The Clausius inequality is a key concept in thermodynamics, setting limits on heat transfer and efficiency in cyclic processes. It states that the integral of heat transfer divided by temperature is always less than or equal to zero for any cycle.

This inequality has major implications for real-world systems like engines and refrigerators. It shows that no heat engine can be 100% efficient and no refrigerator can operate without external work, placing fundamental limits on their performance.

The Clausius Inequality

Clausius inequality and significance

• States for a system undergoing a cyclic process, the integral of $\frac{dQ}{T}$ (heat transfer divided by absolute temperature) is always less than or equal to zero $\oint \frac{dQ}{T} \leq 0$
• Fundamental statement of the second law of thermodynamics establishes direction of heat transfer and limits on efficiency of heat engines (steam turbines) and refrigerators (air conditioners)
• Valid for any cyclic process, both reversible (Carnot cycle) and irreversible (Rankine cycle)

Heat transfer direction in cycles

• Determines the direction of net heat transfer in a cyclic process
• For a reversible cyclic process, the integral of $\frac{dQ}{T}$ equals zero $\oint \frac{dQ}{T} = 0$
• For an irreversible cyclic process, the integral of $\frac{dQ}{T}$ is always less than zero $\oint \frac{dQ}{T} < 0$
• Indicates heat flows from a high-temperature reservoir (combustion chamber) to a low-temperature reservoir (condenser) in a cyclic process

Clausius inequality vs second law

• Mathematical statement of the second law of thermodynamics reinforces idea that heat cannot spontaneously flow from a cold reservoir to a hot reservoir without external work being done on the system
• Implies no heat engine can be 100% efficient, as some heat must always be rejected to a low-temperature reservoir (atmosphere)
• Similarly, no refrigerator can operate without external work input, as heat cannot spontaneously flow from a cold reservoir (inside fridge) to a hot reservoir (room temperature)

Implications for thermal efficiency

• Sets an upper limit on the efficiency of heat engines
  1. The efficiency of a heat engine is always less than the efficiency of a Carnot engine operating between the same two reservoirs
  2. Carnot efficiency: $\eta_{Carnot} = 1 - \frac{T_L}{T_H}$, where $T_L$ and $T_H$ are the absolute temperatures of the low and high-temperature reservoirs
• Sets a lower limit on the coefficient of performance (COP) of refrigerators
  1. The COP of a refrigerator is always less than the COP of a Carnot refrigerator operating between the same two reservoirs
  2. Carnot refrigerator COP: $COP_{Carnot} = \frac{T_L}{T_H - T_L}$
• Demonstrates no real heat engine (internal combustion engine) or refrigerator (heat pump) can achieve the efficiency or COP of a Carnot device, which operates under ideal, reversible conditions