Example 15.4 from College Physics, 15.4 Carnot’s Perfect Heat Engine: The Second Law of Thermodynamics Restated

Maximum Theoretical Efficiency for a Nuclear Reactor

A nuclear power reactor has pressurized water at $300 º C$ . (Higher temperatures are theoretically possible but practically not, due to limitations with materials used in the reactor.) Heat transfer from this water is a complex process (see Figure 15.23). Steam, produced in the steam generator, is used to drive the turbine-generators. Eventually the steam is condensed to water at $27 º C$ and then heated again to start the cycle over. Calculate the maximum theoretical efficiency for a heat engine operating between these two temperatures.

Diagram shows a schematic diagram of a pressurized water nuclear reactor and the steam turbines that convert work into electrical energy. There is a pressure vessel in the middle, dome shaped at the ends. This has a nuclear core in it. The core is a small square in the center of the reactor. Control rods are shown as sticks of equal length attached to the core. The pressure vessel has some coolant tubes passing through it and then back to a steam chamber. These coolant tubes contain a coolant liquid that transports the heat from the pressure vessel to the steam chamber. This whole system is enclosed in another dome shaped containment structure of steel. The water supply to steam chamber and the steam outlet are seen to come out of this chamber. This steam is now shown to run two steam turbines, one a high pressure one and another low pressure one. The turbines are nearly triangular and segmented in shape. The steam turbine in turn generates power using a turbine generator, which is attached to the turbine system. The turbines are again housed in another chamber which gets the steam from the steam chamber and return the steam as water back to the steam chamber with pipes. A coolant tower is shown near the turbine system, which is shown to supply cool water in tubes to the turbine system to cool the steam back to water.

Figure 15.23 Schematic diagram of a pressurized water nuclear reactor and the steam turbines that convert work into electrical energy. Heat exchange is used to generate steam, in part to avoid contamination of the generators with radioactivity. Two turbines are used because this is less expensive than operating a single generator that produces the same amount of electrical energy. The steam is condensed to liquid before being returned to the heat exchanger, to keep exit steam pressure low and aid the flow of steam through the turbines (equivalent to using a lower-temperature cold reservoir). The considerable energy associated with condensation must be dissipated into the local environment; in this example, a cooling tower is used so there is no direct heat transfer to an aquatic environment. (Note that the water going to the cooling tower does not come into contact with the steam flowing over the turbines.)

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Strategy
Since temperatures are given for the hot and cold reservoirs of this heat engine, ${Eff}_{C} = 1 - \frac{T_{c}}{T_{h}}$ can be used to calculate the Carnot (maximum theoretical) efficiency. Those temperatures must first be converted to kelvins.
Solution
The hot and cold reservoir temperatures are given as $300 º C$ and $27.0 º C$ , respectively. In kelvins, then, $T_{h} = 573 K$ and $T_{c} = 300 K$ , so that the maximum efficiency is
${Eff}_{C} = 1 - \frac{T_{c}}{T_{h}} .$
15.35

Thus,

$\begin{array}{l} {Eff}_{C} & = & 1 - \frac{300 K}{573 K} \\ = & 0.476, or 47 . 6% . \end{array}$
15.36
Discussion
A typical nuclear power station’s actual efficiency is about 35%, a little better than 0.7 times the maximum possible value, a tribute to superior engineering. Electrical power stations fired by coal, oil, and natural gas have greater actual efficiencies (about 42%), because their boilers can reach higher temperatures and pressures. The cold reservoir temperature in any of these power stations is limited by the local environment. Figure 15.24 shows (a) the exterior of a nuclear power station and (b) the exterior of a coal-fired power station. Both have cooling towers into which water from the condenser enters the tower near the top and is sprayed downward, cooled by evaporation.

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