A recent project highlighted to me that closed cycle cryogenic cooling systems use far more electrical power to reduce temperatures by a few watts than regular refrigerators. I am sure it must be something to do with the extra difficulty of removing the heat from the liquid nitrogen or similar but I was wondering if you could tell me why this is?
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Yes, for the same refrigeration power, for example, 1 watt, more input power is required as the cold-end temperature is decreased. By using the first and second law of thermodynamics, one can show that the ratio of input power to refrigeration power for a perfect, reversible refrigerator is (Th-Tc)/Tc. We see that as the temperature approaches absolute zero the input power approaches infinity, which is why it is impossible to achieve absolute zero. Actual refrigerators are not perfect, so it takes even more power than the simple expression for an ideal refrigerator. Pumping heat from a lower temperature is somewhat like carrying a bucket of water up from the river in a canyon to the top. The deeper the canyon, the more work it takes to bring it to the top.