Carnot's ideal heat engine
Why no heat engine can be 100% efficient
This explanation is directly inspired by Veritasium’s video on entropy (The Most Misunderstood Concept In Physics”)
Imagine that you have an ideal case of a heat engine, consisting of various parts: a chamber, a piston, and a flywheel. The piston is connected to the flywheel, and the working fluid of the engine is a gas, like air. Now, let’s say that we have two big bars, really they can be anything – so long they have a reservoir of temperature. The first must be hot, and the second must be cold.
When we put a hot bar under the chamber, it will transfer heat as the air comes into contact with the surface of the hot bar. The gas’s temperature increases, things vibrate and fly and bounce around, and expand; just as we learned in the Heat article that the average kinetic energy of a gas’s particles is proportional to its temperature. The hot air’s expansion keeps its temperature constant, while pushing on the piston. The flywheel turns as the piston is pushed upward.
Afterward, we remove the hot bar. The air continues to expand and the temperature begins to decrease, since there is no additional input of heat. In the ideal case, the air eventually cools down to until it is the temperature of the cold bar.
We put the cold bar under.
The flywheel continues to turn, and pushes the piston back down. Since we are increasing the pressure on the air, the total energy that was spread out gets compressed into a small volume. This is because the piston is imparting its own momentum into the gas particles that it comes into contact with. Since all these particles have little mass, what happens? Their velocity increases hugely.
Momentum is the product of mass and velocity. When we transfer a slow-moving but large object’s momentum to a small object, it becomes fast-moving. The compressed air heats up.
But since the air is contact in with the cold bar, heat transfer is happening quickly enough that it becomes isothermal. If we didn’t have the cold bar under, the gas would continue to heat up and resist the compression of the piston. When the piston reaches a little before its maximum extent, we remove the cold bar and the air heats up as the piston reaches its maximum, full compression.
The amount of work done by the piston acting on the gas in the compression stroke, is lesser than that of the amount of work done by the gas on the piston in the expansion stroke. In the compression stroke, since we made the gas isothermal, we effectively removed the resistance of the gas. So, the piston can extend to its maximum for much less effort, and therefore we're able to extract energy from the system, as the "missing work" merely remained in the flywheel.
To repeat the cycle over and over again if we want, we put the hot bar under the chamber again.
Now, in an ideal heat engine – the process is completely reversible. We could run the engine in reverse, and as long we run the same number of cycles in reverse as those that were done forward, no additional input of energy is needed. There was change, and then we reversed it. Nothing changed.
So what is the efficiency of this engine?
You might say 100%, because it’s fully reversible.
But that’s not what happens, even in the ideal case.
To calculate the efficiency, let’s look at what happens to the energy of the flywheel first. We see that it’s increasing by the amount of heat flow from the hot bar into the chamber, minus the heat flowing out of the cold bar. We divide this flywheel energy by the heat input from the hot bar. Since the heat on the hot side is equal to the work done by the gas on the piston, and this will always be greater than the work done on the gas by the piston on the cold side of the cycle, which equals the heat out (the isothermal part of the process).
To increase the efficiency of the engine, let's introduce a new concept: temperature delta. This is the amount of change that either bar can impart in terms of temperature. We can increase the difference in temperature-delta, making either the cold bar colder or the hot bar hotter.
For a 100% efficient engine, we want the difference in delta-T to be as great as possible. Let's talk about absolute zero.
Imagine that the gas is allowed to expand an extreme amount, so much that it cools to the point where all the gas particles stop moving. Then they would exert no pressure on the piston, and it would take no work to compress it on the cold side, so no heat would be lost.
As quantum mechanics taught us, there is always an uncertainty in any particle's momentum (see Heisenberg's uncertainty principle). Therefore there is always a ground state, a minimum level of energy which all the gas particles can be at, but that minimum state is never zero. It's impossible for us to achieve absolute zero, and certainly we can't make the hot side infinitely hot either.
So even with no friction or losses to the environment, it's impossible to make a heat engine 100% efficient. And that's because to return the piston to its original position, we need to remove the resistance of the gas in the compression stroke by dumping heat into the cold bar. So not all the energy stays in the flywheel.