Generalizing Carnot theorem

Question

Motivated by question Can IC engines be modeled as Carnot engines?. I am wondering whether/how Carnot theorem could be generalized to other kinds of devices performing "useful work", such as, e.g.:

• Motor (or generator) fed by a battery
• Nuclear power generators
• Solar cells
• Water wheels

I think that the theorem must be generalized in at least three ways:

• Operating media is neither a gas nor a liquid - that is the reasoning based on isothermic adiabatic expansions might not apply.
• Generalizing the concept of temperature (introducing "effective temperature"?) - e.g., in case of a batter or a water wheel, we do not have two reservoirs with different temperatures to properly speak of, but rather two reservoirs with different (chemical) potential.
• Generalizing the concept of useful work - solar cell and water wheel are not really transferring the energy between two reservoirs - the energy already flows, and the device simply diverts a part of this energy into work. But, since the energy flows anyway, it is not clear whether/how the part of it that is diverted is useful: e.g., how is the current generated by a solar cell is more useful than the heat generated in the surface illuminated (which may be also "useful" in everyday sense.)

Perhaps, there is not much left of Carnot theorem with all these generalizations, and we simply need to consider it as limited to a particular class of phenomena? If so, are there other upper boundaries on converting energy to work (that would be applicable to the devices cited in the beginning?)

Answer 1

The Carnot theorem is very general but we must understand exactly what it means. There is a narrow reading and a broad reading of the theorem:

Narrow Reading of Carnot Theorem

Think of a temperature difference as a voltage difference: if we place a suitable device (an electric motor) between two points that are held at different voltages we can produce useful work. The motor takes advantage of the the spontaneous tendency of electrons to pass from low voltage to higher. This is entirely analogous to a water wheel that extracts work from the spontaneous tendency of water to fall (if it has somewhere to go). In general, we can take advantage of any spontaneous process (wind, waves,solar radiation, forest fires, tsunamis, earthquakes, ...) to produce work. It's only a matter of engineering ingenuity.

The same is true with temperature. Here we are taking advantage of the spontaneous tendency of heat to pass from high temperature to low temperature and the device is a power plant. The Carnot theorem gives the maximum fraction of the heat entering the plant that is converted to work: $$ \frac{\text{work produced}}{\text{heat in}} \leq 1 -\frac{T_\text{low}}{T_\text{high}} = \eta_\text{max} $$ An IC engine may be thought to operate between the high temperature of the ignited fuel and a lower temperature that is that of the surroundings. These two temperatures "squeeze" work out of the heat released through combustion, but the maximum possible amount of work that can be squeezed that way is limited by the Carnot efficiency.

Broader Reading of Carnot Theorem

The most general reading of the Carnot theorem is that the maximum possible work in any process is when the process is conducted reversibly. The derivation of the Carnot efficiency is obtained by running the power plant reversibly. The Carnot power plant has two isothermal steps, heating at $T_\text{high}$, cooling at $T_\text{low}$, and two steps that are reversible adiabatic (one is compression, the other is expansion). The isothermal steps guarantee that there is no temperature gradient between the external temperatures and the plant. This eliminates thermal irreversibilities. The adiabatic steps are strictly mechanical, running them reversibly, for example without vibrations of any kind, avoids wasting work that would otherwise be produced.

This reading can be applied to any process, not only one that involves just two temperatures. By analyzing a process on the basis of the second law (that's the same as saying by the generalize Carnot theorem) we can assess how efficiently a process runs relative to the best case senario that nature permits.

Answer 2

You have asked "whether/how Carnot theorem could be generalized to other kinds of devices performing useful work". Boris Leaf in this article discusses how general potentials, $\pi_i$, not just temperature, associated with conserved extensive quantities, $K_i$, in the Gibbs equation $dU=TdS+\sum_i \pi_idK_i$ also follow Carnot's efficiency formula and a Kelvin's axiom.

With the definition of $\pi'$ as being the potential of the environment (work source) and $\delta g=\pi'dK$, he writes that during a cycle $\oint dK = \oint \frac{-\delta g_{max}}{\pi}=0$, and if $\pi=-const$ then $\oint \delta g_{max}=0$. Consequently for any cycle at constant $\pi$: $g=\oint \delta g \le \oint \delta g_{max}=0$. (I changed his $dg$ to $\delta g$.)

1. "Therefore it is impossible to extract any form of work from a system operating through a cycle at constant value of the corresponding potential. This generalization of the Kelvin postulate is a consequence of the generalized statement of the second law. If all the potentials are constant during a cycle, no work whatsoever can be extracted for use in the surroundings."

Next, in complete analogy to Carnot's calorique equation $\Delta W = (T_1-T_2)\Delta S$, he derives $g_{max}= (\pi_1-\pi_2)\Delta K$ followed by this comment:

2. "This is the expression for the maximum $\pi-K$ work done in the transfer of charge $\Delta K$ from potential $\pi_1$ to $\pi_2$. The quantity $g_{max}$ does not depend on the nature of the system undergoing the cyclic operation; it depends only on the existence of the potential levels $\pi_1$ and $\pi_2$ between which the system operates. It is represented by the area ABCD on the diagram with a sign depending on the relative values of $\pi_1$ and $\pi_2$".

Answer 3

It seems you are looking for Exergy.

"An opportunity for doing useful work exists whenever two systems at different states are placed in communication, for in principle work can be developed as the two are allowed to come into equilibrium. When one of the two systems is a suitable idealized system called an environment and the other is some system of interest, exergy is the maximum theoretical useful work (shaft work or electrical work) obtainable as the systems, interact to equilibrium, heat transfer occurring with the environment only." (Thermal Design and Optimization - Adrian Bejan, George Tsatsaronis og Michael J. Moran)

Answer 4

Operating media is neither a gas nor a liquid

Although in standard treatments we are almost always interested in fluids, in reality, thermodynamics is perfectly fine being applied to solids and other phases too. There is simply no reason why it would not work for other system types.

For example, magnetic refrigeration that is how we get to almost absolute zero, typically works with solids, and follow almost exactly a Carnot cycle too! We also talk about Einstein and Debye model of solids and their phonon contributions to statistical thermodynamics, and some of those integrations to get the entropy, enthalpy, and so forth, are done from low temperature upwards to higher temperatures, and at low enough temperatures almost all materials present as solid forms.

Generalizing the concept of temperature (introducing "effective temperature"?) - e.g., in case of a batter or a water wheel, we do not have two reservoirs with different temperatures to properly speak of, but rather two reservoirs with different (chemical) potential.

It is funny that you mention chemical potential and not realise that it already is the thing that you should be paying attention to. Basically all the intensive variables can come into play, be it temperature, pressure, chemical potential, magnetisation, etc.

I elect to direct attention away from this, because the moment you have a concrete system to consider, the appropriate variables to pay attention to will most likely be trivial to identify, so it is more useful to talk about the next thing:

Generalizing the concept of useful work

Actually, no, you cannot properly define work and heat in general. This is particularly true when you have magnetism in the thermodynamics. Arguments about which forms are true in thermodynamics rages to this day because of magnetism.

But the interesting thing to talk about is the PV cell, and on that we have a lot of stuff we can talk about. In particular, the mechanism of the PV cell operation has to do with the band gap, the electronic states, bands, and so forth. All of these are applications of statistical thermodynamics to solid state physics, and it is necessary to have some kind of temperature distribution for that to work.

Heck, the same thing is true for lasers. For lasers to work, population inversion is necessary, and this is impossible under thermal equilibrium. The only way for it to work, is to have a dynamical system continuously pump more energy than needed into the lasing part, culminating in waste and more. i.e. there is a mini heat engine inside the laser.

Same reasoning as for the fact that when you concentrate the light from the Sun using lenses, you can never concentrate them to get a higher temperature than the surface temperature of the Sun.

All of these things, that seem to violate thermodynamics, actually obey thermodynamics in a broader sense. It is just that we have to expend some effort to tease out what are the appropriate thermodynamics for all of them.