## Rn n r n

/(2) — /°(2) = L ln — dN = L N ln — — (N — N°) . (5.10)

N0 N0 N0

The information is additive and summing both these expressions we get

It is easy to see that this expression and the expression (5.7) do not fundamentally differ from each other.

### 5.6. Exergy of solar radiation

Solar and heat radiation fluxes (incoming and outgoing), which are incident and reflected from a surface, are described by the continuous spectral distributions Em (n) and Eout(v), where n is a spectral frequency or, if the whole of the spectrum is divided into n spectral intervals [n,, ni+1 ], i = 1,..., n, their discrete analogies E™ and Eout:

Since the energy of one photon is equal to hv, where h is the Planck constant, Em(v)/hv = n0(v) is the number of photons with the frequency n ("green" photons, "red" photons, etc.) incoming to a single surface during a time unit. Correspondingly, Eout(v)/hv = n(v) is the number of outgoing photons with the frequency n reflected and irradiated by the surface. Then rv+1 rv+1

Thus, photons are distinguished from each other by their frequencies. On the other hand, we can consider each photon with the frequency n as the sum of nt identical particles where t is some characteristic time of the considered process. The energy of each particle is equal to H/t. Note that it is not necessary to seek some physical sense in our virtual particles; this is simply a methodological way to apply the results described below.

In accordance with our assumption the radiation fluxes E'0 and E1 can be presented in the form of some flows of particles, q0 = Jn+1 n0 (n)nt dv and q, = Jn+1 n(v)vT dv, so that Ein = (h/T)q0 and E°ut = (h/T)q, . ' '

Keeping in mind that solar radiation, not including energy, is also bringing information, we apply the result of Section 5.5 to this case. According to the information concept, the outgoing radiation contains the whole information about the active surface, which interacts with incoming radiation. We also assume that the transformation of distribution (E°,..., E°) into (E°,..., E°) takes place very fast in comparison with the characteristic time of radiation fluxes. By setting instant values N0 and N, in Eq. (5.7) to be equal to their flows qi0 and qi and moving to the energy units we obtain the expression for the power of exergy of the solar radiation (Svirezhev and Steinborn, 2001)

Note that later on we shall often use the expressions: energy, exergy (Ex), radiation, information, etc. instead of power, power of exergy (PEx), radiation, information, etc.

flow and fluxes. Introducing exergy in such a way we can apply the general thermodynamic concepts to the process of interaction between solar radiation and some reflecting, transforming and absorbing surface (e.g. vegetation).

The sum R = J]= 1 E - 1 Ej = Ein - Eout, which is the difference between the total incoming and outgoing radiation, is called the radiation balance. In other words, R is the amount of incoming solar energy which is absorbed by a surface (soil, water, vegetation, etc.). Introducing the frequencies pn = E0/Eln and = Ej/Eout, and keeping in mind Kullback's measure, the expression for exergy can be rewritten in the form where K = 1 pout ln(pout/pn) is Kullback's measure. One can see that the value of exergy can be considered as a function of two independent variables, K and R, and one external parameter, Em. The function Ex = Ex(R, K) for fixed Em monotonously increases with increasing K, and has the minimum Excr = Excr(Rcr, K) = Eln(1 - e-K) with respect to R at Rcr = Ein(1 - e-K). It is clear that Ex(0, K) = EinK, Ex(Ein, K) = Ein and Ex(0, 0) = 0.

Undoubtedly, all main conclusions would be obtained from the analysis of this expression but we can do it in a more elegant way if we use its dimensionless form. In addition, the transition to dimensionless variables implies no dependence on the external parameter Ein :

We introduce two new definitions. The ratio rR = R/Eln is named the radiation efficiency. It is obvious that this value describes a fraction of the total energy absorbed by the surface. (Note: if the radiation balance is negative, rR must also be negative.) Analogously, the ratio rEx = Ex/Eln is called the exergy efficiency. By keeping in mind that exergy is a measure of the "useful" work which a system is able to perform, we can say that rEx is an efficiency coefficient of some "radiative" machine, which is our active surface. The working process of this machine is an interaction of incoming radiation with the active surface. Dividing both sides of Eq. (6.3) by Eln and using the notations introduced above we get

The value rEx is also a function of two independent variables, rR and K, but it does not depend on a parameter. The function rEx monotonously increases with increasing K and it has the minimum rEEx = VEx(rRr, K) = 1 - e-K with respect to rR at rR = 1 - e-K (see Fig. 5.3). It is obvious that rEx(0, K) = K, rjuxd, K) = 1 and

In order to analyse some properties of the function Hex(hR, K) we consider its two-dimensional sections for two values of K (Fig. 5.4). Let K = K1 then the minimum (rRRr)1 divides the curve rEx into two branches (ab and bc): the first, where rR > (rRR)1 and rEx < rR, and the second, where rR < (rRRr)1 and rEx > rR.

The shape of this curve and the inequalities which define its different branches allow us to formulate the following hypothesis:

Fig. 5.3. The exergy efficiency hEx as a function of two variables: hR and K.

1. On the right-hand branch (a-b) the active surface operates as a classic thermodynamic machine, performing mainly mechanical or chemical work.

2. On the left-hand branch (b-c) the active surface operates as an information machine, producing mainly information.

If the radiation (energy) balance was a measure of the mechanical work of the thermodynamic machine, then the value of Kullback's measure, K, could be a measure of