Can Any Refrigerator Surpass a Carnot Refrigerator in Efficiency?

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SUMMARY

No refrigerator can surpass the efficiency of a Carnot refrigerator when operating between two thermal reservoirs at the same temperatures. This conclusion is grounded in the second law of thermodynamics, which states that a perfect refrigerator, requiring no work to transfer heat from a low-temperature region to a high-temperature region, is unattainable. The coefficient of performance (COP) for an ideal refrigerator is defined as COP = QL / W, where W represents the work done. The ideal Carnot refrigerator's COP can be expressed as COPideal = (TL / TH) / [1 - (TL / TH)], confirming its maximum efficiency.

PREREQUISITES
  • Understanding of the second law of thermodynamics
  • Familiarity with the concept of coefficient of performance (COP)
  • Knowledge of the first law of thermodynamics
  • Basic principles of heat transfer and thermodynamic cycles
NEXT STEPS
  • Study the derivation of the Carnot cycle and its implications for thermodynamic efficiency
  • Explore real-world applications of Carnot refrigerators in refrigeration technology
  • Investigate alternative refrigeration cycles and their efficiencies compared to Carnot refrigerators
  • Learn about the limitations of thermodynamic systems and the concept of irreversibility
USEFUL FOR

Students of thermodynamics, engineers in refrigeration and HVAC fields, and anyone interested in the principles of energy efficiency in thermal systems.

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Homework Statement



Show that no refrigerator operating between two reservoirs at a given temperature can have higher co-efficient of performance than a Carnot refrigerator operating between the same two reservoirs.

Homework Equations


The Attempt at a Solution



Please check if I am correct


A perfect refrigerator is one in which no work is required to take heat from the low temperature region to the high temp. region.
This is not possible according to the second law of Thermodynamics
The coefficient of performance of a refrigerator
COP = QL / W
where W = work done
from the first law of thermodynamics we can write
COPideal = QL / ( QH - QL )
= TL / ( TH - TL )
= ( TL / TH) / [ 1 - ( TL / TH) ]
= ( TL / TH) / eideal
= analagous to an ideal Carnot refrigerator
 
Last edited:
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I'd prove this by contradiction.
 

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