Rates of entropy-production

From ( A3.2.12). it is seen that the rate of entropy production is given by... 

Consider an isotropic fluid in which viscous phenomena are neglected. Concentrations and temperature are non-unifonn in this system. The rate of entropy production may be written... 

When a process is completely reversible, the equahty holds, and the lost work is zero. For irreversible processes the inequality holds, and the lost work, that is, the energy that becomes unavailable for work, is positive. The engineering significance of this result is clear The greater the irreversibility of a process, the greater the rate of entropy production and the greater the amount of energy that becomes unavailable for work. Thus, every irreversibility carries with it a price. 

In Equation 2-114, is the rate of entropy production within the control volume symbols with dots refer to the time rate of change of the quantity in question. The second law requires that the rate of entropy production be positive. 

Because the rate of entropy production is negative, the device violates the second law and is therefore impossible. Note that the device would be theoretically possible if the final pressure were specified as 400 psia or less by the inventor. That is, at = 400 psia, T = 40°F, h = 290 Btu/lb, and S, = 1.25 Btu/lb °R, the entropy production rate would be... 

A reaction at steady state is not in equilibrium. Nor is it a closed system, as it is continuously fed by fresh reactants, which keep the entropy lower than it would be at equilibrium. In this case the deviation from equilibrium is described by the rate of entropy increase, dS/dt, also referred to as entropy production. It can be shown that a reaction at steady state possesses a minimum rate of entropy production, and, when perturbed, it will return to this state, which is dictated by the rate at which reactants are fed to the system [R.A. van Santen and J.W. Niemantsverdriet, Chemical Kinetics and Catalysis (1995), Plenum, New York]. Hence, steady states settle for the smallest deviation from equilibrium possible under the given conditions. Steady state reactions in industry satisfy these conditions and are operated in a regime where linear non-equilibrium thermodynamics holds. Nonlinear non-equilibrium thermodynamics, however, represents a regime where explosions and uncontrolled oscillations may arise. Obviously, industry wants to avoid such situations ... 

A significant question is whether the asymmetric contribution to the transport matrix is zero or nonzero. That is, is there any coupling between the transport of variables of opposite parity The question will recur in the discussion of the rate of entropy production later. The earlier analysis cannot decide the issue, since can be zero or nonzero in the earlier results. But some insight can be gained into the possible behavior of the system from the following analysis. 

As stressed at the end of the preceding section, there is no proof that the asymmetric part of the transport matrix vanishes. Casimir [24], no doubt motivated by his observation about the rate of entropy production, on p. 348 asserted that the antisymmetric component of the transport matrix had no observable physical consequence and could be set to zero. However, the present results show that the function makes an important and generally nonnegligible contribution to the dynamics of the steady state even if it does not contribute to the rate of first entropy production. 

It should be clear that the most likely or physical rate of first entropy production is neither minimal nor maximal these would correspond to values of the heat flux of oc. The conventional first entropy does not provide any variational principle for heat flow, or for nonequilibrium dynamics more generally. This is consistent with the introductory remarks about the second law of equilibrium thermodynamics, Eq. (1), namely, that this law and the first entropy that in invokes are independent of time. In the literature one finds claims for both extreme theorems some claim that the rate of entropy production is... 

The rate of entropy production is always positive in the present case, since transport processes are irreversible in nature, i.e. always connected with irreversible losses (dissipation) of energy. 

Fig. 11.7 A diagram representing the development of our ecosystem. Time is along the axis of the cone with separation of oxidised chemicals in the environment and reduced chemicals in increasing numbers of chemotypes, see text. The Darwinian tree of species evolution fits into the cone and has linear connectivity while the ecological cone is continuously filled. The upper side-figure indicates the extent of each zone and the species in it. The lower side-figure shows the increase of use of energy, the rate of entropy production, with time.

Response to the driving force is defined as the rate of change of the extensive parameter Xk, i.e. the flux Jk = (dXk/dt). The flux therefore stops when the affinity vanishes and non-zero affinity produces flux. The product of affinity and associated flux corresponds to a rate of entropy change and the sum over all k represents the rate of entropy production,... 

The rate of entropy production per unit volume may then be written as... 

The affinities that define the rate of entropy production in continuous systems are therefore gradients of intensive parameters (in entropy representation) rather than discrete differences. For instance, the affinities associated with the z-components of energy and matter flow for constituent k, in this notation would be... 

This contraction rate has been identified in these models as the rate of entropy production [49], which proves the formula (101) in this case as well. 

For a one dimensional isothermal sample of (fixed) unit area cross section, the rate of entropy production o is therefore... 

If we take the Gibbs-Duhem equation, ck flk = 0 into account, the rate of entropy production is... 

The rate of entropy change and the local rate of entropy production can be inferred by invoking equilibrium thermodynamic variations and the assumption of local equilibrium. 

The rate of entropy production, (Eq. 2.19), for one-dimensional diffusion becomes... 

Theorem 16. If 5 r and 0lf = y r are two equivalent sets of reactions over the same species, their rates of entropy production are the same. 

As we stated earlier, diS > 0. Generalizing the concept of affinity of chemical reactions, the rate of entropy production can be written as7... 

The parameters controlling the rate of entropy production in the tower are now obvious the vapor flow rate V (a function of the reflux ratio), the inlet and outlet mole (or mass) fractions, and the relationship between yA and y g (a function of the reflux ratio and the relative volatility). 

---