## Heat Engines and the Carnot Cycle

A heat engine is defined by four requirements. It

1. Receives heat from a high-temperature source (e.g., coal furnace, nuclear reactor)

2. Converts part of this heat to work (e.g., by a turbine)

3. Rejects the remaining waste heat to a low-temperature sink (e.g., atmosphere, river)

### 4. Operates on a thermodynamic cycle

The steam power plant fits most closely the definition of a heat engine, receiving heat from an external combustion chamber, extracting work by a turbine, and rejecting heat to a condenser. In contrast, internal combustion engines such as spark-ignition and diesel engines as well as gas turbines are "open" and therefore do not satisfy the requirement of operating on a thermodynamic cycle because the working fluid is continually replaced; the combustion gas is exhausted into the atmosphere and a fresh charge of air is scavenged for the next mechanical cycle.

For a cyclic process, the initial and final states are identical, so the First Law relation of Eq. (3.2) involves only the heat input and work output terms, as in Eq. (3.17).

Therefore, the net work that the system can perform is related to the net heat flow that enters the system.

According to the definition of a heat engine, at least two thermal reservoirs are involved in the cycle, with heat entering the system from the high-temperature reservoir and heat exiting the system to the low-temperature reservoir. The net work, Wnet, in Eq. (3.19) is the difference between the heat input, Qin, and heat output, Qout, of the system.

The thermal efficiency, nth, of a heat engine is determined by the amount of work converted from the amount of energy input into the system.

III I III

FIGURE 3.3 The four stages of the reversible Carnot cycle: (1-2) isothermal expansion, (2-3) adiabatic expansion, (3-4) isothermal compression, (4-1) adiabatic compression.

_ ^^net _ Qin Qout _ i Qout / o nth = ~Q~ ~ ~Q- " 1 ~Q~ ( )

The greater the net work of the system, the higher the conversion efficiency. To achieve the maximum possible work in a heat engine, all of the processes should be conducted in a reversible manner. An idealized cycle proposed by the French scientist, Sadi Carnot, in 1824, involves four reversible processes and is known as the Carnot cycle; it is depicted in Fig. 3.3 with a piston in a cylinder and in Fig. 3.4 with steady-flow devices.

Because heat addition and heat rejection are performed reversibly and isothermally, Eq. (3.15), modified to give Eq. (3.21), can be used to determine the efficiency of the cycle. Equation (3.22) is the heat addition step (step 1-2), and Eq. (3.23) is the heat rejection step (step 3-4).

The cycle is represented as a thermodynamic cycle in Fig. 3.5, which shows that during the reversible adiabatic processes, steps 2-3 and 4-1, the entropy remains the same. The figure also shows that the difference between the heat input, Qin, and the heat output, Qout, is the net work, Wnet, indicated by the shaded area.

FIGURE 3.5 The Carnot cycle shown on a T-s diagram.

The entropy terms in Eqs. (3.22) and (3.23) are identical, and substituting the expressions into Eq. (3.20) results in Eq. (3.24), the thermal efficiency for the Carnot cycle. Because all of the processes are reversible, the Carnot cycle is the most efficient heat engine, and Eq. (3.24) is the maximum possible conversion efficiency for any heat engine.

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