Clausius Inequality

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Since the concept “reversibility” plays a crucial role in the thermodynamics, it is necessary for us to find an approach to determine whether the process reversible or not. And the most well-known Clausius Inequality provides us this possibility.

Image that we have a reversible engine which consists of more than two heat reservoirs.  We know from the Carnot Principles that between the same reservoirs (one is a constant temperature source and the other a constant temperature sink), the efficiency of a reversible heat engine is always greater than the efficiency of an irreversible one.(for further explanation about Carnot Principles, please click here:https://thermodynamics-engineer.com/carnot-principals/)

Hence, for the maximum efficiency, Carnot efficiency, all the heat transfer must take place at highest and lowest reservoir temperatures. In other words, heat is absorbed from temperature source with TH and released to temperature sink with TL by heat engine during each sub process.

Now we draw an arbitrary reversible cycle process operated by the engine in p-v diagram (1→2→1).

pv diagram clausius

We can divide this cycle process into infinite small Carnot process (a→b→c→d):

a→b: reversible isothermal expansion at T1; Heat δq1 is absorbed.

b→c: reversible adiabatic expansion.

c→d: reversible isothermal compression at T2; Heat δq2 is released.

b→c: reversible adiabatic compression.

We calculate the maximum efficiency, the Carnot efficiency:

therm 1 ungleichung(Pay attention here, δq2<0)

So we obtain an equation:

therm 2 ungleichungAs mentioned above, for the whole reversible process 1→2→1, we have:

therm 3 ungleichung

Now 1→2 is still reversible process, but 2→1 is supposed to be irreversible process. So also according to the Carnot Principles:

therm 4 ungleichungFor the whole irreversible cycle process:

therm 5 ungleichungTherefore, if we combine the both statements for reversible and irreversible processes, we get:

therm 6 ungleichungwhere

  • “<” for irreversible process
  • “=” for reversible process

This inequality is known as Clausius Inequality.

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