An ideal gas undergoes a cycle between heat reservoirs at 600 K and 300 K, returning to its initial state. It absorbs 100 J of heat from the hot reservoir.
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この機関では、受け取った熱の何％を仕事として取り出せるか？
出力部の摩擦で1周ごとに10 Jが熱になると、効率は何％か？
The engine’s operation is related to the equations
14 processes complete the cycleIsothermal→Adiabatic→Isothermal→Adiabatic……
2The heat absorbed during isothermal expansion at the higher temperature is expressed in terms of workQH=∫V₁V₂ p dV……
3Integration gives the heat absorbed at the higher temperatureQH=n R TH ln(V₂V₁)……
4The heat transferred to the cold reservoir is calculatedQC=n R TC ln(V₃V₄)……
5The 2 adiabatic processes show that the volume ratios for isothermal expansion and compression are equalV₂V₁=V₃V₄……
6The volume ratios express the heat ratio in terms of the temperature ratioQCQH=TCTH……
7The efficiency is calculatede=1 − TCTH……
8In reverse, the engine acts as a refrigeratorQH=W+QC……
9An engine more efficient than a Carnot engine would violate the second lawe≦1 − TCTH(Heat engines operating between the same hotand cold reservoirs)……
10The loss due to friction in the output mechanism is calculatedefric=W − WfricQH＜1 − TCTH……
Further study: entropy and applications to real machines
11Dividing heat by absolute temperature leads to entropyFor any reversible cycle∮δQrevT=0⇒dS=δQrevT……
12Ideal upper limits are calculated for power generation and air conditioningUpper limit on power-generation efficiency65％/Upper limit on the coefficient of performancefor air conditioning37.5 times……
What this chapter has shown
The second law prevents a heat engine that returns to its initial state after each cycle from converting all the heat it absorbs into work. Between the same hot and cold reservoirs, the reversible-engine efficiency 1 − TC/TH is the upper limit. It cannot reach 100％ if the cold reservoir has a positive absolute temperature.
50％Upper limit at 600K and 300K
When friction converts 10 J of work into heat each cycle, the efficiency falls to 40％.
In everyday lifeIn a thermal power plant, the heat received from the fuel that is not converted into work is released to a colder reservoir, such as seawater.
Detailed summary
The maximum efficiency of a heat engine depends only on the ratio of the temperatures at which it absorbs and rejects heat and cannot exceed 1 − TC/TH. Exceeding this limit would violate the second law of thermodynamics. Irreversible processes, such as friction, reduce the efficiency further. Calculations for the Carnot cycle lead to the result that integrating the heat transferred divided by temperature along a reversible path gives the change in a state function called entropy.