DEx X ln dNi X

This means that in moving from thermodynamic equilibrium the system accumulates exergy. It is natural that in the course of inverse spontaneous transition to the equilibrium the system loses exergy. However, inequality (4.4) must not necessarily be fulfilled far from equilibrium. Later on we shall show that some additional conditions are necessary for this.

It is very important that this definition of exergy does not at all require that the system must be in a dynamic equilibrium with the environment during the whole path, as is required in Section 5.1. Equilibrium at the beginning of movement is sufficient, and formally it is not necessary that the equilibrium is a thermodynamic one. Moreover, it is not necessary that the starting point is a dynamic equilibrium: it is sufficient if the system has the equilibrium somewhere else. We shall discuss this (and many other things) below.

Let us consider the following model of a "chemical" system. We assume that the dynamics (kinematics) of the model are described by the system of ordinary differential equations for concentrations (specific numbers) c = {c,- $ 0}, i = 1,..., n dci ti \ -7- =Ji(c1; •■•, cn; «1, •■•, Am); dt

where the vector of parameters a describes either the state of the environment or the characteristics of exchange between the system and its environment. However, very often we do not know the concrete form of the right sides in Eq. (4.5); we only know that system (4.5) has a single stable equilibrium c* = {c* $ 0}, where either its stability (see Chapter 6) or its position depends on parameters. Therefore, let us consider two cases.

Case 1. Let the parameters be a°, and the system is in the stable equilibrium c* (a°) = c° . The change a° ! a1 leads to the loss of its stability, but its position is not changed. We identify the stable state c° with thermodynamic equilibrium of the system with its environment, when c° = cenv (the latter is the vector of concentrations in the environment). Then the change a° ! a1 may be interpreted as the following: Ecodemon opens a lid at time t°, and some additional flows of energy come into the system by performing some work on it. The system starts to go away from the state c° . From the viewpoint of dynamical theory it implies that any small fluctuation leads the system out of the equilibrium, and c° ! c(t) for t > t° . The transition is accompanied with a dissipation of energy, which is calculated as (see Chapter 3)

Here the expression RT° ln[c°/ci(t)] gives the value of affinity for the transition c° ! ci(t) where c° = constant . This is the affinity of diffusion processes transporting a matter from the environment into the system.

The negativeness of Diss(c° ! c), i.e. the total value of dissipated energy, implies that the system transition from some initial state to a current one is not spontaneous, but forced. We know that all spontaneous processes of return to a stable equilibrium after relatively small internal fluctuations are always accompanied by an increase in entropy. In this case entropy of our open system decreases, since the system consumes free energy from the environment.

In accordance with one of the definitions of exergy, it is equal to the absolute value of the entropy decrease multiplied by the temperature of environment, i.e. Ex(c, c°) = —Diss(c° ! c), where the equilibrium c° is considered as a reference state. For calculation of exergy we used formula Eq. (4.3), in which Ni and N° were replaced by ci and c°, respectively. We also replaced the upper index ( * ) by (0).

We can also say that the system moving far from the equilibrium c° accumulates the exergy, so that at moment t its exergy, i.e. its potential ability to perform some useful work, is equal to Ex(c(t), c°) = —Diss(c° ! c(t)) .

Let us assume that at some moment t = t1 our Ecodemon closes the lid, by the same token stopping the energy flow, i.e. it switches a1 again to a°, so that the equilibrium c° becomes stable again. The system starts to move to the state c°, but the movement is spontaneous, not forced, since the inflow of free energy is stopped. Entropy of the system increases. In the course of the transition c(t1) = c1 ! c° the system can do some work on the environment, decreasing the entropy of the latter. However, if to take into account that at time t1 the system has accumulated storage of low-entropy "good" energy in the form of exergy, which will be dissipated when c1 ! c°, then the transition can be considered as a forced one, and the dissipation of energy is calculated analogously

1 c0

(as was shown above). Since the affinity for c ! c0 is RT0 lnlc/c0], fi n c dc n F0 c

Diss^1 ! c0) = RTo\ Xln -0 dCi dt = KT, X In -0 dc-

By comparing Eqs. (4.6) and (4.7) we see that Diss(c! ! c0) = Diss(c0 ! c1) = —Ex(c1, c0). All these equalities imply that

1. Work done on the system by its external environment in the process of forced transition c0 ! c1 is equal to lA011 = —Diss(c0 ! c1).

2. In the course of the transition the system accumulates exergy, which is equal to Ex(c1; c0) = lA011 = —Diss(c0 ! c1).

3. When the system is closed, it returns spontaneously to the stable state c0; the return is accompanied by the dissipation of exergy in the process of performing work by the system. The transition must be finished in the state c0: namely, at this moment the work lA10l = —Diss(c1 ! c0) = Ex(c1, c0) and the whole storage of exergy will become exhausted.

However, there are some remarks about the last point. Strictly speaking, the equilibrium c0 is attained at infinite time; at finite time the system can come into some vicinity of the equilibrium so that the cycle cannot be closed. As a result, the accumulated exergy does not dissipate fully, and its small part remains within the system before a new cycle starts. In fact, entropy of environment increases by the corresponding quantity. In order to "save" the Energy Conservation Law we can say that the conserved exergy is used by Ecodemon for opening and closing of the lid. By the same token Ecodemon adjusts the evolution of our system. The problem is how to estimate this value if we do not know the concrete form of dynamic equation (4.5). To get inside the problem, let us consider another model.

Case 2. Unlike Case 1, we assume that the system has a single stable equilibrium, continuously depending on the parameters a. Let the vector a = a0 before the moment t0 and the system is in the stable equilibrium c*(a°) = c0. At the moment t0 the parameters are shifted from a0 to a1 . The operation was realised by Ecodemon very quickly, so that the state c0 does not manage to change. As a result the state c0 ceases to be in equilibrium and the system begins to move to the new stable equilibrium c*(a1) = c1. In this case the affinity for the transition c(t) ! c1 (c(t) is a current state) is equal to RT0 ln[cI/cI1], and the total dissipation of energy for the transition c0 ! c1 is ri n c dc n Diss1(c0 ! c1)= RT0 X ln —j--- dt = -RT0X

Generally, as we mentioned above, the transition requires infinite time, but if a finite time t1 is sufficiently large to get into a small vicinity of c1, then with a sufficient accuracy we can assume that the equilibrium c1 is attained in the finite time. In other words, in the course of t1 the transition process c0 ! c1 must be established.

Let at the moment t? the vector a be again shifted from a1 to a0. In this case the equilibrium c1 ceases to be equilibrium, and the system begins to move to the old stable equilibrium c0. The total energy dissipated in the course of transition c1 ! c0 is described by Eq. (4.7), and its absolute value is formally equal to Ex(c?, c0).

The latter may be interpreted in the following manner: the exergy is equal to the minimal work, which must be performed in order to kill the system or to destroy it. Note (this is very important) that the work cannot be done on the system directly; it must be done on its environment. In other words, we cannot destroy the system by a direct impact, but we have to change the environment in a hostile (for the system) way. For that a mechanical work may be used and the system will be destroyed mechanically. Or, this could be a result of a change of the chemical status of the environment (for instance, pollution).

Let us consider the following example. Assume that some poisonous substance impacts on a living system. In order to "poison" the system we have to increase the poison concentration from the basic concentration in a "normal" environment, c0, to the mortal concentration, cm: For that we have to perform work Ach against the gradient of chemical potentials, RT0 ln(cm/c0): Ach = RT0[cm ln(c"/c0) - (cm - c°)J. Then the exergy of the living system (per unique volume) will be equal to Ach: Since the basic concentration for the normal condition, c0 is usually very low, the term ln(cm/c0) would be sufficiently large and, as a result, the exergy can be also sufficiently large.

By comparing the expressions (4.7) and (4.8) we can see that they differ from each other, i.e. the total energy dissipated in the course of the first half of the cycle when the system moves far from its initial state differs from the same value calculated for the second half when the system returns to its initial state. This is a principal distinction between Case 1, when both values are equal, and Case 2. The reason for the distinction is that in Case 1 the system is simply moving far from some equilibrium, which is saved, while in Case 2 a new stable equilibrium appears far from the initial state, which furthermore is ceasing to be equilibrium. The phenomenon of the appearance of a new (dynamic) equilibrium far from thermodynamic one is a typical character of non-linearity, and we have the right to expect some new effects in such a type of cycles.

So, from the "exergetic" point of view, the system moving away from its initial state to the new equilibrium accumulates the exergy Ex(c0, c1), and dissipates the exergy Ex(c?, c0) returning to the initial state. In Case 2 the cycle is closed incompletely: a residual of exergy remains in the system. The residual may be both positive and negative. In the first case this exergy may be used as a "push" for the next cycle, Ecodemon's interference is not needed and the system begins its slow evolution (of course, by means of increasing the entropy of environment). In other words, if SEx > 0, then the system possesses an ability to evolution. In the opposite case, if SEx < 0, then the system does not have this ability.

What are the orders of magnitude of the exergy and its residual? In order to answer the question visually we are restricted to (1) the case of one variable, c(t), and (2) the case of small deviation, c1 - c0 = Ac, lAcl << c0. Expanding the expressions for Ex(c0, c1) and

SEx into a series in the power of Ac and restricting only the first terms of expansion we get:

It is clear that when we calculate the exergy, we are not leaving the Onsager-Prigogine world with its weak quadratic non-linearity, while the exergy residual is a phenomenon of another, more non-linear world. In practice the phenomenon is unnoticeable when the system is in a vicinity of thermodynamic equilibrium, and the long-term evolution is needed in order to reveal this phenomenon's significant role.

All these results have biological interpretation. Let the system be a living organism and the exergy cycle be its life cycle, which is passed from "dust" to "dust", from detritus to detritus. Then the exergy residual can be interpreted as an amount of information, which must be saved to start the next life cycle, i.e. to maintain evolution. This is nothing more or less than the genetic information contained in the genome. "Soma is mortal, genome is immortal" (Weismann).

Finally, we would like to call your attention to the following: there is a fundamental difference between the types of Gedanken experiment in classic thermodynamics and here. If in classic thermodynamics in order to change the state of the system we perform the work on the system, then here in order to obtain the same result we must perform the work on the system's environment.

Was this article helpful?

0 0

Post a comment