal entropy production this represents the time-related entropy growth generated within the system (djS/df). The internal entropy production is the most important quantity in the thermodynamics of irreversible systems and reaches its maximum when the system is in a stationary state. The equation for the entropy production is then ... 

Fig. 9.3 Entropy-time diagram of an evolution process. If a negative fluctuation of the internal entropy production o occurs in a system, the controlling stationary state is terminated. An instability occurs, starting from which a new stable state is taken up. The change in the internal entropy is always negative (5Si < 0). The new stationary state has a lower entropy, i.e., the order of the system is increased (Eigen, 1971a, b)...

The second law of thermodynamics asserts that the total entropy 5 of a system may change in time because of exchanges with its environment and internal entropy production which is vanishing at equilibrium and positive out of equilibrium [5]... 

Fig. 1. Open systems Internal entropy production. d-,S a 0. dcS is the exchange of entropy with the environment.

From the general entropy balance equation dS= dJS+ dxS, we conclude that for an incompressible and isothermal process, we have deS d,S. This relation shows the equality between the dissipated heat flow and internal entropy production and hence the loss of power is q = Eloss. Therefore, Eq. (b) becomes... 

For constant V and S ys, the external work variation, dw is dw < —dE. For a reversible process, the external work is —dE, the maximum external woric available with dV = 0 and dS y. For an irreversible process, the available work is less than the maximum. In terms of the internal entropy production, equation 26 becomes... 

Again comparing with equation 25a, we note that E /, drii = —T djS. The internal entropy production is a result of change in the composition of the system. At equilibrium, the drii vanish, as does the entropy production within the system. For a fixed temperature and pressure system, dG = E /t, drii = -TdiS. 

To describe the state of a reaction in a phase, we need to know the stoichiometric coefficients, j, and the chemical potential, pi, for each species in the reaction. For reaction equilibrium, the quantity AG = E Vi pi = 0 (as is T diS). For a possible, or spontaneous, reaction, AG < 0. For multireaction systems, complete equilibrium corresponds to dG = 0 for the system, that is, the Gibbs energy of the phase is a minimum. The total internal entropy production must vanish for the entire system. Similar consideration apply to multiphase systems. An expression analogous to equation 39 for dE, but for fixed T and p conditions, is ... 

The idea of the production of entropy plays a central role in the present work. The entropy of a system, which is an extensive quantity relating to the system as a whole, can vary for two reasons and for two reasons only either by a transport of entropy to or from the outside world across the boundary of the system, or by the production of entropy by irreversible phenomena taking place within the system. If d S denotes the amount of entropy received from the surroundings in a given time interval, and diS is the internal entropy production during the same time, then... 

From the above two cases it is clear that the rate of internal entropy production per unit volume is the sum of the products of the flux and the conjugate driving force. This may he expressed as... 

Another Way of Looking at Entropy, by Daniel Hershey in Chemical Engineering Education (1989, summer, p. 154), discusses entropy and aging. Write an expression for entropy production in the hnman body that is consistent with the following statement by Hershey The internal entropy production in living systems is a consequence of several irreversible chemical reactions which constitute the chemistry of life. ... 

Entropy production for a non-equilibrium close to equilibrium is estimated with the help of Gibbs equation with the objective to estimate internal entropy production a = dj5 /df which is needed for characterization of fluxes J and forces X since as we shall later that cr can be expressed as sum of product of fluxes and forces. To illustrate this point, we consider a discontinuous system involving two chambers separated by a barrier but maintained at different temperatures Tj and T. In the present case, heat flow only occurs on account of force generated due to temperature difference (Fig. 2.4). 

The flows passing through the membrane would be perpendicular to the surface and will have the same value at all points on the surface, provided the membrane is homogeneous. Since the internal entropy production occurs on passage to the membrane, we can also evaluate the net entropy production by integration over the thickness of the membrane by using the following expression ... 

This can be applied to the continuum as understood from the definition of s (D.72). That is, the change of entropy ds is divided into two parts as ds = ds +ds, where ds is a result of energy exchange, and ds is the internal entropy production. For the reversible process ds = 0 and for the irreversible process ds > 0. Thus the Second Law of Thermodynamics for the continuum can be written as... 

The rate per unit volume of the internal entropy production, using the assumption of local equilibrium, is given by... 

In the previous section we have seen that the stationary states in the linear regime are also states that extremize the internal entropy production. We shall now consider the stability of these states, and also show that the entropy production is minimized. In Chapter 14 we saw that the fluctuations near the equilibrium state decrease the entropy and that the irreversible processes drive the system back to the equilibrium state of maximum entropy. As the system approaches the state of equilibrium, the entropy production approaches zero. The approach to equilibrium can be described not only as a steady increase in entropy to its maximum value but also as a steady decrease in entropy production to zero. It is this latter approach that naturally extends to the linear regime, close to equilibrium. 

An increase of U can be effected by heat dQ flowing into the system from the outside or by mechanical work dW done onto the system. The change of entropy may be split up into a portion —d S which is equal to dQ/T and into the internal entropy production d S caused by irreversible processes as for instance by stress relaxation or generation of heat through friction ... 

First we want to compute the internal entropy production diS/dt using Eqs.(7.78) and (7.105). There are no external forces, the interior of the system is homogeneous in the concentrations as well as temperature, and viscosity effects (off diagonal elements of the pressure tensor) are negligible. This means that only the last two terms in Eq. (7.105) must be included in the calculation. We begin with the activities for the two reactions ... 

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