# Maximum efficiency of heat engines not operating between two reservoirs

By considering Carnot's theorem, we know that the maximum efficiency of an engine operating between 2 reservoirs with different temperatures are given by $1$$-$$T_1$/$T_2$.

Consider the case when we want to build an engine operating between a reservoir and a (lower temperature) body by transferring heat from the reservoir to the body. Since the body is of a finite size and has a heat capacity, the temperature of the body changes throughout the process. What will be the maximum efficiency in this case?

## 1 Answer

Here, the efficiency is not constant but rather changes continuously as the finite lower-temperature body heats up. The maximum efficiency occurs at the first moment of operation and subsequently decreases to zero. Ultimately, the finite body is heated to the temperature of the high-temperature reservoir. Is this the information you were looking for?

EDIT: Furthermore, Carnot's theorem continues to hold because the efficiency is never greater than $1-T_\mathrm{low}/T_\mathrm{high}$. As discussed in the comments below, you can maintain reversibility (in this hypothetical and idealized thought experiment) if you lessen the impact of each cycle so that the temperature increase of the finite body per cycle is arbitrarily small. However, $T_\mathrm{low}$ will continue to grow.

However, if you abandon reversibility and allow the cycle to heat up the finite body by some finite amount, then entropy is generated (specifically, $\Delta S=C\ln T^\prime_\mathrm{low}/T_\mathrm{low}$, where $T^\prime_\mathrm{low}$ is the increased temperature and $C$ is the heat capacity of the finite body). This then is no longer a Carnot engine.

• If the engine transfer a finite amount of energy to the body in one cycle, the body will heat up and its temperature will change. Then how does Carnot's expression 1-T1/T2 hold since T1 will not be well defined over a single cycle? – user148792 May 11 '18 at 16:45
• In this idealization of perfect reversibility and maximum efficiency (which is the whole point of the Carnot engine thought experiment), we would optimize our heat engine to minimize the cycle impact by rejecting a suitably small amount of work that the finite body would not heat up appreciably. Thus, the temperature change of the finite body would be negligible over a single cycle. – Chemomechanics May 11 '18 at 17:07
• I've posted a follow-up question inspired by your response: physics.stackexchange.com/questions/405337/… I'm wondering if you could help out – user148792 May 11 '18 at 18:05
• It may help to recall that entropy is generated (and reversibility is violated) any time energy moves down a gradient (e.g., a temperature difference). In the Carnot engine, we assume that these gradients can be made arbitrarily small. The act of heating a finite body from a lower temperature to a higher one violates the Carnot postulate. – Chemomechanics May 11 '18 at 18:49
• This paper and its references may be of interest as you incorporate the entropy increase from heating a finite body. – Chemomechanics May 11 '18 at 18:50