Physics Rediscovered #12: The perfect engine
And why we cannot have it
The temperature is rising as we are progressing in our journey into thermodynamical hell.
In the previous episode we’ve witnessed how the idea of heat baffled the best minds of the 17th and 18th century. We saw the rise and fall of phlogiston, we sniffed various vapors, and discovered oxygen. We also made water, but not in the natural sense.
Phlogiston, having been a highly flawed theory, burned to ashes in the fires of experimentation and from it rose the theory of caloric, imagined by Antoine Laurent de Lavoisier. This new substance of heat, soon established itself as the leading paradigm of the 18th and 19th century.
However, dark clouds were gathering over caloric. One of the first lightnings struck when count Rumford attacked it with his cannons, to staggering effect. His experimental barrage undermined caloric in favor of heat being the results of motion.
Rumford was one of many in the every growing “heat is motion“ camp. Yet, despite the damage, caloric held firm. Rumford and his likeminded friends had to wait patiently.
For now…
Preparing the runway
If you were to ask caloric evangelists at the time if abandoning that idea seemed likely, they would have probably denied it and not for no reason, mind you.
Caloric seemed to explain a lot of things - from keeping you coffee warm to creating and on top of that scored an incredible victory when pretty boy Nicolas Léonard Sadi Carnot decided to imagine the perfect engine using caloric as a principle.
Sadi Carnot, was a military officer, engineer and totally into steam engines. These were found all over the place at the beginning of the 19th century and even combustion engines were a thing then. Many of those were crappy, however. They wasted fuel left and right, resulting is poor performance. This didn’t stop steamboats from swimming and train from running, though.
Carnot looked at the landscape of engines and thought that it’s nice and all, but can we do better?
We will make the answer obvious, but remember that no idea exists in a vacuum. You don’t just wake up one day and say: “ah yes, the jet engine, of course“. Carnot’s complex idea grew over time and so will our understanding. Remember that complexity is evolved, not created so all you creationists can suck it.
Let’s lay some ground work first.
By Carnot’s time, multiple experiments have confirmed that gasses expand as their temperature increases and compress when it decreases. Conversely, forced compression and expansion affects the temperature. It didn’t take long to figure out that this principle can be used to push and pull things around and do some work. This is really the basic idea behind an engine.
Carnot thought that if he was supposed to come up with the best engine ever, he would need to take that idea to the extreme. This means that he couldn’t afford temperature changes that weren’t associated with expansion or compression and therefore, useful work. Anything else would be a waste.
Remember that Carnot might’ve had intuitions, but didn’t know about the conservation of energy as we understand it today. He also didn’t consider the atomic structure of matter and its often random, statistical behavior. He thought of it all in the paradigm of the caloric.
For added immersion, we will also try to mimic Carnot and think in terms of caloric rather than modern ideas of energy.
Ok, enough foreshadowing. Let’s start building.
Prepare for liftoff
We will make our engine out of a piston enclosed in a cylinder. The whole thing is perfectly isolated, unless we say otherwise, so that we are in full control of what happens. Under the piston there’s air, which will be the working substance. The piston can move up and down without friction inside the cylinder and affect whatever it is connected to, like wheels or something.
Ok, that’s the mechanics done. Now we need something to power our contraption and for a perfect engine we will require a perfect heat source. This means something that can maintain its temperature at all times, and therefore provide an infinite amount of heat. Here’s what we have so far:
Ok, is this enough? Let’s fire up what we have and see what happens. Also, let’s use specific numbers to keep track of things. Let’s say that the air in the cylinder is at 20°C and that our perfect heat source is always at 100°C.
The cylinder is perfectly isolated, but for a moment we will allow heat to come in from the heat source and heat up the gas and… Whoa! Wait just a minute. We almost blew it.
It might be very tempting to start heating up the gas just like that, but it would be a mistake. Why? Because of the significant temperature difference between it and the heat source. Such heating would results in losses. Don’t worry, all will be made clear.
We will tackle this problem with an abstract trick. Carnot knew this trick, from a combination of experimentation and intuition. We are going to make the heating process really slow. By taking our sweet time we are trying to keep the process of heating is as reversible as possible.
What’s so special about reversibility? It ensures that the process is efficient as possible.
To understand this, consider an irreversible process. An example of that is when you drop a glass on the floor. The glass shatters into a million pieces and cannot be unshattered ever again. You cannot shoot the pieces back on the same trajectories so that they magically glue themselves together and bounce back up on the table.
This is not due to technological limitations. This is because it all happened too fast and was too chaotic to turn back the clock. You could try to pick up the pieces, glue them together into a glass and put it back on the table yourself, but it just wouldn’t be the same glass. You would lose something in the process.
But now let’s say that we recorded the whole thing in super slow motion and we can go through the recording frame by frame. If between two frames pieces are flying in some direction, we could take the energy of each piece from the second frame, apply it backwards and that would bring the whole picture back to the first frame. The smaller the time distance between the frames, the more accurate we could be. In this sense the process is reversible on very small, though unrealistic scales.
Heating up a cold gas to a high temperature suddenly is similar. It would be impossible to cool it down in reverse. The animation below should help you understand what I’m all about. It’s cheating a bit because it shows gas particles, but check it out nonetheless for an extra 42% of intuition:
The apparent change, when the gas is almost as hot as the heat source, is barely noticeable. The closer the temperature of the gas is to the heat source, the less pronounced the difference is and we are approaching reversibility.
Ok! With that, we are ready.
Going up
With all the above considerations, we are going to start with a gas that is almost at the temperature of the heat source, so almost 100° C. We’ll say that somebody heated it up for us beforehand. This may sound like cheating but it’s not and I’ll explain why later.
The piston is currently at rest. We plug in the heat source to the cylinder and the air is starting to heat up. The air would go from 99.9999 to 100° C, expand a little, which would drop the temperature a little, but the presence of the heat source does not allow it. Approximately, the temperature of the gas does not change and we label this process as isothermal.
As a result, the piston moves up and that, in turn, moves the wheels or whatever it is connected to. The whole thing is super slow in order to ensure reversibility.
Good! We have movement.
At some point we need to stop though. Our engine is not infinite in size so the piston cannot move up forever. Let’s then remove the heat source.
Ok, the source removed and the cylinder sealed. Now we can… wait a minute… The piston is still moving…
Are you surprised the piston is still moving? Well, Carnot wasn’t. In fact, he might’ve been counting on exactly that. He saw similar behaviors in experiments in the past. This seemly spontaneous expansion after heating we label as adiabatic expansion. The name is weird, but who cares. What it stands for is that the process involves no heat transfer, whatsoever.
This was no obvious thing for 19th century physicists. The caloric theory had trouble explaining why this happens exactly. One could argue that we’ve just pumped the air full of caloric and, if you recall from the previous episode, caloric is self-repellent. This means it will want to escape from the air, but also drag the air with it in the process.
Anyway, Carnot didn’t seem to give a damn. He was an engineer through and through, and detailed explanations of the nature of caloric did not interest him. He just focused on the practical. All he cared about is that the pressure resulting from an adiabatic expansion will push the piston further. This time, however, there is no heat source to maintain the temperature of the air so, as it expands, its temperature drops.
This will continue until there is not enough kick in the caloric to push on the piston upwards. Mind you, that this is a much faster and, perhaps surprisingly, a reversible process.
How does this make sense? Well, the cylinder is perfectly isolated at this stage so nothing is gaining or losing heat. No caloric transfer or anything. We can move the piston up and down all we want and not lose anything while doing so.
Ok, so the piston is at the very top and let’s say the air is at a little above 20° C here. We can take a look at what we have so far:
Excellent! We have pushed the piston to its max and that has, in turn, pushed some wheels, maybe. Now we want to do it again. There is a problem, though.
The piston is stuck.
Going down
If we want an engine operating on a repeatable cycle we need to get the piston back somehow. What do we know so far?
The air did its maximum amount of pushing on the piston. The weight of the piston is pressing down on the air, but is balanced by the air’s pressure. In realistic conditions we would have some heat transfer between the air and the walls of the cylinder. That would cause the air to decrease in temperature and compress, allowing the piston to move down. However, this has no place in our perfect engine, no sir.
What are we supposed to do now?
We could eject the air, right? The piston would fall down, but we would have to inject new air and heat it up to make the whole thing work again. That sound terribly wasteful, if you ask me.
We can do better - let’s cool the air.
To do that, we need to add a heat sink to our project. Just like the heat source, our sink can maintain constant temperature forever, no matter how much heat it absorbs.
What should be the temperature of our heat sink? The gas is at 20.0001°C right now, so we could do something cold like 0°C to cool it real good. However, if you are paying attention, then you’ll notice that this is the same problem as with suddenly heating up the gas. There’s just too much difference in the temperature. We need to do it the same way as we did it with heating - extremely gradually or, again, isothermally.
In this sense, I think we have no choice. We need to set the heat sink at almost the temperature of the gas in its current state, but a tiny but lower, say 20° C. Once we allow for contact between the gas and the heat sink, the gas will want to drop its temperature a bit, which will cause a bit of compression, which in turn will raise the temperature a bit, and so on.
After all this hassle, the piston is finally moving down.
We have almost completed the cycle, but there’s one more obstacle to overcome. We’ve started the whole thing with the gas almost at the temperature of the heat source, or about 100° C. Right now it has about 20° C. Does that mean we need to reheat it? Well, no and you can probably guess what we’re going to do about that. We’ll let the Universe do its thing.
Our first stage was isothermal heating, followed by adiabatic expansion, followed by isothermal cooling. What’s next in the pattern? That’s right, adiabatic compression.
Why? Well, so far, we were consistently cooling the gas at 20° C, removing all the caloric from it. This makes the gas “weak” in terms of pressure. It won’t be able to support the weight of the piston. Let’s, then remove the heat sink.
That piston is now almost in free fall. It presses on the cold gas and this compression raises its temperature, quickly. This continues until the gas reaches it’s initial position and initial temperature of almost 100°C. Yes, we’ve measured the dimensions of out engine and timed everything perfectly to make that happen, exactly. Here’s the final effect:
There we go. The piston is back to it’s initial position and the cycle is complete. We are ready to go again and thus, we have a working engine.
Unrealistic standards
This was more or less Carnot’s thinking. You can take a look yourself here.
In his mind he invented the perfect engine. All other engines will be substandard in comparison, because his has no losses anywhere. All the heat, that went in from the heat source, did the maximum effort it could pushing the piston before some of it went out into the heat sink. Not only heat was necessary to make an engine go - that was obvious. It was actually the flow of heat from hot to cold that made all the difference.
So how good is this perfect engine?
We can measure that by simply asking: how much did I pay for what I’m getting? In other words, how much work did the gas do on the piston for the amount of heat I had to put in.
We will call this the efficiency of the engine. It’s often labeled with the Greek letter η. The derivations of this efficiency are easily accessible, so no need to bother here. However, let’s build an intuition.
In the example we had here, the difference between the heat source and the heat sink was significant. This made the piston go all the way up. This time, let’s say that the difference between the heat source and sink is minimal, like 1° C.
Nothing else changes. The process is exactly the same. However, with such small temperature differences, you can imagine that the piston would move up only minimally before attaining the temperature of the heat sink. Then it would immediately need to go back down. That’s likely not enough to move any wheels on the other side. In this case the work done by the gas is negligible.
If I now make the difference larger, say 2° C, the movement would be more noticeable. It seems that the temperature difference is what really drives this thing. In fact, after more rigouristic analysis we would get:
Here, the subscripts C and H refer to the cold sink and the hot source, respectively. So in order to make the bestest engine, we would need to make the temperature differences between the source and sink as large possible. That is more easily said than done, but I’ll let the engineers worry about that.
However, no engineer is a miracle worker. The cylinder in our engine was perfectly isolated - good luck with that. The piston moved without friction - one can dream. Also, our isothermal processes had to be really slow. In all honesty they would have to be infinitely slow - are you busy this weekend?
With all that, the Carnot engine is an unrealistic standard to achieve, which lowers the self-esteem of all engineers to this day.
There’s one more thing here. At the beginning, I stated that someone would initially heat up the gas in the cylinder for us, to the temperature of the heat source. This is obviously an energy expenditure so how can we just ignore it without cheating? Well, our engine is going to perform many, many cycles. This means a lot of heat transferred and work done. The initial, single cost of heating the gas is negligible in comparison.
More than an engine
It may not be obvious, but Carnot invented so much more than an engine. If you run the process in reverse, you get… a refrigerator!
Here’s, how to cool your beer in 4 easy steps:
As the gas sits at 100° C, you pull the piston up quickly. You get an adiabatic expansion and you cool the gas to 19.9999° C.
You plug in the heat sink at 20° C, which makes the heat sink give away heat to the gas. All this happens isothermally and the piston moves slowly up.
By now the piston is at the very top. You remove the heat sink and push on the piston like its nobody’s business. This adiabatic compression heats it up the gas to 100.0001° C.
Finally you plug in the heat source, which temperature is 100° C. The gas slowly gives away heat to the heat source as the piston gradually falls down.
How is this a refrigerator? Notice, that this time we are taking heat from the colder sink and dumping it in the hotter source. Normally, that wouldn’t happen, but this time we’ve put in some work and presto. That cold sink is none other than the inside of your fridge.
But wait! There’s more!
Imagine for a moment that the heat source is your home and the heat sink is the dreaded outside. What takes heat from the outside and warms the inside?
A heat pump, that’s what.
With all these marvelous applications, Carnot’s work was praised by every scientist and engineer as soon as it was published.
Nah… none of that happened.
Sure, the work was published, but for no one to read it. Carnot was detached from academic circles and worked in his own, private space. The lack of prominent connections ensured that his “Reflections on the Motive Power of Fire“ was noticed by no one.
Fortunately, Carnot was rediscovered a few decades later, by notable physicists of the time. Unfortunately, he was no longer around to see it.
Parallels in life
I find Carnot’s story and treatment of ideas reflected in our lives. If you are someone trying to make something for yourself, then this section is for you.
If you have an idea for a project or a business and you’re worried that you might not have the right idea - don’t. Carnot devised the perfect engine that we all accept today based on a theory that we don’t - the caloric. Sometimes wrong assumptions can lead to good results.
Also, don’t worry if what you have is far from perfect. Carnot’s perfect engine is a standard, which no one can ever achieve. Everyone knows this, but we still aim at creating real engines, which aspire to Carnot’s standard. This is good and healthy as long as that perspective is maintained. Only when we forget about it, it becomes self-destructive.
Finally, if you’re doing something worthwhile - tell everyone around you. Tell your family, tell your friends, tell your coworkers and their mothers, and tell their dogs too. Give others a chance to hear your name.
With that, it’s time to go. Don’t let the caloric out the window and I’ll see you in the next episode of Physics Rediscovered.
You might enjoy reading Truesdell's "The Tragicomical History of Thermodynamics, 1822–1854" and his and Bharatha's "The Concepts and Logic of Classical Thermodynamics as a Theory of Heat Engines".