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Thermal Efficiency & The Second Law of Thermodynamics

Patrick Ford
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Hey guys, so in the last couple videos, we were introduced to heat engines. And in some problems you'll have to calculate something called the thermal efficiency of the engine. So that's what I want to introduce you in this video and we're also going to see what the second law of thermodynamics is. Let's check this out here and we're gonna work a bunch of examples out together. So remember that heat engines, basically what they do is they produce work using the heat energy that is flowing through the engine from hot reservoir, too cold. So you have some heat that flows in right from hot to cold. And then basically the engine takes some of that and produces some work some useful energy. Now, an engine's thermal efficiency is basically just a measure of how good it is at doing that, how good it is from producing work from heat, different engines, like a gasoline engine versus a diesel engine operate on different cycles and they have different efficiencies. The letter that we'll use for the sufficiency is the lower case letter E. And basically the equations are pretty straightforward. One of them you'll need to know is W over Q. H. The way I like to think about this is this w is the work that gets put out by the engine. Q. Is what you put into the engine. So the efficiency is how much you get out divided by, how much you put in. Now, this is just a ratio. So sometimes your professors might want to multiply this by 100%. That's really up to you. Now, there's actually one other equation that you need to know and it's because we can rewrite this w here as Q H minus Q. C. And if you go ahead and do that and simplify, you'll end up with this equation over here, one minus Q. H. Sorry Qc over Q. H. It's always the smaller number. Over the bigger number. That's basically all there is to it guys, let's take a look at some examples here. So basically what we're gonna do for these diagrams is calculate whatever variables are missing. And this first example here we have 1000 jewels that gets put in zero work that gets produced and 1000 jewels that gets put out. We want to calculate the efficiency here. So the efficiency remember is just the amount of work that you get out divided by the amount of heat that you put in, that's just going to be zero, divided by 1000. And if you multiply this, you're just gonna get 0% which is actually a pretty terrible engine, you would put in a bunch of heat energy into this and you will get no useful work out of it kind of sucks. Let's move on to the second one here. Now we have the amount of heat that flows in the amount of heat that flows out, but now we actually don't know the amount of work that gets produced. So if we want to use the same equation here, heat efficiency equals W over Q. H. What happens is we don't know what this work is now, we could find it because remember this number here is always the difference between these two numbers or we can actually just take a more direct approach and actually use this equation that's over here, because that has the two cues which we do know. So it really just depends on which variables, you know which ones you're given. You can take a more direct approach here, so this is gonna be one minus Q. C over Q. H. And basically this is going to be one minus the 600 joules divided by 1000. It's always the small number over the bigger one. So if you work this out, you're gonna get is 0.4. Now again, some professors might multiply by 100 or make you multiplied by 100 and therefore the x efficiency will be expressed as 40%. And that's the answer. Let's move on to part C. Now, in part C we know the work and we know the efficiency of these engines but we actually don't know the heat that goes in and the heat that comes out. So we're going to calculate this now, you might be tempted to write down the equation W equals Q H minus Q C. That's the first one that we learned, we actually can't use this because we have two unknown variables. We can't use this instead. We're gonna use another equation. The efficiency equation because we have E. And W. So this E. Here is equal to W over Q. H. Remember. So now we want to do is we want to calculate this QH here, that's our missing variable. So we're gonna basically trade places with these two variables, and Q. H. is going to be the work divided by the efficiency. So this is gonna be 600 divided by uh and this is going to be 0.3 and what you're gonna get here is 2000 jewels. So 2000 jewels of heat energy was pumped into this heat engine. That's your cue. H. Now, basically, now that we have this qh we can actually go back and use this equation here and calculate the waste heat that gets expelled. This is going to be 2000 minus 600 you're gonna end up with 1400 jewels that leaves the engine and now last but not least in this engine over here, we have all of the heat that gets pumped in and then all of it gets converted into work. You have 1000 and 1000. So the efficiency here is going to be equals w over Q. H. That's just gonna be 1000 over 1000. And that's basically just going to equal 100%. Now, even though there is nothing in this equation that prevents this from happening. This kind of engine is actually impossible to make because doing so violates what's called the Second law of thermodynamics. So let's talk about that. We've worked a lot with the first law of thermodynamics. The second one unfortunately isn't really an equation so much as it is. A bunch of statements. Now, what's one statement? The first thing we're gonna talk about here is that it's actually impossible. It's impossible to take an engine and have it convert all of the heat into work with 100% efficiency. It's impossible to design an engine that basically spits out as much work as the heat that you plug into it. Engines must expel waste heat to the cold reservoir. This statement here is known as the kelvin or the engine statement of the Second law of thermodynamics. It's a really conceptual one. But basically the idea here is that if you could design an engine that takes all of the heat energy and then converts it into work, you could basically build an engine that actually propels or sustains itself using its own work. This is called a perpetual motion machine. And everything in physics tells us that you absolutely cannot do this. You can't just keep this process going forever. You have to have heat that goes to the cold reservoir and so they can start the cycle all over again. Anyway, so that's it for this one. Guys, let me know if you have any questions