 Problem 4. A completely hypothetical thermodynamics instructor drives a 2004 Dodge Intrepid with a lot of character. During one of his bi-weekly performance tests, he measures the power output of the engine to be 220 horsepower. He measures that the burning fuel is producing 390.6 kilojoules of heat energy every second. So in terms of what I actually know here, I know that the net power output is going to be referring, or the power output of the engine is referring to the net power output. If I were modeling this engine as a heat engine, which it is, I know that I'm getting energy in the form of heat. So I have a Q in, a rate of heat transfer in, and that's allowing me to produce some amount of net power output. That is the goal of my engine. Fortunately, my engine doesn't do that very well, but a hypothetical engine. And then it can't convert all of that heat to work. So some amount of that heat is exhausted. And I know that the 220 horsepower is going to be my net power output. And then I also know Q.in. And that's because it's reasonable for me to assume that all of the energy produced in the form of heat by burning the fuel is going into the engine. The combustion is actually happening inside the engine. And if I assume that it's running steadily, the piston walls have probably already heated up and they're shedding some heat to the environment through the engine itself, but I'm just ignoring that for the moment. So I'm saying that all of the heat energy produced by the fuel is my heat input to my engine. And this is phrased a little bizarrely, but this is 390.6 kilojoules of heat energy per second. So this is 390 kilojoules per second, which is a kilowatt. So this would be 390 kilowatts. So this is a unit of power. This is also a unit of power. So I could figure out Q out by recognizing that Q out is just whatever heat didn't get turned into work. So my Q out, which is what I'm looking for for the answer to part A is Q in or Q dot in, excuse me, minus my net power output. And I got this by looking at the heat or the heat engine diagram over here. But I also could have come up with it by doing an energy balance. That's what this is. If I did an energy balance on this engine, I recognize that it's operating steadily. So I could take this and divide all the terms by dt. And I'd get E dot and E dot. So it's steady. Nothing changes with time. Therefore, this is zero. So I'm left with E dot in the energy, the rate of energy entering the system is equal to E dot out the rate of energy exiting the system. Now, is this system open or closed? Well, in this case, I'm going to be treating it as if it were a closed system. So in reality, I do have different air entering as the engine is operating. It's taking in some air and it's exhaling some exhaust. But for the moment, I'm treating the combustion process as if it's just heat input to air that is unchanged chemically. So I'm treating it as if the same air is just going around in the cycle. That's more of an analysis for Thermo 2. But for the moment, I'm treating it as being closed. So the only types of heat input and output I could have are heat and work. So this would be Q dot in plus work dot in is equal to Q dot out plus work dot out. And then I know that my net power output is defined as being my W dot out minus my W dot in. So if I were to rearrange this equation in terms of net power output, I would have Q in Q dot in is equal to Q dot out plus workout minus work in, which is net power output. So this equation is the same as this equation. Just rearrange for Q dot out. So this is an energy balance. Anyway, I know that Q dot out is going to be Q dot in minus net power output. And 390 kilowatts and 220 horsepower are both valid representations of power. They're both units of power, but I can't subtract them without having dimensional homogeneity. So I need to convert kilowatts to horsepower or horsepower to kilowatts. And I'm going to operate in metric here. So I'm going to convert my horsepower into kilowatts. So I have 390 kilowatts minus 220 horsepower. And then to get the conversion from horsepower into kilowatts, I'm going to go back to my textbook, which is here. And I have various conversions for energy transfer rate, which is power. So I know that one horsepower is 0.7457 kilowatts. I could also use this row here, which is one kilowatt equals 1.341 horsepower. I'm actually going to use this value just because I know it already 1.341 horsepower in one kilowatt. So 1.341 horsepower per one kilowatt. And again, I could have just written this value of my 0.7457. I could have just written that in the numerator and it would have been mathematically the same thing. Anyway, I'm getting a little bit off on a tangent here. 390 kilowatts. My horsepower and horsepower are going to cancel minus 220 divided by 1.341. So that would be 220 divided by 1.341. Excuse me, 220 divided by 1.341. What did those letters come from? That's a mystery. 164.057 kilowatts. So 390 minus 164.057 gives me 200 and 25.943 kilowatts. A kilowatts, awesome. So that's actually my answer for part A. Whatever heat came in, which didn't get converted into net power output, leaves. Now part B, the thermal efficiency of the engine. Well, this is a heat engine. So the thermal efficiency is going to still be the desired output per heat. So desired output, remember any efficiency is the desired output divided by the input that you need to achieve that output. In this case, this is thermal efficiency. Therefore, it's referring to the heat input. So I could write this as being my net power output. That's the goal of this engine is to produce net power output. And the heat input then would be q dot in. But I could just have easily written this as w over q without the dots. I could have written this as specific net power output or specific net work output over the specific heat input. But the capital letter dot form is going to be most convenient here because I know my net power output and I know my rate of heat input. So this would be 220 horsepower, which I have already converted into kilowatts. That's 164.057 kilowatts. 164.057 kilowatts divided by q dot in, which is 390 kilowatts. My kilowatts and kilowatts cancel and I'll get my thermal efficiency. So this would be 164. Why did I just, there we go. Divided by 390. That's 0.4207. 0.4207. Oh, that's where the B and H came from. I see. It's the letters that I used to switch back and forth between moving the screen around and writing stuff on the screen. It makes a lot of sense. Anyway, I get an answer of 42-ish percent for the thermal efficiency of my engine. Makes sense. My engine is the answer to the question of life, the universe, and everything.