Heat Engines, Refrigerators and Heat Pumps

A cyclic engine receives heat QH from a hot source, does work W and rejects heat QC to a colder sink. The first law gives W=QH−QC. The second law rules out a cyclic device whose only effect is to turn all heat taken from a single reservoir into work.

A reversible engine operating between absolute temperatures TH and TC has the maximum possible efficiency between those reservoirs. Real engines do worse because of friction, finite-temperature heat transfer and other irreversible processes. Increasing the hot temperature or reducing the cold temperature raises the ideal limit.

A refrigerator uses work input to remove heat QC from a cold region and reject QH to warmer surroundings. A heat pump uses the same basic process but the useful output is QH delivered to the warm region. For either device, QH=QC+W.

Coefficient of performance compares the desired heat transfer with work input. It can exceed one because the device moves heat as well as converting work to heat. It is therefore different from a heat-engine efficiency.

For reversible operation the refrigerator COP is TC/(TH−TC), and the heat-pump COP is TH/(TH−TC). A small temperature difference gives a large ideal COP; real performance is lower. These formulae are bounds for an ideal reversible machine, not guarantees for any appliance.

Combined heat and power uses some otherwise rejected heat for space or water heating, increasing the fraction of fuel energy used for a purpose. It does not violate the heat-engine efficiency limit: useful heat and work are counted as different outputs.

Try each problem before opening the worked solution. State the physical model, show substitutions with consistent units, and explain what your answer means.