Carnot cycle reservoir

Stirling engines also have the maximum theoretical possible efficiency because their power cycle (their theoretical pressure volume diagram) matches the Carnot cycle. The Carnot cycle, first described by the French physicist Sadi Carnot, determines the maximum theoretical efficiency of any heat engine operating between a hot and a cold reservoir. The Carnot efficiency formula is... 

Carnot efficiency is one of the cornerstones of thermodynamics. This concept was derived by Carnot from the impossibility of a perpetuum mobile of the second kind [ 1]. It was used by Clausius to define the most basic state function of thermodynamics, namely the entropy [2]. The Carnot cycle deals with the extraction, during one full cycle, of an amount of work W from an amount of heat Q, flowing from a hot reservoir (temperature Ti) into a cold reservoir (temperature T2 < T ). The efficiency r] for doing so obeys the following inequality ... 

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. 

If the Carnot cycle for a heat engine is carried out in the reverse direction, the result will be either a Carnot heat pump or a Carnot refrigerator. Such a cycle is shown in Fig. 1.5. Using the same graphical explanation that was used in the Carnot heat engine, the heat added from the low-temperature reservoir at Tl is area 1-4-5-6-1 g4i is the amount of heat added to the Carnot cycle from a low-temperature thermal reservoir. 

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

The Carnot cycle is the ideal cycle only for the conditions of constant-temperature hot and cold surrounding thermal reservoirs. However, such conditions do not exist for fuel-burning engines. For these engines, the... 

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. 

We continue with a reversible heat engine operating in a Carnot cycle, but center our attention on the working substance rather than on the entire system consisting of the heat engine, the work reservoir, and the two heat reservoirs. For such a cycle we can write... 

The Carnot cycle is the operation of an idealised (reversible) engine in which heat transferred from a hot reservoir, is partly converted into work, and partly discarded to a cold reservoir. 

The Carnot cycle engine achieves what we are looking for, a conversion of heat into work, with return of the engine to its initial state. We note, however, that in order to complete the cycle, we have paid a price. In the isothermal compression at Tc, some of the work produced in the expansion has to be used up to compress the system, finding its way into heat at the cold reservoir temperature. 

An important measure of the quality of an engine is its efficiency, [the fraction of the energy that it removes from a high temperature reservoir (the heat term in step I) that it converts into work].7 For the Carnot cycle engine to work as efficiently as possible, the heat transfers should be reversible. Thus, the heat transferred to the system in step I should be from a heat reservoir at temperature Th, and the heat transferred from the system in step III should be to a reservoir at Tc. From Table 2, we see that the efficiency of a Carnot cycle engine is... 

The Carnot cycle engine is actually the only reversible engine that we can design with two heat reservoirs. We see that because of the need to reject heat when returning the engine to its initial state, the engine cannot operate with unit efficiency. In Chapter 3, we will elevate this observation to one of the basic tenets of thermodynamics—the second law. 

Rationalize the statement The Carnot cycle engine is the only reversible engine that operates with just two heat reservoirs. ... 

We have included the summation because there may be more than one heat reservoir in the surroundings that is involved in the process. For example, in the Carnot cycle engine, we remove heat from a hot reservoir and deposit heat in a cold reservoir. 

In Chapter 2, we have analyzed one particular type of heat engine, the reversible Carnot cycle engine with an ideal gas as the working substance, and found that its efficiency is e = 1 — Tc/Th. For both practical and theoretical reasons, we ask if it is possible, with the same two heat reservoirs, to design an engine that achieves a higher efficiency than the reversible Carnot cycle, ideal gas engine. What can thermodynamics tell us about this possibility ... 

A more realistic cycle than the Carnot cycle is a modified cycle taking into account the processes time of heat transfer between the system and its surroundings, in which the working temperatures are different of those its reservoirs [1], obtaining the efficiency t]CAN =1 Jtc / TH, first found in references [2] and [3], and known as Curzon-Ahlbom-Novikov-Chambadal efficiency. At present, the duration of heat transfer processes is important. Based on this model, at the end of the last century, a theory was developed as an extension of classical equilibrium thermodynamics, the finite time thermodynamics, in which the duration of the exchange processes heat becomes important. 

If the reversed Carnot cycle were coupled to the irreversible machine and driven by it, there would be a net amount of heat <2i Qi > 0 flowing into the reservoir from this coupled machine. There being no net work production, this amount of heat is obtained from the low-temperature reservoir at T2. The coupled machine would be pumping heat from a lower temperature to a higher temperature while producing no other changes in the surroundings. But such a result is impossible by the second law. 

It is possible to devise a cyclic process so that J%y is positive that is, such that after the cycle, masses are truly higher in the surroundings. It can be done in complicated ways using reservoirs at many different temperatures, or it can be done using only two reservoirs at different temperatures, as in the Carnot cycle. However, experience has shown that it is not possible to build such an engine using only one heat reservoir (compare with Section 7.6). Thus, if... 

Two systems ( ) and ("), not necessarily reservoirs, but big systems, can be thermally coupled by a Carnot cycle. This is in practice a heat pump, or if it runs in the reverse direction, a heat engine. In this case, we have S + S" = 0, thus if the systems are big, for one revolution of the Carnot cycle d5 -I- d5" = 0. This constraint does not imply that T = T". Namely, the Carnot engine is an active device that releases or donates energy. In the Carnot engine, the change of entropy S is not directly connected with the change of entropy S", even when the cross balance d5 -F d5" = 0 over a full turn holds. 

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