The Efficiency of Engines and Refrigerators

A heat engine converts a heat transfer into useful work, or vice versa. A two-reservoir engine operates between two reservoirs: it draws heat Qc from the hot reservoir, gives back part of it, Qf, to the cold reservoir, and converts the difference into work W. The first law applied to a complete cycle requires Qc = W + Qf, since the internal energy of the working fluid returns to its initial value after a cycle.

The efficiency of an engine is naturally defined as the ratio of what we get out to what we put in: eta = W / Qc, or equivalently eta = 1 - Qf/Qc. The second law imposes an unbeatable theoretical limit, reached only by a perfectly reversible cycle such as Carnot's: eta is always less than or equal to 1 - Tf/Tc, where the temperatures are expressed in kelvins.

Let us check with an example: an engine operates between Tc = 500 K and Tf = 300 K. Its Carnot efficiency is eta = 1 - 300/500 = 1 - 0.6 = 0.4, or 40%. No real engine between these two temperatures can do better, because of friction and irreversibilities, which always degrade part of the useful energy.

The refrigerator works in reverse: work W is supplied to it to extract heat Qf from the cold interior and reject Qc = W + Qf into the warmer room. Its effectiveness, called the coefficient of performance, is defined by COP = Qf/W, and can exceed 1, which violates no law: it is not a conventional energy efficiency, but a ratio between two transfers.

Classic pitfall: believing that 100% efficiency would merely be hard to achieve. In reality, as long as Tf is not zero kelvin, an efficiency of 1 is strictly forbidden by the second law, according to Carnot's theorem, whatever the technology used.

        (hot reservoir, Tc)
                |
                | Qc   (heat drawn from the hot reservoir)
                v
          +-----------+
          | (engine)  | -------> W   (useful work supplied)
          +-----------+
                |
                | Qf   (heat rejected to the cold reservoir)
                v
        (cold reservoir, Tf)

  Balance: Qc = W + Qf     (first law applied to the engine)
  Efficiency: eta = W / Qc = 1 - Qf/Qc  <=  1 - Tf/Tc   (Carnot limit)

  --- reverse cycle: the refrigerator ---
        (cold reservoir, Tf)
                |
                | Qf   (heat drawn from the cold contents)
                v
          +------------------+
   W -->  | (refrigerator)   |
          +------------------+
                |
                | Qc   (heat rejected to the room, the hot reservoir)
                v
        (hot reservoir, Tc)

  Balance: Qc = W + Qf ;  effectiveness: COP = Qf / W  (can exceed 1)