L TH2 TL2 r

two results and generalizing for the sum of all such cycles, then Q = 0 j . In the limit as A ^ 0, the adiabatic lines come closer thus making the heat quantities infinitesimal resulting in

0 > which is the important Clausius theorem.

rev, isothermal rev, isothermal isothermal

Fig. 2.3 Development of the Carnot cycle.

isothermal

Fig. 2.3 Development of the Carnot cycle. Fig. 2.4 Development of Clausius' statement and entropy.

Now consider two reversible processes R1 and R2 starting from the initial state i and ending at the final state f, Fig. 2.5. Since they are reversible, it is possible to change the sense of R2. Since R1 and R2 now form a reversible cycle, then i

and j | + | _ 01 This results in the most general relation for the integral in a reversible process J J L = J J ^Q I =...= J JI which says v that if a reversible path is chosen, the path itself is not important so long as the process starts at i and ends at f. The quantity is, therefore, given by the end states and not the path. As is the case in the first law of thermodynamics,

[<SQ -jSW =<< dU ] and Id dU = 0 since internal energy is a thermodynamic property, then property, then is a thermodynamic property and is called entropy, S. Fig. 2.5 A reversible cycle.

Therefore,

or for an infinitesimal process,

= dS> that forms the mathematical formulation of the second law of ther-

modynamics. It is, therefore, seen that there is a similarity between the two laws of thermodynamics and their definition of internal energy and entropy. To further extend this discussion to the inequality of Clausius, consider the fact that all heat engines operating between a given high temperature source, TH, and a lower temperature sink of Tl, none can have a higher efficiency than the Carnot engine.

Thus using the figure above, but this time having the process at TH to be irrevers-

ible, then the result obtained is

""Lrev

-"HIRREV J

QH J rev

Using the fact that

• is for reversible energy transfers, then

 m \ T < 1 - L V T H J

Transposing and keeping in mind that there is a negative sign, the result becomes

Using the definition of entropy as given above, , which states that in all real processes

entropy increases and the equality is only for the reversible process. This further reduces the result to what is expected, the inequality of Clausius, the fact

If the entropy changes of the system are added to the entropy changes occurring in the surroundings as a result of the changes in the system, the sum represents the total changes of the system and the surroundings and is called the entropy change of the universe or entropy generation, a. For a reversible process,

let SQrev amount of energy be absorbed by the system. Then dS.

system

0 since it has been put into the system. Since this energy has to be given up by the surroundings, then dS d. = SQrev j <0. As a result, the entropy generation is (dSuniverse = da= 0). Thus, when a reversible process is performed, the entropy of the universe remains unchanged.

The second law representation of the systems that we are interested in is as follows:

Closed system:

Open system:

Filling system: Emptying system:

vTj/

Q"

The product of the environment temperature T0 with A a is called the irreversibility, Iie, of the process. Therefore, either entropy generation or irrever-

sibility can be used to discuss the given situation. It is not important which choice is made. It is this irreversibility that gives rise to pollution and degradation of the environment leading to unsustainability of the present quality of life in nature.

2.9 Exergy

Exergy or available energy is the capacity to perform useful work with a given amount of energy. This can also be considered as the taxation of energy by nature. What nature is saying that although we may have an amount of energy that we should be able to use, the portion of that energy between the lowest available temperature T0[K] and 0[K] is the amount that is taken out by nature before any useful work can be obtained, (T0AS). Unlike the government's share of our income as taxes, this amount is not negotiable. Therefore, this amount goes back to nature, unfortunately in the form of pollution. This result also states that if we want to produce useful products, such as electricity or transportation, we agree to pollute the environment.

Thus to be conscious of our responsibilities to our and future generations, better conversion technologies and conservation seem to be the only immediate solutions since any power generation MUST produce pollution of some sort that we may not be able to accept. Intelligent use of resources, therefore, must happen; otherwise the consequences are not very desirable.

The exergy formulation for the systems that we are interested in are as follows:

Closed system:

Open system:

ToAa j 