For a bit of light

Carnot heat pump

We consider the standard Carnot-cycle machine, which can be thought of as having a piston moving within a cylinder, and having the following characteristics:

  • A perfect seal, so that no atoms escape from the working fluid as the piston moves to expand or compress it.
  • Perfect lubrication, so that there is no friction.
  • An ideal-gas for the working fluid.
  • Perfect thermal connection at any time either to one or to none of two reservoirs, which are at two different temperatures, with perfect thermal insulation isolating it from all other heat transfers.
  • The piston moves back and forth repeatedly, in a cycle of alternating "isothermal" and "adiabatic" expansions and compressions, according to the PV diagram shown below:

By definition, the isothermal segments (AB and CD) occur when there is perfect thermal contact between the working fluid and one of the reservoirs, so that whatever heat is needed to maintain constant temperature will flow into or out of the working fluid, from or to the reservoir.

By definition, the adiabatic segments (BC and DA) occur when there is perfect thermal insulation between the working fluid and the rest of the universe, including both reservoirs, thereby preventing the flow of any heat into or out of the working fluid.

The isothermal curves (but not the adiabatic curves) are hyperbolas, according to PV = nRT. The enclosed area (and therefore the mechanical work done) will depend on the two temperatures ("height") and on the amount of heat transferred, which depends in turn on the extent of the isothermal compression or expansion ("width"), during which heat must be transferred to maintain the constant temperature.

We will denote the heat transferred to or from the high-temperature reservoir (during the transition between points A and B) as Qh.

We will denote the heat transferred to or from the low-temperature reservoir (during the transition between points C and D) as Qc.

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