The Carnot Cycle

The transfer of heat between two bodies requires a difference in their temperatures. The reversible transfer of heat between two bodies would require their temperatures to be the same. The question then arises of how we may reversibly add a quantity of heat to or remove a quantity of heat from a system over a temperature range. In order to do so we assume that we have an infinitely large number of heat reservoirs whose temperatures differ infinitesimally. Then, by bringing the system into thermal contact with these reservoirs successively and allowing thermal equilibrium to be obtained in each step, we approach a reversible process. 

We next carry out an adiabatic reversible expansion from B to C in which the additional quantity of work, W2, represented by the area BCV3V2 is done on the system (a negative value). During this expansion the cylinder is enclosed by an adiabatic envelope. No heat is transferred to the fluid, and its temperature decreases to 0X. 

We now compress the fluid isothermally and reversibly from C to D at the temperature 01. In this compression the cylinder is in thermal contact with the heat reservoir having this temperature. A quantity of work equal to the area CDV4V3 is done and a quantity of heat is transferred from the fluid to the heat reservoir. However, because of our original definition of the symbols W and Q, we state that the work, W3, is done by the surroundings on the fluid (a positive quantity) and the quantity of heat, Qu is transferred from the heat reservoir to the fluid (a negative quantity). The point D or V4 is not independent. The adiabatic AD is determined by the point A and the isotherm CD is determined by the point C. Point D is the intersection of the two curves. 

Finally, we compress the fluid adiabatically and reversibly from D to A, thus completing the cycle. In this compression the work, WA, is done by the surroundings on the fluid (a positive quantity) and no heat is transferred from or to the system. 

It is important to remember that the numerical value of Q is negative. 

The line 2-3 corresponds to adiabatic gas expansion (Q = 0) with no contact with the heat reservoir. Then + Aj-s = 0. Therefore, the work done by gas along this line is 

When it reaches tanperature Tj (at the end of the adiabatic expansion), the gas transfers to the refrigerator heat 

Then along the line 4-1, the gas is compressed so that its temperature acquires the initial temperature T. Since the heat exchange along this line is absent, the work of the external forces A 4 i equals the increment of the internal energy AUj.j = v Cy Then the 

The total work performed by gas in one cycle is the sum of work produced on each segment 

After some transformation, the expression adopts the form  

In the form in which it has been expressed thus far, the second law is not a statement that can be applied conveniently to chemical problems. We wish to use the second law of thermodynamics to establish a criterion by which we can determine whether a chemical reaction or a phase change will proceed spontaneously. Such a criterion would be available if we could obtain a function that had the following two characteristics. 

It should be a thermodynamic property that is, its value should depend only on the state of the system and not on the particular path by which the state has been reached. 

It should change in a characteristic manner (for example, always increase) when a reaction proceeds spontaneously. 

The Camot engine is a device by which a working substance can exchange mechanical work with its surroundings and can exchange heat with two heat reservoirs. 

To maintain isothermal conditions during this process, a quantity of heat qi is absorbed from a high-temperature heat reservoir operating at T2. Since AC/ = 0 for this isothermal expansion, qz — -in, so that 

To maintain isothermal conditions, a quantity of heat q given by 

Note that the sum of the adiabatic work in steps (2) and (4) is zero. Also, since the two adiabats span the same temperature range, equation (3.76) can be used to show that 

Substitution for K4 jVy from this equation into equation (3.84) and dividing by 72 given by equation (3.79) gives the efficiency 77 of conversion of heat into work for the Carnot cycle as 

But this is exactly the same as equation (2.82) that we obtained earlier 

Starting at point (1), the fluid is compressed without heat loss (adiabatically) or mechanical loss to point (2). The absolute temperature rises from to T2 during this compression. The fluid expands at constant temperature without losses to point (3) as it takes heat Q ) from a reservoir at temperature (T ), It then expands without heat or mechanical loss to point (4) as the temperature of the fluid drops to Tj. The fluid is compressed adiabatically back to point (1) at constant temperature as it rejects heat (Qj) to a second reservoir having a constant temperature (Tj). From points (2) to (3) and (3) to (4), work equal to Q2 is delivered to an external system, but from (4) to (1) and (1) to (2), work equal to Qi is taken from an external system. The net work done is Q2 - 2i) and the efficiency of the process (e ) is  

However, the Kelvin temperature scale is based on the fact that  

This is the maximum efficiency possible for any cycle operating between absolute temperatures T ) and T2). 

In the annals of science Carnot s work is almost unique, for it had no discernible predecessors and was built up from the assemblage of unordered opinions and problems, concepts, theories and measurements that were available at the time. Carnot, in short, was not standing on the shoulders of giants he sawfurther than his contemporaries because he had a much clearer vision (Cardwell, 1971, p.l93). 

Indeed we cannot (explain Carnot s thought process), for he was one of the most truly original thinkers in the whole history of science. It might not be wholly inappropriate to say that in the independenceofhis spirit he was English, and in the rigor and clarity of his mind he was French but in the depth and scope of his insight he transcends all such classifications (Ibid, p.211). 

The piston has now completed its cycle. I have drawn a picture of this cycle in Fig. 29.2.1 have plotted the pressure inside the cylinder against 

The gas is pushed out into the discharge line, at a pressure equal to the discharge-line pressure. 

The piston reaches bottom dead center, and reverses its direction. 

The gas left in the end of the cylinder expands—and depressurizes— until the suction-line pressure is reached. 

New gas is taken into the cylinder until the piston returns to top dead center. 

Pressure-volume diagram for the Carnot cycle. AB and CD are isolherrris, and BC and DA are adiabatics (no heat transfer). 

Let us consider these four steps in further detail In particular we want to know the AU values for each step, the amounts of heat absorbed (q) and the work done (w). The expressions for these quantities are summarized in Table 5.1, 

Step A 5 B is the reversible isothermal expansion at Th. On p. 172 we showed that for the isothermal expansion of an ideal gas there is no change of internal energy  

In equation (4.69) we showed that the work done on the system in an isothermal reversible process 

Step B - C involves surrounding the cylinder with an insulating jacket and allowing the system to expand reversibly and adiabatically to a volume of V. Since the process is adiabatic 

In 1824 a French engineer, Sadi Carnot, investigated the principles governing the transformation of thermal energy, heat, into mechanical energy, work. He based his discussion on a cyclical transformation of a system that is now called the Carnot cycle. The 

Carnot cycle consists of four reversible steps, and therefore is a reversible cycle. A system is subjected consecutively to the reversible changes in state  

Step 1. Isothermal expansion. Step 3. Isothermal compression. 

Step 2. Adiabatic expansion. Step 4. Adiabatic compression. 

Since the mass of the system is fixed, the state can be described by any two of the three variables T, p, V. A system of this sort that produces only heat and work effects in the surroundings is called a heat engine. A heat reservoir is a system that has the same temperature everywhere within it this temperature is unaffected by the transfer of any desired quantity of heat into or out of the reservoir. 

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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... 

If the Carnot cycle is used to calculate the work embedded in the thermal flows with the assumption that the heat-transfer coefficient, U, is constant and the process temperature is much greater than AT, a simple derivation yields the following ... 

The Carnot refrigeratiou cycle is reversible and consists of adiabatic (iseutropic due to reversible character) compression (1-2), isothermal rejection of heat (2-3), adiabatic expansion (3-4) and isothermal addition of heat (4-1). The temperature-entropy diagram is shown in Fig. 11-70. The Carnot cycle is an unattainable ideal which serves as a standard of comparison and it provides a convenient guide to the temperatures that should be maintained to achieve maximum effectiveness. 

The Intercooled Regenerative Reheat Cycle The Carnot cycle is the optimum cycle between two temperatures, and all cycles try to approach this optimum. Maximum thermal efficiency is achieved by approaching the isothermal compression and expansion of the Carnot cycle or by intercoohng in compression and reheating in the expansion process. The intercooled regenerative reheat cycle approaches this optimum cycle in a practical fashion. This cycle achieves the maximum efficiency and work output of any of the cycles described to this point. With the insertion of an intercooler in the compressor, the pressure ratio for maximum efficiency moves to a much higher ratio, as indicated in Fig. 29-36. 

The thermal efficiency of the process (QE) should be compared with a thermodynamically ideal Carnot cycle, which can be done by comparing the respective indicator diagrams. These show the variation of temperamre, volume and pressure in the combustion chamber during the operating cycle. In the Carnot cycle one mole of gas is subjected to alternate isothermal and adiabatic compression or expansion at two temperatures. By die first law of thermodynamics the isothermal work done on (compression) or by the gas (expansion) is accompanied by the absorption or evolution of heat (Figure 2.2). 

Figure 2.2 The indicator diagrams for the Carnot and the Otto engines. The Carnot cycle operates between the two temperatures Tj and T2 only, whereas the Otto cycle undergoes a temperature increase as a result of combustion.

All the Carnot cycles making up the simple gas turbine cycle have the same efficiency. Likewise, all of the Carnot cycles into which the cycle a-b-c-2-a might similarly be divided have a common value of efficiency lower than the Carnot cycles which comprise cycle 1-2-3-4-1. Thus, the addition of an intercooler, which adds a-b-c-2-a to the simple cycle, lowers the efficiency of the cycle. 

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 his search for high efficiency, the designer of a gas turbine power plant will attempt to emulate these features of the Carnot cycle. 

In Chapter 1, the gas turbine plant was considered briefly in relation to an ideal plant based on the Carnot cycle. From the simple analysis in Section 1.4, it was explained that the closed cycle gas turbine failed to match the Carnot plant in thermal efficiency because of... 

It was pointed out in Chapter I that the desire for higher maximum temperature (T nx) in thermodynamic cycles, coupled with low heat rejection temperature (Tmin), is essentially based on attempting to emulate the Carnot cycle, in which the efficiency increases with... 

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... 

Applying the first law of thermodynamics to the Carnot cycle gives... 

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 ... 

An estimate of the efficiency of a heat engine working between two temperatures T and T - can be obtained by assuming the Carnot cycle is used. By combining the results from applying the first and second laws to the Carnot cycle, the Carnot efficiency e, may be written ... 

The maximum theoretical work Wn, obtainable from a system was derived by Carnot who considered the transformation of heat energy into work when a perfect gas in a cylinder with a piston was taken through a reversible cycle (the Carnot cycle), in which the system was almost at equilibrium during each step of the cycle. It was shown that... 

A concept of the cycle of thermodynamic processes, introduced later than the Carnot cycle. Modifications of the Rankine cycle are of practical importance in boiler design, in relating the successive thermodynamic changes as water is converted to steam, expands and converted to mechanical energy in a turbine, then condenses and returns to the boiler. 

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 the next chapter, we will return to the Carnot cycle, describe it quantitatively for an ideal gas with constant heat capacity as the working fluid in the engine, and show that the thermodynamic temperature defined through equation (2.34) or (2.35) is proportional to the absolute temperature, defined through the ideal gas equation pVm = RT. The proportionality constant between the two scales can be set equal to one, so that temperatures on the two scales are the same. That is, 7 °Absolute) = T(Kelvin).r... 

We remember from the earlier discussion of the Carnot cycle that an empirical temperature scale is based on some arbitrary physical property (such as density, electrical resistance, magnetic susceptibility, etc.) that changes in a way that is continuous and single valued. 

To summarize, the Carnot cycle or the Caratheodory principle leads to an integrating denominator that converts the inexact differential 8qrev into an exact differential. This integrating denominator can assume an infinite number of forms, one of which is the thermodynamic (Kelvin) temperature T that is equal to the ideal gas (absolute) temperature. The result is... 

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... 

In summary, the Carnot cycle can be used to define the thermodynamic temperature (see Section 2.2b), show that this thermodynamic temperature is an integrating denominator that converts the inexact differential bq into an exact differential of the entropy dS, and show that this thermodynamic temperature is the same as the absolute temperature obtained from the ideal gas. This hypothetical engine is indeed a useful one to consider. 

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