3.4 The Carnot cycle. Carnot's theorem

 

In 1824 the French engineer S. Carnot published a paper which later provided the theoretical fundamentals of heat engines. In this work Carnot investigated the cycle of a heat engine (named after him later on) which is of paramount importance for thermodynamics.

The Carnot cycle is realized by a working medium between two heat sources, a high- and a low-temperature source, in the following way (Fig. 3.4). The working medium, whose properties at the initial point 1 of the cycle are temperature T1 specific volume v1 and pressure p1, receives heat from the high-temperature source (whose temperature is designated by Th.t.) with Th.t. > T1. The working medium (gas) expands, performing work (for in­stance, displaces a piston in a cylinder). The process of adding heat to the working medium can be visualized as occurring in a way that the temperature of the working medium remains constant (i.e. the decrease in the tem­perature of the expanding gas is compensated for by adding heat from the outside). In other words, an isothermal process is being affected, T1 = const. After the gas expands to some state (point 2), the addition of heat to the gas terminates and further expansion of the gas proceeds without heat addition, following an adiabatic line. In the course of the adiabatic expansion the temperature of the gas decreases, since no energy is added to the gas from the outside, and, consequently, work is done only at the expense of the internal energy of the gas[1]. After the gas reaches some state 3 (the temperature of the gas in this state will be denoted T2), the process of expansion involving the performance of work terminates and the working medium returns into its initial state. The gas is compressed at the expense of work transferred from some external source, and heat is rejected from the gas while it undergoes compression, with the heat being transferred to a low-temperature source (heat sink) which is at a temperature T1.t. < T2. Heat is removed from the gas in a way that the temperature of the compressed gas remains constant, i.e. the gas undergoes isothermal compression along the isotherm T2 = const. After the gas comes into the state at point 4, lying on one adiabat with the initial point of the cycle 1, the rejection of heat ceases. Further adiabatic compression of the gas takes place until the gas returns into point 1. Thus, the Carnot cycle consists of two isothermal processes and two adiabatic processes. The work done by the gas during expansion is represented on the p-v diagram by the area under the line 1-2-3; the work spent to compress the gas, by the area under line 3-4-1; the useful work delivered to an outside consumer during the cycle is represented by area 1-2-3-4-1.

 

 

Fig. 3.4

 

It will be recalled that the amount of heat added to the working medium from the high-temperature source is denoted by Q1, and the amount of heat rejected from the working medium to the low-temperature source by Q2.

Since heat is added from the high-temperature source to the working medium in the process 1-2 at a finite temperature difference Th.t.— T1, and heat is rejected from the working medium to the low-temperature source during the process 3-4 at a finite temperature difference T2 — Tl.t., these processes are irreversible. As was mentioned above, the irreversibility of processes can be reduced almost to zero if the difference between the temperatures of the working medium and the heat source is infinitely small:

 

                                                                                                                       (3.14)

 

                                                                                                                       (3.15)

 

Thus, in the isothermal process 1-2 the temperature of the gas is below the temperature of the high-temperature source by the infinitesimal dT, and during the isothermal process 3-4 the temperature of the gas exceeds the temperature of the low-temperature source (sink) by the infinitesimal value dT.

If this condition is satisfied and the gas expands during the process 2-3 and is compressed during the process 4-1 without friction, the considered cycle becomes reversible. Indeed, consider a Carnot cycle affected by the same working medium between the same heat sources in the reverse direction, as illustrated in Fig. 3.5. The compressed gas whose state is represented on the p-v diagram by point 1, expands along the adiabat 1-4 performing work (displacing a piston). During adiabatic expansion the temperature of the gas decreases. After this the gas, undergoing adiabatic expansion, reaches point 4, at which its temperature (denoted by, TII) is below the temperature of the low-temperature source by dT:

 

                                                                                                                      (3.16)

 

and the adiabatic process terminates. Then, the gas undergoes isothermal expansion along the line 4-3 (TII = const) and in the course of this process the gas removes heat from the low-temperature source. Further, at the expense of the work received from some external machine the gas is compressed adiabatically to the state corresponding to point 2. During the process of adiabatic expansion the temperature of the gas rises. The state 2 is selected so that in this state the temperature of the gas (denoted by TI) will be higher than the temperature of the high-temperature source by dT:

 

                                                                                                                            (3.17)

 

 

Fig. 3.5

 

Further, the gas undergoes isothermal compression following the line TI = const, and the heat released during compression is removed, or rejected, to the high-temperature source, resulting in that the gas returns to the initial point 1. Comparing Eqs. (3.14) and (3.15) with Eqs. (3.16) and (3.17), we see that T1 = TI and T2 = TII accurate to infinitesimals, i.e., in a reverse cycle the state of the working medium changes in the same temperature interval as in the forward cycle. The work done by the expanding gas is represented by the area under the curve 1-4-3, and compression work is equal to the area under curve 3-2-1, and, consequently, the difference between expansion work and compression work is represented by the area 1-4-3-2-1. Denote this difference in the two kinds of work by —Lc, the minus sign indicating that the work is transferred from an external source. The reverse cycle considered results in that an amount of heat Q1 is removed from the low-temperature source and added to the high-temperature source. The high-temperature source also receives heat equivalent to the external work Lc. Thus, the high-temperature source receives a total amount of heat Q1 = Q2 + Lc. Since, as it was shown above, T1 = TI and T2 = TII, the heat removed from the low-temperature source during the reverse cycle, Q2, is equal to the amount of heat added to this source during the forward cycle. Accordingly, during the reverse cycle the amount of heat added to the high-temperature source, Q1, is equal to the amount of heat removed from this source during the forward cycle. Hence, the work expended by the external source for the reverse cycle to be realized, is equal exactly to the work transferred to the external consumer during the forward cycle.

Thus, we performed a reverse Carnot cycle, following the path of the forward cycle, i.e. the forward cycle is reversible.

The reversibility of the cycle is due to the fact that the temperatures of the high-temperature source and of the working medium are equal (to an infinitesimal) during the isothermal process between points 1 and 2, and also because the temperatures of the low-temperature source and of the working medium are equal during the isothermal process from 3 to 4. If the difference between the temperatures of the heat source and the working medium was finite, the cycle would be irreversible.

In the reversible cycle heat Q2 is reversibly transferred from the high-temperature source to the low-temperature source. Thus, a reversible cycle can be considered as a method for affecting reversible transport of heat from a body at a higher temperature (heat source) to a body at a lower temperature (heat sink), and vice versa. But if the cycle is irreversible, the process of heat transfer from the heat source to the heat sink is irreversible. The degree of irreversibility of the transfer of heat from the heat source to the heat sink is the greater, the larger the difference between the temperatures of the high-temperature source and the working medium and between the temperatures of the working medium and the low-temperature source. It is clear that the maximum degree of irreversibility corresponds to the transfer of heat from the heat source to the heat sink without the performance of work. In this connection, consider the thermal efficiency of the Carnot cycle. In accordance with the definition given above, the thermal efficiency of any cycle is expressed by the ratio

 

                                                             

 

Assume that the working medium (substance) is an ideal gas with a constant heat capacity. Since the internal energy u of an ideal gas depends only on temperature, it follows that

 

                                                               

 

With account taken of the above relationship, the mathematical formulation of the first law of thermodynamics for an ideal gas can be presented as

 

                                                                                                                 (3.18)

 

whence, for an isothermal process (T = const, i.e. dT = 0) we get:

 

                                                                                                                                  (3.19)

 

Since, in accordance with Clapeyron's equation, for an ideal gas

 

                                                                 

 

Eq. (3.19) takes the following form:

 

                                                                                                                              (3.20)

 

whence

 

                                                                                                                        (3.21)

 

where the subscripts "I" and "II" pertain to the initial and final points of a process, respectively. If an ideal gas expands in an isothermal process, i.e. vII > vI, then, as can be seen from equation (3.21), qI-II > 0, meaning that heat should be added to the gas so that its temperature remains constant in the process of isothermal expansion. But if compression is involved, i.e. vII < vI, then  qI-II < 0.

It follows from the mathematical formulation of the first law of thermodynamics for an ideal gas (Eq. 3.18) that in an adiabatic process (dq = 0)

 

                                                                                                                       (3.22)

 

Dividing this equation by Clapeyron's equation term wise, we get:

 

                                                                                                                           (3.23)

 

Since [see Eq. (2.54)]

 

                                                                   

 

 we get:

 

                                                                                                                     (3.24)

 

where k denotes the ratio of heat capacities cp and cv.

Inasmuch as we consider an ideal gas whose heat capacity does not depend on temperature, k will be constant and independent of temperature. Taking this fact into account, we find that integration of Eq. (3.24) yields

 

                                                                                                                      (3.25)

 

Let us use these relationships to calculate the thermal efficiency of the Carnot cycle.

As applied to the isothermal processes in the Carnot cycle, with account taken of Eq. (3.21) the formulas for q1 and q2 are

 

                                                                                                                    (3.26)

 

                                                                                                                    (3.27)

 

Substituting Eqs. (3.26) and (3.27) in the equation expressing the thermal efficiency of a cycle, we get:

 

                                                                                                      (3.28)

 

Inasmuch as in the Carnot cycle the adiabatic processes 2-3 and 4-1 proceed under the same temperatures T1 and T2, in accordance with Eq. (3.25) we can write for each of these adiabats:

 

                                                                                                                       (3.29)

 

and

 

                                                                                                                      (3.30)

 

consequently,

 

                                                                                                                              (3.31)

 

whence, the expression for the thermal efficiency of the Carnot cycle is

 

                                                                                                                       (3.32)

 

As can be seen from Eq. (3.32), the thermal efficiency ηT depends on T1 and T2, with the efficiency being the higher the greater the difference between T1 and T2 is. The thermal efficiency of the Carnot cycle becomes equal to unity in two practically unattainable cases: either when T1 = ∞ or when T2 = 0.

Compare now the thermal efficiencies of the reversible and irreversible Carnot cycles with an ideal gas, realized between the same heat sources which are at temperatures Th.t. and Tl.t. In accordance with what was said above, it is clear that to calculate the thermal efficiency of a reversible Carnot cycle we must substitute T1 = Th.t. and T2 = Tl.t.. (accurate to infinitesimals) in Eq. (3.32):

 

                                                                                                                (3.33)

 

where  is the thermal efficiency of a reversible Carnot cycle.

In an irreversible Carnot cycle there is a finite difference between the temperatures of the heat sources and of the working medium:

 

                                                                                                                    (3.34)

 

                                                                                                                    (3.35)

 

The working temperature interval of the cycle diminishes since T1 < Th.t. and T2 > Tl.t. (Fig. 3.6). With account taken of these relationships, we obtain from Eq. (3.33):

 

                                                                                        (3.36)

 

where  is the thermal efficiency of an irreversible Carnot cycle.

Comparing (3.36) and (3.33), we can see that

 

                                                                                                                      (3.37)

 

or the thermal efficiency of an irreversible Carnot cycle is smaller than that of a reversible Carnot cycle. Note that this conclusion was obtained only for a Carnot cycle involving an ideal gas with a constant heat capacity.

 

 

Fig. 3.6

 

It should be stressed that the inequality (3.37) was derived with account being taken of only the external irreversibility of the cycle, i.e. the finite temperature difference between the working medium and the heat sources. Actual cycles also involve other factors which bring about, in addition, the internal irreversibility of cycles: processes of friction, the absence of mechanical equilibrium in component elements of an engine, etc. All these circumstances result in an additional decrease of the useful work of the cycle, q1 - q2, and consequently, to a further decrease of the thermal efficiency of the cycle.

It is clear that not only the Carnot cycle but also any other cycle can be visualized as being reversible (Fig. 3.7).[2] The condition determining the possibility of reversing these cycles is the same as for the Carnot cycle: an infinitesimal difference between the temperatures of the working medium and of the heat sources. It is essential to note here if the processes of heat addition and heat rejection are nonisothermal and, consequently, the temperature of the working medium changes continuously during the process of heat transfer between the medium and the heat source, the situation becomes still more complicated: even if at the initial point of the process the difference between the temperatures of the working medium and of the heat source is infinitely small, then, inasmuch as the temperature of the working medium varies with a change in its state, a finite temperature difference will set in between the working medium and the heat source and the process of heat transfer will therefore be irreversible. This difficulty is dealt with by introducing the concept of an infinitely great number of heat sources.

 

 

Fig. 3.7

 

What has been said is elucidated in Fig. 3.8. Any reversible cycle of arbitrary shape can be visualized as a totality of elementary Carnot cycles, consisting of two adiabats and two isotherms, with heat being added and rejected in any of the elementary cycles only along corresponding isotherms. As was already stated, for this to be accomplished there must be an infinite number of heat sources, and also an infinite number of elementary cycles must be realized. The totality of the elementary Carnot cycles is wholly equivalent to the original arbitrary reversible cycle. Indeed, inasmuch as the adiabatic lines of compression and expansion of each elementary Carnot cycle are infinitely close to each other, the processes of heat addition and rejection can be considered as being isothermal. Inasmuch as each of the adiabats, excluding the two outer ones, serves in total two times as the path of a cycle, and is passed in opposite directions, the work output of the cycle, when replaced by elementary Carnot cycles, does not change.

 

 

Fig. 3.8

 

From the point of view of the number of heat sources the reversible Carnot cycle is most effective, compared with any other reversible cycle: for the cycle to be realized only two heat sources must be available, inasmuch as heat is transferred from the high-temperature source to the working medium and from the latter heat is rejected to the low-temperature source in isothermal processes.

Let us proceed now to the proof of the following important thesis, known as the Carnot theorem[3]:

The thermal efficiency of a reversible Carnot cycle, realized between two heat sources, does not depend on the properties of the working medium used to realize the cycle.

The proof of this theorem is based on the rule of contraries. Consider the reversible cycle of a heat engine realized between two heat sources with the aid of some working medium; denote the thermal efficiency of this cycle by . Further consider the reversible cycle of a heat engine realized between the same heat sources but with the aid of another working medium; assume that the thermal efficiency of this engine (denoted by ) differs from the thermal efficiency of the first engine and, for the sake of definiteness, suppose that

 

                                                                                                                                (3.38)

 

The quantities pertaining to the first heat engine will have one prime, and those referring to the second engine, two primes.

Further assume that the second engine operates in a forward cycle, i.e. performs work  removing heat  from the high-temperature source and transferring heat  to the low-temperature source. At the same time, operating in a reverse cycle between the same heat sources, the first engine removes heat  from the low-temperature source, at the expense of work  received from some external source of work, and transfers heat  to the high-temperature source. The schematic layout of these cycles, realized between the same heat sources, is shown in Fig. 3.9. If the amount of the working medium used in one of these engines is given, we can always choose an amount of the working medium for the other engine so that  will be equal to . It is clear that the thermal efficiency in no way depends on the amount of the working medium involved in the cycle. It will be the same both for a cycle with 1 kg of the working medium and for a cycle involving 1000 kg of this working medium.

 

 

Fig. 3.9

 

In accordance with Eq. (3.10), the work transferred to the working medium in the cycle proceeding in the first engine is equal to

 

                                                                                                                       (3.39)

 

the work resulting from the cyclic operation of the second engine is

 

                                                                                                                    (3.39a)

 

Since we assumed that   and  it follows that

 

                                                                                     

 

or, which is the same,

 

                                                                                                                                                              (3.40)

 

where ΔL is the difference between the work output of the cycle realized in the second engine and the amount of external work added in the cycle realized in the first engine.

Comparing the expressions for the thermal efficiency of the cycles realized in the first and second engines, i.e.

 

                                                                                                                    (3.41)

 

                                                                                                                (3.41a)

 

and taking into account inequality (3.38), we obtain:

 

                                                                                                          (3.42)

 

Since

 

                                                               

 

Eq. (3.42) reduces to

 

                                                       

 

and further

 

                                                                                                                            (3.43)

 

Hence, in the reverse cycle the amount of heat removed from the low-temperature source is greater that the amount of heat added to the low-temperature source upon realization of the forward cycle. As regards the high-temperature source, since

 

                                                               

 

the amount of energy stored in the high-temperature source does not change after the realization of the forward and reverse cycles.

Thus, we arrive at the conclusion that the realization of the two reversible cycles considered, i.e. the forward and reverse cycles, results in the heat  being removed from the low-temperature source and, whence, work ( ) equivalent to this amount of heat[4] is produced. The thermal state of the high-temperature source undergoes no changes, however.

This conclusion violates the second law of thermodynamics as stated by Planck. Consequently, the initial prerequisite that the thermal efficiencies of the considered reversible cycles, which are realized with different working media between the same heat sources, are different, is erroneous. In this way the Carnot theorem is proved.

This theorem is true for reversible cycles realized between two heat sources.

We can prove this theorem in another way. Select the amounts of the working medium for the first and second engines so that the work done by the second engine in the forward cycle is equal to the work consumed by the first engine, operating in the reverse cycle:

 

                                                                                                                            (3.44)

 

or, which is the same,

 

                                                                                                            (3.44a)

 

If this is true, the work required to realize the reverse cycle can be transferred to the first engine not from some external source of work but from the second engine, producing an amount of work equal exactly to the amount of work consumed by the first engine.

Since, according to Eq. (3.41),

 

                                                       

 

and, in accordance with Eq. (3.41a),

 

                                                     

 

with account taken of Eq. (3.44), from Eq. (3.38) we get:

 

                                                                                                                            (3.45)

 

i.e. during the realization of the reverse cycle more heat is transferred to the high-temperature source than is removed from it during the forward cycle:

 

                                                                                                                    (3.46)

 

With account taken of this inequality, it follows from Eq. (3.44a) that

 

                                                               

 

i.e. during the realization of the reverse cycle more heat is removed from the low-temperature source than it receives in the forward cycle:

 

                                                                                                                  (3.47)

 

From equation (3.44a) it follows that

 

                                                                                                                          (3.48)

 

Thus, we arrived at the conclusion that while the two reversible cycles considered (the forward and the reverse) are realized, heat flows spontaneously and without any work expenditure (since ), from a body at a lower temperature (the low-temperature source) to a body at a higher temperature (the high-temperature source). This conclusion is also erroneous, since it violates Clausius' statement of the second law of thermodynamics.

Thus, in accordance with the Carnot theorem, the thermal efficiency of any reversible cycle realized between two heat sources does not depend on the properties of the working medium involved. Consequently, all conclusions that were previously drawn on the basis of the reversible Carnot cycle for an ideal gas with a constant heat capacity, are true for a reversible Carnot cycle with any working medium. In particular, the previously obtained expression (3.32) for the thermal efficiency of a cycle,  is applicable to any reversible Carnot cycle.

 

 

 


[1] For instance, for an ideal gas for which du = cvdT, in the absence of heat exchange between the working medium and sources (dq = 0), it follows from the mathematical statement of the first law of thermodynamics that  i.e. the temperature of the gas actually decreases during adiabatic expansion, and increases during the process of adiabatic compression. Adiabatic processes shall be considered in greater detail in Sec. 7.4. In Chapter 7 it will be shown, in particular, that on the p-v diagram the adiabatic line is steeper than the isothermal line.

 

[2] Various cycles of heat engines and of refrigerating machines are considered in Chapters 9-13.

[3] This theorem was formulated and proved by S. Carnot in his work "Réflexions sur la puissance motive de feu" [English translation: Carnot, S. Reflections on the Motive Power of Fire, New York, Dover (1960)] published in 1824 (i.e. before the law of equivalence of heat and work has been formulated and generally accepted). Carnot based all his reasonings on the then generally accepted theory of the phlogiston. This, however, in no way affects the general nature of the results obtained by Carnot.

[4] Since  and  it follows from Eq. (3.40) that  From the equality  it follows that  or