A refrigerator is a device that uses work to transfer energy from a low-temperature reservoir to a high-temperature reservoir as the device continuously repeats a set series of thermodynamic processes. In a household refrigerator, for example, work is done by an electrical compressor to transfer energy from the food storage compartment (a low-temperature reservoir) to the room (a high-temperature reservoir).
Air conditioners and heat pumps are also refrigerators. The differences are only in the nature of the high- and low-temperature reservoirs. For an air conditioner, the low-temperature reservoir is the room that is to be cooled and the high-temperature reservoir is the (presumably warmer) outdoors. A heat pump is an air conditioner that can be operated in reverse to heat a room; the room is the high-temperature reservoir, and heat is transferred to it from the (presumably cooler) outdoors.
Let us consider an ideal refrigerator:
In an ideal refrigerator, all processes are reversible and no wasteful energy transfers occur as a result of, say, friction and turbulence.
Figure 20-13 shows the basic elements of an ideal refrigerator. Note that its operation is the reverse of how the Carnot engine of Fig. 20-7 operates. In other words, all the energy transfers, as either heat or work, are reversed from those of a Carnot engine. We can call such an ideal refrigerator a Carnot refrigerator.
The designer of a refrigerator would like to extract as much energy |QL| as possible from the low-temperature reservoir (what we want) for the least amount of work |W| (what we pay for). A measure of the efficiency of a refrigerator, then, is
where K is called the coefficient of performance. For a Carnot refrigerator, the first law of thermodynamics gives |W| = |QH| − |QL|, where |QH| is the magnitude of the energy transferred as heat to the high-temperature reservoir. Equation 20-12 then becomes
Because a Carnot refrigerator is a Carnot engine operating in reverse, we can combine Eq. 20-8 with Eq. 20-13; after some algebra we find
For typical room air conditioners, K ≈ 2.5. For household refrigerators, K ≈ 5. Perversely, the value of K is higher the closer the temperatures of the two reservoirs are to each other. That is why heat pumps are more effective in temperate climates than in climates where the outside temperature is much lower than the desired inside temperature.
Fig. 20-13 The elements of a refrigerator. The two black arrow-heads on the central loop suggest the working substance operating in a cycle, as if on a p–V plot. Energy is transferred as heat QL to the working substance from the low-temperature reservoir. Energy is transferred as heat QH to the high-temperature reservoir from the working substance. Work W is done on the refrigerator (on the working substance) by something in the environment.
It would be nice to own a refrigerator that did not require some input of work—that is, one that would run without being plugged in. Figure 20-14 represents another “inventor’s dream,” a perfect refrigerator that transfers energy as heat Q from a cold reservoir to a warm reservoir without the need for work. Because the unit operates in cycles, the entropy of the working substance does not change during a complete cycle. The entropies of the two reservoirs, however, do change: The entropy change for the cold reservoir is −|Q|/TL, and that for the warm reservoir is +|Q|/TH. Thus, the net entropy change for the entire system is
Because TH > TL, the right side of this equation is negative and thus the net change in entropy per cycle for the closed system refrigerator + reservoirs is also negative. Because such a decrease in entropy violates the second law of thermodynamics (Eq. 20-5), a perfect refrigerator does not exist. (If you want your refrigerator to operate, you must plug it in.)
This result leads us to another (equivalent) formulation of the second law of thermodynamics:
Fig. 20-14 The elements of a perfect refrigerator—that is, one that transfers energy from a low-temperature reservoir to a high-temperature reservoir without any input of work.
No series of processes is possible whose sole result is the transfer of energy as heat from a reservoir at a given temperature to a reservoir at a higher temperature.
In short, there are no perfect refrigerators.
CHECK POINT 4 You wish to increase the coefficient of performance of an ideal refrigerator. You can do so by (a) running the cold chamber at a slightly higher temperature, (b) running the cold chamber at a slightly lower temperature, (c) moving the unit to a slightly warmer room, or (d) moving it to a slightly cooler room. The magnitudes of the temperature changes are to be the same in all four cases. List the changes according to the resulting coefficients of performance, greatest first.
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