In his introduction to thermophysics, Daniel Schroeder writes the following about the process in a heat engine:

Only part of the energy absorbed as heat can be converted to work. The reason is, that the heat, as it flows in, brings along entropy, which must somehow be disposed of before the cycle can start over. To get rid of the entropy, every heat engine must dump some waste heat into its environment.

Why must the extra entropy be disposed?

In my understanding, we could gather more and more extra entropy, while converting all the heat into work until the entropy reaches its maximum. Then a state of equilibrium would be reached and no more energy could be withdrawn from the system according to the second law of thermodynamics. But then we could just go to the next machine and do the same, always converting all heat into work.

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    How much energy do you have to add to your entropy-storage-machine in order to create additional states to hold the additional entropy? – immibis Oct 15 at 23:44
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    And what are you going to do with the full entropy-storage-machines? Recycle them by dumping the entropy out of them, hopefully. – immibis Oct 16 at 4:15
  • I'm not interested in the practical use, but in the fact if it is possible to transform all heat into work, even if it is only for a short amount of time. – JoKli Oct 16 at 4:18
up vote 17 down vote accepted

We are talking about cycles here. After a complete cycle the system must be right back where it started. Since entropy is a state variable, it must then be that after one complete cycle the entropy is at its "initial" value. By the second law this must mean that the entropy has to "go somewhere else". If you "gathered more and more" entropy, then it is no longer a cycle.

If you did want to do what you propose of replacing engines then it would be extremely inefficient. You would get one "run" out of the process and then have to get a new engine (I am not sure how this would actually work). It is much better to use the same engine on a cycle.

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    Why does it have to be a cycle? Can't it be more of a linear path where the entropy just increases until its maximum? – JoKli Oct 15 at 12:21
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    @jokli - imagine you have a car going on gasoline. Probably you know, that it has around 2000 -3000 rotations per minute. it is perfect example of heat engine. So with your proposal you have to use no more than, say, 1000 cycles and change engine. Which means 3 times per minute! I am writing this because maybe it is somehow abstract for you, but this is the same for train, car, plane. But of course there are engines working exactly as you are proposing: rockets. This is single run engine. – kakaz Oct 15 at 14:38
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    @JoKli In fact you can. However, in real engines that maximum is achieved in one cycle - that would be after moving a few meters a car or a locomotive. Changing the engine every few meters may be thermodynamically sound but rather unpractical. – Pere Oct 15 at 14:50
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    A one cycle engine - rocket? oh wait, SpaceX reuses liquid fuel rockets... Solid fuel rocket then? It could be argued that the Shuttle's SRBs were reused, which would then entail a cycle. Do we need a rocket that self destructs? – Baldrickk Oct 15 at 15:08
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    Isn't a rocket just burning fuel until there is nothing left? That wouldn't be the same as my suggestion, where the engine runs until it has the same entropy as the heat reservoir. But it's probably not very useful to think about practical realisations at this level of abstraction. – JoKli Oct 16 at 4:16

In my understanding, we could gather more and more extra entropy, while converting all the heat into work

You can't store entropy while still converting all the heat into work. Storing an amount of entropy $dS$ requires that you also store an amount of energy $TdS$, where $T$ is the temperature of the object you're storing the entropy in.

But then we could just go to the next machine and do the same, always converting all heat into work.

You can retain some of the entropy internally inside your heat engine rather than expelling it into some external reservoir such as a river or the air. Let's say you have a tank of water that stays inside your heat engine until you throw the heat engine away. You store entropy in this tank of water, which requires heating the water. There are two issues here: (1) As the water tank gets hotter, the energy cost of storing energy in it, $TdS$, gets worse and worse. (2) The tank is no different from an external heat reservoir. You can keep it inside the box that holds your engine, but that doesn't matter. Our description of a heat engine abstracts away questions like where the low-temperature reservoir is physically located. The only real difference is that we normally idealize the low-temperature reservoir as an infinite resource, whose temperature never changes, while the tank is actually finite, and therefore worse thermodynamically because it heats up.

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    I thought, that if I transfer heat from a hot object to a cold one, I create extra entropy, as the multiplicity of ways to store energy increases. How can there be an amount of energy corresponding to this extra entropy? Wouldn't that create energy out of nowhere? – JoKli Oct 15 at 14:32

The reason you need to dump the heat is because engines, by their definition, operate in a cycle. They return to a previous configuration before continuing on. So your solution of just using an unlimited number of one-shot devices is theoretically possible. It just wouldn't be called an engine. It also wouldn't be practical. One might, however, consider the big bang itself to be the ultimate one-shot device!

Engines also want this cycle to go in one direction, so we have to engineer them to do so. In theory, one could have a device which simply goes in either direction without dumping heat. We've built such devices -- little turbines that operate at the molecular scale where random molecular motion causes things to bump one way or another. However, we can't make them do work (we tried). To get work out of them, we need to know which way around the cycle they will go.

To make that happen, we target two equilibrium, rather than one. One equilibrium is at maximum entropy, such as at the fullest expansion of a piston. Once we get there, we need to reset the machine, targeting a second equilibrium (such as with a piston at its most compressed). As we do this, we have to dump the heat because this second equilibrium is not the highest entropy state with all that heat in the system. We have to get rid of the heat before this second equilibrium is achieved.

Now you are allowed to use the heat to drive another engine. This is called a multi-stage engine and they are used in many power plants. They can be more efficient than a single stage engine. However, the laws of thermodynamics provide a hard limit on how efficient they can be, no matter how many stages you use. The resulting limit on efficiency is defined by Carnot's Theorem, and depends on the temperatures of the hot source and the cold sink. (Note: only heat engines have this limit. Other devices, such as fuel cells, do not operate as a heat engine, so they can achieve higher efficiency)

The ultimate example of this is a Matiroshka Brain. This is a fantastic megastructure wrapped around a star to get as much usable work out of the fusion engine as possible. It is a massive heat engine which has a tremendously large number of stages (thousands to millions), where the waste heat from each stage is the hot source for the next stage. The result could theoretically get close to the ultimate ideal heat engine.

For a Matrioshka Brain around our star (the sun), we can calculate its efficiency. The sun is roughly 5800K on its surface, so that's our hot temperature. Our low temperature is the background radiation of empty space, which is a mighty frigid 2.725. Plugging this into Carnot's equation, $\eta_{max}=1-\frac{T_C}{T_H}$, we get a maximum efficiency of 99.95%. These brains can be amazingly efficient, but they can never avoid the slow march of entropy!

If you consider a structure which absorbs extra heat as part of the engine, then you can certainly make an engine which can work for a certain time without disposing of waste heat. A car engine doesn't require any cooling when it just has started, and will work for a few minutes before it overheats.

Note that the engine will not become more efficient while doing so. It will still deliver only a fraction of energy you put in as mechanical work, and the rest of the energy will be used to raise its temperature. So, your idea not improve the efficiency of the engine while drastically reducing its lifetime.

Single run engine, working on heat is used and known: it is rocket engine. It burn some components ( fluids, solids ) in order to obtain thrust. But it's very expensive method of going anywhere, and not so useful as you may expect. It is well designed just for it: one time run.

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