Reversible processes Carnot cycle

The Carnot cycle is formulated directly from the second law of thermodynamics. It is a perfectly reversible, adiabatic cycle consisting of two constant entropy processes and two constant temperature processes. It defines the ultimate efficiency for any process operating between two temperatures. The coefficient of performance (COP) of the reverse Carnot cycle (refrigerator) is expressed as... 

The second law of thermodynamics may be used to show that a cyclic heat power plant (or cyclic heat engine) achieves maximum efficiency by operating on a reversible cycle called the Carnot cycle for a given (maximum) temperature of supply (T ax) and given (minimum) temperature of heat rejection (T jn). Such a Carnot power plant receives all its heat (Qq) at the maximum temperature (i.e. Tq = and rejects all its heat (Q ) at the minimum temperature (i.e. 7 = 7, in) the other processes are reversible and adiabatic and therefore isentropic (see the temperature-entropy diagram of Fig. 1.8). Its thermal efficiency is... 

In the ultimate version of the reheated and intercooled reversible cycle [CICICIC- HTHTHT- XJr, both the compression and expansion are divided into a large number of small processes, and a heat exchanger is also used (Fig. 3.6). Then the efficiency approaches that of a Carnot cycle since all the heat is supplied at the maximum temperature Tr = T ax and all the heat is rejected at the minimum temperature = r,nin. 

Because the gas in the Carnot cycle starts and ends at the same state, the system s entropy does not change during a cycle. Now apply the second law to the universe for the case of the Carnot cycle. Because the processes are reversible, the entropy of the universe does not change by Equation 2b. This can be written ... 

Theorem.—A process yields the maximum amount of available energy when it is conducted reversibly.—Proof. If the change is isothermal, this is a consequence of Moutier s theorem, for the system could be brought back to the initial state by a reversible process, and, by the second law, no work must be obtained in the whole cycle. If it is non-isothermal, we may suppose it to be constructed of a very large number of very small isothermal and adiabatic processes, which may be combined with another corresponding set of perfectlyJ reversible isothermal and adiabatic processes, so that a complete cycle is formed out of a very large number of infinitesimal Carnot s cycles (Fig 11). 

Figure 2.10 (a) A schematic Carnot cycle in which isotherms at empirical temperatures 6 and 62 alternate with adiabatics in a reversible closed path. The enclosed area gives the net work produced in the cycle, (b) The area enclosed by a reversible cyclic process can be approximated by the zig-zag closed path of the isothermal and adiabatic lines of many small Carnot cycles. 

Consider any reversible cyclic process that involves the exchange of heat and work. Again, the net area enclosed by the cycle on a p-V plot gives the work. This work can be approximated by taking the areas enclosed within a series of Carnot cycles that overlap the area enclosed by the cycle as closely as possible as shown in Figure 2.10b. For each of the Carnot cycles, the sum of the q/T terms... 

In Chapter 2 (Section 2.2a) we qualitatively described the Carnot cycle, but were not able to quantitatively represent the process on a p— V diagram because we did not know the pressure-volume relationship for a reversible adiabatic process. We now know this relationship (see section 3.3c), and in Figure 3.3, we compare a series of p-V adiabats with different starting temperatures for an... 

Figure 3.4 Carnot cycle for the expansion and compression of an ideal gas. Isotherms alternate with adiabats in a reversible closed path. The shaded area enclosed by the curves gives the net work in the cyclic process.

Considering the concepts of reversible processes, a reversible cycle can be carried out for given thermal reservoirs at temperatures and Tl. The Carnot heat engine cycle on a p-V diagram and a T-S diagram, as shown in Fig. 1.4 is composed of the following four reversible processes ... 

During process 4-1, heat is transferred isothermally from the working substance to the low-temperature reservoir at Tl. This process is accomplished reversibly by bringing the system in contact with the low-temperature reservoir whose temperature is equal to or infinitesimally lower than that of the working substance. The amount of heat transfer during the process is 641= f TdS = Ti Si — S4), which can be represented by the area 1-4-5-6-1 Q41 is the amount of heat removed from the Carnot cycle to a low-temperature thermal reservoir. 

A reversible isothermal heat-transfer process between the Carnot cycle and its surrounding thermal reservoirs is impossible to achieve... 

Suppose a thermally driven heat pump operates at temperatures Th, Tm, and T1. At the high temperature Th, heat Qh is supplied to the heat pump at temperature Tm, heat is generated by the heat pump and at the temperature T1, heat Qi is extracted from a low temperature source. For a Carnot cycle (reversible process) the following relations hold ... 

CARNOT CYCLE. An ideal cycle or four reversible changes in the physical condition of a substance, useful in thermodynamic theory. Starting with specified values of die variable temperature, specific volume, and pressure, the substance undergoes, in succession, an isothermal (constant temperature) expansion, an adiabatic expansion (see also Adiabatic Process), and an isothermal compression to such a point that a further adiabatic compression will return the substance to its original condition. These changes are represented on the volume-pressure diagram respectively by ub. he. ctl. and da in Fig. I. Or the cycle may he reversed ad c h a. 

So far only the energy requirement for a process in the form of work has been considered. Freezing, vapor compression, and reverse osmosis processes are examples of processes that require a work input. There are, however, other important processes, such as multiple-effect evaporation and flash evaporation, for which the energy input is in the form of heat. How does one relate the energy requirement of these processes to the minimum work of separation One method is to convert the heat requirement to a work equivalent by means of the Carnot cycle. If T is the absolute temperature of the heat source and T0 the heat-sink temperature, then one can use the familiar relation... 

The cell temperature T c is again the temperature T of the process environment. The work wtcc produced by the Carnot cycle CC increases with higher Tpc and the work in/ i crev produced by FC decreases with lower 7jc as already expected. The work wtSyst of the system is independent of 7 if (or nearly independent in the case of the simplified process). The FC operates reversibly in both cases but the Carnot cycle CC does not operate completely reversible in the simplified process caused by the fact that a small part of the waste heat of FC is needed to heat air and fuel. The practical benefit of this combined fuel cell-heat reference cycle is the opportunity for using exergetic efficiencies to describe the operation of real cycles with this very simple model. The needed exergetic efficiency f is defined as... 

The combination of a SOFC with a heat engine allows highest electric efficiencies. This is caused by the comparable low entropy production within high temperature fuel cells. Generally, the combination of a reversible fuel cell and a reversible heat engine, as represented by the Carnot cycle, results in a reversible process at any operating temperature of the fuel cell. This combination can be used as refer-... 

In the development of the second law and the definition of the entropy function, we use the phenomenological approach as we did for the first law. First, the concept of reversible and irreversible processes is developed. The Carnot cycle is used as an example of a reversible heat engine, and the results obtained from the study of the Carnot cycle are generalized and shown to be the same for all reversible heat engines. The relations obtained permit the definition of a thermodynamic temperature scale. Finally, the entropy function is defined and its properties are discussed. 

In Section 3.3 we concluded that an isolated system can be returned to its original state only when all processes that take place within the system are reversible otherwise, in attempting a cyclic process, at least one work reservoir within the isolated system will have done work and some heat reservoir, also within the isolated system, will have absorbed a quantity of heat. We sought a monotonically varying function that describes these results. The reversible Carnot cycle was introduced to investigate the properties of reversible cycles, and the generality of the results has been shown in the preceding sections. We now introduce the entropy function. 

Figure 4.3 Reversible Carnot cycle, showing steps (1) reversible isothermal expansion at th (2) reversible adiabatic expansion and cooling from th to tc (3) reversible isothermal compression at tc (4) reversible adiabatic compression and heating back to the original starting point. The total area of the Carnot cycle, P dV, is the net useful work w performed in the cyclic process (see text).

The above relate to Figure A.2, which shows an enhanced version of Figure A.l, designed to allow operation of the cell at any selected high temperature and pressure. Isentropic circulators are incorporated to generate the increased conditions. The cell generates heat which is passed without temperature difference to a Carnot cycle to generate power, a reversible process free from the irreversibility of combustion. 

For a single reversible process between two sets of fixed conditions, the work is independent of the reversible path. However, in a network of reversible processes, such as Figure A.l, alteration of the pressure and temperature of the isothermal enclosure alters the pressure ratio of, for example, the fuel isothermal expander. The power output of Figure A.l is therefore variable and not a constant, merely because it is reversible. The maximum power, the fuel chemical exergy, is obtained from an electrochemical reaction at standard temperature, Tq, and sum of reactant and product pressures, Pg, with isothermal expanders only and without a Carnot cycle. 

The processes tlrat occur as tire working fluid flows around the cycle of Fig. 8.1 are represented by lines on tire TS diagram of Fig. 8.2. The sequence of lines shown coirfomrs to a Carnot cycle. Step 1 - 2 is tire vaporization process taking place in the boiler, wherein saturated liquid water absorbs heat at tire constant temperahire Th, and produces saturated vapor. Step 2 3 is a reversible, adiabatic expansion of saturated vapor into the two-phase... 

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