# Heat Engines & Efficiency

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• 0:00 What is a Heat Engine?
• 2:10 Efficiency of a Heat Engine
• 4:20 Calculation Example
• 5:20 Lesson Summary

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Lesson Transcript
Instructor: David Wood

David has taught Honors Physics, AP Physics, IB Physics and general science courses. He has a Masters in Education, and a Bachelors in Physics.

After watching this video lesson, you will be able to explain what a heat engine is, how it works from a thermodynamic perspective, and complete some simple calculations involving the efficiency of a heat engine. A short quiz will follow.

## What is a Heat Engine?

Whenever I fly in a plane, I like to take the seat near the wing. There are several reasons for this: being in the center of the plane you're less likely to get airsick, and some even argue that it's safer there. But one of the reasons is my interest in physics. Just a few meters away are the jet engines, a marvel of human engineering and knowledge. And jet engines, like all mechanical (or non-electric) engines, are just one example of a heat engine.

The second law of thermodynamics tells us that heat only spontaneously goes from hot places to cold places, never the other way around. A heat engine is the general term for any engine that uses this transfer of heat to extract useful work; in most cases, to create physical motion. This is how car engines, jet engines, and the original steam engines all work.

This is the kind of diagram that is used to describe this process. A heat engine will have a hot reservoir, a cold reservoir, and as heat energy transfers from hot to cold, work will be extracted. This is exactly the opposite process to what happens in a refrigerator, where we put work (or energy) in through electricity and force heat to transfer the opposite way; force the cold reservoir to get colder so that our food doesn't go bad. When heat goes the natural way (hot to cold), we can extract work. When heat goes the unnatural way (cold to hot), we have to do work.

This diagram also shows some of the algebra we use for heat engines. QH is the heat energy leaving the hot reservoir (measured in Joules), QC is the heat energy entering the cold reservoir (also measured in Joules), and W is the useful work extracted from the heat engine (again, that's measured in Joules).

Conservation of energy says that energy is neither created nor destroyed; it only moves from one place to another. So this means that QH must be equal to the sum of W and QC. Or in other words, the energy entering the heat engine must be equal to the work extracted, plus the heat energy lost to the cold reservoir. Whatever heat goes in, the same amount of energy must come out.

## Efficiency of a Heat Engine

To make a heat engine work and continue working, you have to keep the hot reservoir nice and hot. That takes a lot of energy. It's therefore super important that heat engines are as efficient as possible. A perfectly efficient heat engine would be one where all the heat energy you put in to keep the hot reservoir hot is completely transferred into work, where no energy is absorbed by the cold reservoir at all. And it turns out, this can never happen in the real world. Some heat energy is always lost, and no process is perfectly efficient.

If we want to calculate the efficiency of a heat engine, we need to figure out what proportion of the heat energy we put into the hot reservoir comes out as work. So that would be work, W, divided by QH, the heat we put into the heat engine. If 100% of the energy we put in came out as work, it would be 100% efficient and W would be equal to QH. That means that W divided by QH would equal 1. This is a decimal, so 1 means 100%. If you want it as a percentage, you can just multiply by 100.

But what if we don't know how much work was extracted? What if all we know is how much heat was put into the hot reservoir, and how much heat ended up in the cold reservoir? In that case, we'll need a different efficiency equation. Because of conservation of energy, we know that the energy that goes into the heat engine must equal the energy that goes out. So QH must be equal to W + QC. If we rearrange this equation to make W the subject, we see that W (work) is equal to QH - QC. This makes sense because it says that the work extracted from a heat engine is equal to the difference between the energy that goes in from the hot reservoir and the energy that goes out via the cold reservoir. Whatever the difference between these two numbers is the energy that was extracted as work.

We can substitute QH - QC into our previous efficiency equation and then we'll see that the efficiency of a heat engine is also equal to QH - QC divided by QH. So, depending on what information we are given, we can use either of these two equations to figure out the efficiency of a heat engine.

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