Example Problem with Complete Solution

7A-2 : Properties of Cyclic Integrals 3 pts
Consider the heat engine shown below. The cyclic integral of dQ is greater than zero. Does this violate the Clausius Inequality? Explain.
Read : This problem is designed to make you think very carefully about how to use a cyclic integral. The key is that the temperature is not the same for both of the heat transfer interactions in this cycle.
Diagram: See the problem statement.
Given: TH 400 K QH 450 kJ
TC 300 K QC -350 kJ
W -100 kJ
Find: Does the cycle violate the Clausius Inequality?
Assumptions: None.
Equations / Data / Solve:
The Clausius Inequality is:
Eqn 1
Because the cycle only exchanges heat with the hot and cold thermal reservoirs, the integrals can be simplified:
Eqn 2
Eqn 3
100 kJ
-0.04167 kJ/K
This cycle is irreversible because the cyclic integral in the Clausius Inequality is less than zero.
It is true that the cyclic integral of dQ > 0. But the Clausius Inequality is still satisfied.
Confusion about the cyclic integrals sometimes arises if you mistakenly pull T out of the cyclic integral.
Eqn 4
You could only pull T out of the cyclic integral if ALL of the heat exchange across the system boundary went to or from reservoirs that were ALL at the SAME temperature.
That is almost never going to happen. It is definitely not the case in this problem as the hot and cold reservoirs are at 400 K and 300 K, respectively.
Verify: None.
Answers : Yes, the cycle does indeed satisfy the Clausius Inequality.