What are (a) the thermal efficiency and (b) the heat extracted from the hot reservoir for the heat engine shown in FIGURE EX21.16?

Question

Accepted Answer

Hey everyone. So this problem is dealing with heat engines and PV diagrams. Let's see what it's asking us. A mechanical engineer designed a thermal engine that operates using the cycle shown in the graph below to determine the heat extracted denoted by Q of H from the heat source and the thermal efficiency denoted by A. So this graph is a PV diagram where we have P on the Y axis and that's our pressure in pascals. And then on the X axis, we have the hour volume. And so the thermal engine operating cycle goes from A to BB to C and then C back to a in a triangle. And we're also showing here that from the B to C part of the cycle, we are exhausting 20 jewels to the environment or to outside of that cycle. And from the C to a part of the cycle, we have an exhaust of 120 jewels. So our multiple choice answers here are a 120 jewels and 13% B 120 jewels and 15% C jewels and 13% or D 160 jewels and 15%. So the key to solving this problem is we're calling our heat extracted equation, which is Q of H is equal to W our work plus Q of C our exhaust. So that's our extracted heat, that's equal to our work plus our exhaust heat. And then we need to find thermal efficiency. And the second part of the problem, so we can recall that equation where thermal efficiency A is equal to work divided by our extracted heat. And so the first step to solving this problem is recognizing how we're going to find both work and our exhaust heat. So our exhaust heat comes directly from our diagram where we're shown that we, from A to B, we don't lose any heat. There's no heat exhausted. But from B to C, we do have 20 jewels and then from C to A, we have 120 Jews. And so we sum that and we get 140 Jews is our heat exhaust. To find our work. We can recall that using the PV diagram, our work is equal to the area inside the cycle. And so we're dealing with a triangle here. So the area of a triangle we can recall is one half multiplied by the base multiplied by the height. This is for a right triangle which we have here. And so our base goes from 0. 20.060 and those units are tend to the negative 2 m cued. And so the delta from 0. to 0.02 is of course, 0.04 times 10 to the negative 2 m cubed is the length of our base. And then the height of our triangle goes from 1.5 times 10 to the fifth pascals to 2.5 times 10 to the fifth pascals. So 2.5 minus 1.5 is of course one. And so that will be one times 10 to the fifth baskets for our height, plug that into our calculator and we come up with a work of 20 tools. So now we go back to our first equation. So we have Q some age is equal to our work. That's 20 Jews plus our exhaust heat, which is 140 Jews and we get jewels. And so that is the answer to part one. When we look at our multiple choice answers, that means we can eliminate answers A and B. So now we need to solve for the thermal efficiency where again, Ada is equal to work divided by Q of H. So we've solved for work, that's 20 Jews. And we just solved for Q of H, that's 160 Jews. We plug that into our calculator and we get 0. or 13%. And that is the answer to the second part. And so the correct answer for this problem is answer choice c so that's all we have for this one, we'll see you in the next video.

What are (a) the thermal efficiency and (b) the heat extracted from the hot reservoir for the heat engine shown in FIGURE EX21.16?

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Key Concepts

Thermal Efficiency

Heat Reservoirs

First Law of Thermodynamics