# 5.3 Complete the demonstration of the equivalence of the Clausius

```5.3 Complete the demonstration of the equivalence of the Clausius and Kelvin–Planck statements of
the second law given in Sec. 5.1 by showing that a violation of the Kelvin–Planck statement implies
a violation of the Clausius statement.
5.12 Using the Kelvin–Planck statement of the second law of thermodynamics, demonstrate the
following corollaries:
(a) The coefficient of performance of an irreversible refrigeration cycle is always less than the
coefficient of performance of a reversible refrigeration cycle when both exchange energy by heat
transfer with the same two reservoirs.
(b) All reversible refrigeration cycles operating between the same two reservoirs have the same
coefficient of performance.
(c) The coefficient of performance of an irreversible heat pump cycle is always less than the
coefficient of performance of a reversible heat pump cycle when both exchange energy by heat
transfer with the same two reservoirs.
(d) All reversible heat pump cycles operating between the same two reservoirs have the same
coefficient of performance.
5.18 The data listed below are claimed for a power cycle operating between reservoirs at 527oC and
27oC. For each case, determine if any principles of thermodynamics would be violated.
(a) QH =700 kJ, Wcycle =400 kJ, QC =300 kJ.
(b) QH =640 kJ, Wcycle =400 kJ, QC =240 kJ.
(c) QH=640 kJ, Wcycle =400 kJ, QC =200 kJ.
5.19 A refrigeration cycle operating between two reservoirs receives energy QC from a cold reservoir
at TC =280 K and rejects energy QH to a hot reservoir at TH =320 K. For each of the following cases
determine whether the cycle operates reversibly, irreversibly, or is impossible:
(a) QC =1500 kJ, Wcycle=150 kJ.
(b) QC =1400 kJ, QH =1600 kJ.
(c) QH =1600 kJ, Wcycle =400 kJ.
(d) β=5.
5.23 A power cycle operates between a reservoir at temperature T and a lower-temperature reservoir
at 280 K. At steady state, the cycle develops 40 kW of power while rejecting 1000 kJ/min of energy
by heat transfer to the cold reservoir. Determine the minimum theoretical value for T, in K.
5.29 Ocean temperature energy conversion (OTEC) power plants generate power by utilizing the
naturally occurring decrease with depth of the temperature of ocean water. Near Florida, the ocean
surface temperature is 27 oC, while at a depth of 700 m the temperature is 7 oC.
(a) Determine the maximum thermal efficiency for any power cycle operating between these
temperatures.
(b) The thermal efficiency of existing OTEC plants is approximately 2 percent. Compare this with
the result of part (a) and comment.
5.33 An inventor claims to have developed a refrigeration cycle that requires a net power input of 1.2
kW to remove 25,000 kJ/h of energy by heat transfer from a reservoir at -30 oC and discharge energy
by heat transfer to a reservoir at 20 oC. There are no other energy transfers with the surroundings.
Evaluate this claim.
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