卡诺热机效率 Carnot efficiency — 公式推导、符号与局限

\[\eta = \frac{-w}{q_h} = 1 - \frac{T_c}{T_h},\qquad \frac{q_h}{T_h} + \frac{q_c}{T_c} = 0\]

制冷机与热泵的性能系数coefficients of performance

\[\varepsilon_\text{制冷} = \frac{T_c}{T_h - T_c},\qquad \varepsilon_\text{热泵} = \frac{T_h}{T_h - T_c}\]\[\varepsilon_\text{refrigerator} = \frac{T_c}{T_h - T_c},\qquad \varepsilon_\text{heat pump} = \frac{T_h}{T_h - T_c}\]

含义Meaning

工作于两个热源之间的热机，可逆卡诺循环效率最高，且只取决于两热源的温度（卡诺，1824）。它引出了热力学温标和熵的概念。

Between two reservoirs the reversible Carnot cycle is most efficient, depending only on their temperatures (Carnot, 1824). It led to the thermodynamic temperature scale and to entropy.

符号说明Symbols

符号Symbol	含义 Meaning	单位 Unit
\(T_h, T_c\)	高温、低温热源温度hot and cold reservoir temperatures	K
\(q_h, q_c\)	从高温源吸热、向低温源放热（q_c < 0）heat from hot and to cold reservoir	J

推导Derivation

卡诺循环：等温膨胀（T_h 吸热）→ 绝热膨胀 → 等温压缩（T_c 放热）→ 绝热压缩。
Isothermal expansion, adiabatic expansion, isothermal compression, adiabatic compression.
两等温步 \(q_h = nRT_h\ln(V_2/V_1)\)，\(q_c = nRT_c\ln(V_4/V_3)\)；两绝热步使 \(V_3/V_4 = V_2/V_1\)。
Adiabats make V₃/V₄ = V₂/V₁.
故 \(q_h/T_h = -q_c/T_c\)：可逆循环中 Σq/T = 0，定义出状态函数熵。
So Σq/T = 0 over the cycle, defining entropy.
效率 \(\eta = (q_h + q_c)/q_h = 1 - T_c/T_h\)；任何实际热机效率更低（第二定律）。
Efficiency 1 − T_c/T_h; real engines do worse.

适用条件与局限Assumptions & limitations

可逆极限无法实现；实际电厂效率约为卡诺效率的 60–70%。
An unreachable reversible limit.

例题Worked example

蒸汽轮机 T_h = 823 K、T_c = 300 K：η_max = 1 − 300/823 = 63.5%；实际约 40%。

Steam turbine 823/300 K: 63.5 % maximum, ~40 % in practice.