Explain how real machines can never recover the full amount of energy they were supplied with?.

1 Answer
Nov 23, 2016

Sounds like efficiency to me. In general, efficiency is defined as the work output acquired from some work input.

where:

  • ww is work.
  • q_HqH is the heat flow into the engine from the hot reservoir.
  • q_CqC is the heat flow from the engine into the cold reservoir.
  • T_HTH is the temperature of the hot reservoir.
  • T_CTC is the temperature of the cold reservoir.

The ideal machine can achieve 100%100% efficiency, and the work it performs is 100%100% reversible. Real machines, however, can never quite achieve 100%100% efficiency, and the work they do is nearly 100%100% reversible.

As I derived above, a cyclic process has a change in entropy of DeltaS = 0, and if heat flow q is reversible, then DeltaS = q_"rev"/T.

If we separate the cyclic process into two processes, then

q_H/T_H + q_C/T_C = 0

Also from the above, I calculated that

color(blue)(e) = |w|/q_H = (q_H + q_C)/q_H = color(blue)(1 + q_C/q_H).

q_C flows out of the engine, and thus, q_C <= 0. When q_C = 0, the efficiency is 100%.

Or, using the first relation:

color(blue)(e) = |w|/q_H = (q_H + q_C)/q_H = color(blue)(1 - T_C/T_H).

So, we could even say that since all absolute temperatures are positive (i.e. when in "K"), as they are in this equation, 0 <= e <= 1. When T_C = T_H, e = 0, and when T_C = "0 K" (which has yet to be accomplished!), or T_H is absurdly large, e ~~ 1.

Therefore, the efficiency can never be more than 100%, and for real machines, the efficiency is effectively never exactly 100%.