6G-1 : | Efficiency and Coefficient of Performance of Carnot Cycles | 4 pts |
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A Carnot
Cycle operates between thermal
reservoirs at 55oC and 560oC. Calculate… a.) The thermal efficiency, h, if it is a power cycle b.) The COP if it is a refrigerator c.) The COP if it is a heat pump |
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Read : | This is a straightforward application of the definitions of efficiency and coefficient of performance. | |||||||||||||
Given: | TH | 560 | oC | TC | 55 | oC | ||||||||
TH | 833.15 | K | TC | 328.15 | K | |||||||||
Find: | h | ??? | COPR | ??? | COPHP | ??? | ||||||||
Diagram: | Not necessary for this problem. | |||||||||||||
Assumptions: | None. | |||||||||||||
Equations / Data / Solve: | ||||||||||||||
Part a.) | The thermal efficiency of a Carnot Cycle depends only on the temperatures of the thermal reservoirs with which it interacts. The equation that defines this relationship is : | |||||||||||||
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Eqn 1 | |||||||||||||
Just be sure to use absolute temperature in Eqn 1 ! In this case, convert to Kelvin. Temperatures in Rankine will work also. | ||||||||||||||
h | 60.6% | |||||||||||||
Part b.) | The coefficient of performance of a Carnot Refrigeration Cycle also depends only on the temperatures of the thermal reservoirs with which it interacts. The equation that defines this relationship is : | |||||||||||||
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Eqn 2 | |||||||||||||
Using T in Kelvin yields : | COPR | 0.6498 | ||||||||||||
This is an exceptionally BAD COPR because it is less than 1. This isn't terribly surprising when you consider that the refrigerator must reject heat to a thermal reservoir at 560oC !! | ||||||||||||||
Part c.) | The coefficient of performance of a Carnot Heat Pump Cycle also depends only on the temperatures of the thermal reservoirs with which it interacts. The equation that defines this relationship is : | |||||||||||||
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Eqn 3 | |||||||||||||
Using T in Kelvin yields : | COPHP | 1.6498 | ||||||||||||
This is a BAD COPHP because it is just barely greater than 1. This isn't terribly surprising when you consider that the heat pump must put out heat to a reservoir at 560oC !! | ||||||||||||||
Notice also that : |
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Eqn 4 | ||||||||||||
This is always true for Carnot Cycles. | ||||||||||||||
Verify: | No assumptions to verify that were not given in the problem statement. | |||||||||||||
Answers : | h | 60.6% | COPR | 0.650 | COPHP | 1.65 |