## Info

s, kJikg-K

s, kJikg-K

Fig. 4.5 Process diagrams for the power cycle of Fig 4.4 on the EOS of water

4.1.3 The First law for heat engines

In all four prior illustrations of heat engines, the 1st law is:

Because of the cyclic nature of the system, there is no change in internal energy or any other property of the working fluid in each cycle.

### 4.2 The Second Law applied to heat-engine cycles

The following qualitative constraints on the heat-engine cycles discussed in Sect. 4.1 were discovered in the nineteenth century and eventually led to the concept of entropy. These constraints restrict the functioning of cycles more severely than does the 1st law.

Both are equivalent statements of the Second law, and are ultimately based on empirical evidence. By comparison to Fig. 4.1, the disallowed cyclical engines are shown in Fig. 4.6

Impossible (Kelvin-Planck)

Hoi reservoir Th

Hoi reservoir Th

Cold Reservoir

Cold Reservoir

Impossible (Clausius)

Fig. 4.6 Power cycles inadmissible by the 2nd law

Kelvin-Planck: No cycle can produce net work with only a single thermal reservoir.

This is a formal phrasing of a fact that has been noted earlier: heat cannot be completely converted to work. The Kelvin-Planck statement says that QL in the power cycle Fig. 4.1 cannot be zero.

Clausius: No cycle can only transfer of heat from a cold reservoir to a hot reservoir

This statement essentially prohibits heat from flowing from cold to hot bodies, something that was demonstrated (also using the Second law) in Sect. 1.9. With reference to Fig. 4.1, the Clausius statement says that the direction of QH and QL cannot be reversed and at the same time W set equal to zero. This version of the Second law does not, however, entirely prohibit transfer of heat from a cold body to a hot body; it simply requires that external work must be expended in order to do so.

The two statements of the 2nd law appear to be quite different, but in fact they are equivalent. This equivalence can be shown with the aid of Fig. 4.7, which contains heat engine A and heat pump B operating between the same hot and cold reservoirs.

Fig. 4.7 Demonstration of the equivalence the two statements of the 2nd Law

The devices are arranged so that the heat withdrawn from the cold reservoir by pump B is the same magnitude as the heat rejected by engine A. Therefore, the two cancel and the cold reservoir experiences no net heat transfer. Cycle B violates the Clausius statement. It remains to show that the combination of cycle A and cycle B violates the Kelvin-Planck statement.

Cycle A receives a quantity of heat QH from the hot reservoir that is greater than delivered by cycle B. By the 1st law applied to the system within the dashed box, Qh - Ql units of heat are completely converted to work without rejecting heat to the cold reservoir, in violation of the Kelvin-Planck statement. The device in Fig. 4.7, which is known as a perpetual-motion machine of the second kind, thus fails both the Clausius and Kelvin-Planck versions of the 2nd law.

### 4.3 The Carnot Cycle

If the Kelvin-Planck form of the Second law prohibits complete conversion of heat to work, what then is the maximum value of the efficiency of the cycle shown in Fig. 4.1?

The cycle efficiency is defined by:

Qh Qh where the second equality arises from elimination of W using Eq (4.3).

The maximum-efficiency cycle is the one containing the four steps listed in Sect. 4.1 with the additional restriction that the heat engine be reversible; heat exchange with the reservoirs must take place over infinitesimally small temperature differences, all motions must be frictionless, and rapid compression of the working fluid is prohibited. This idealized heat engine was first investigated by Carnot in 1824, and the cycle and its efficiency bear his name. Carnot showed that the efficiency of this cycle is simply related to the temperature of the hot and cold reservoirs, and moreover, that no other engine operating between the same two temperatures can have a greater efficiency.

Determination of the efficiency of the Carnot cycle starts from the form of the Second law given by Eq(1.13) for reversible operation:

ASengine + ASsurroundings = 0

The thermodynamic system in this usage is the Carnot engine and the surroundings are the two reservoirs. The surroundings also contain a mechanism for receiving the work performed by the Carnot engine but this mechanism is not involved in exchange of entropy. Since the working fluid returns to its original state after each cycle, its entropy change is zero. Because all aspects of the heat engine are reversible, no entropy is generated from this device. Consequently, ASengine = 0, and by the above equation,

ASsurroundings = 0 as weU.

The null entropy change of the surroundings consists of two canceling terms, each of the form given by Eq (1.9). The hot reservoir delivers entropy in the amount QH/TH to the engine and the reject heat transfers entropy equal to QL/TL to the cold reservoir. If the surroundings do not experience an entropy change as a result of operation of the Carnot engine, these two entropy flows must be equal, or:

Qh Ql

Combining this result with Eq (4.4) gives the Carnot efficiency: