Problem

A coal-fired electrical power station has an effi…

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Problem 32 Hard Difficulty

32. Practical steam engines utilize $450^{\circ} \mathrm{C}$ steam, which is later exhausted at $270^{\circ} \mathrm{C}$ (a) What is the maximum efficiency that such a heat engine can have? (b) since $270^{\circ} \mathrm{C}$ steam is still quite hot, a second steam engine is sometimes operated using the exhaust of the first. What is the maximum efficiency of the second engine if its exhaust has a
temperature of $150^{\circ} \mathrm{C} ?(\mathrm{c})$ What is the overall efficiency of the two engines? (d) Show that this is the same efficiency as a single Carnot engine operating between $450^{\circ} \mathrm{C}$ and $150^{\circ} \mathrm{C} .$ Explicitly show how you follow the steps in the Problem-Solving Strategies for Thermodynamics.

Answer

A. 24.9$\%$
B. 22.1$\%$
C. 41.5$\%$
D. $=41.5 \%$

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Physics 101 Mechanics

College Physics for AP® Courses

Chapter 15

Thermodynamics

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Christina K.

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Watch More Solved Questions in Chapter 15

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Problem 15

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Problem 32

Problem 33

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Video Transcript

this is problem 32 Chapter 15 on were asked, What's the maximum efficiency that this engine can have? And then we're asked, What's the maximum energy and efficiency for a second engine? And then we're asked, What is the overall efficiency of the two engines on? Then? We're supposed to show that this is the same efficiency as a single engine, um, and show our steps. So for part A where we are trying to find the maximum efficiency, we have our efficiency equation here, Um, and so we can go ahead and use this a former formula to go ahead and solve for the efficiency of the first engine. Ah, And so in our case, we actually need to convert to Kelvin on dso N. C. One for engine one is gonna be equal to one minus the temperature of our cooling reservoir, which is gonna be 170 degrees Celsius. And then we'll add 2 73.15 which is just the conversion factor to get it into Kelvin on then, 450 which is the hot temperature plus 273.15 which just converts it to Kelvin. Uh, so this is actually gonna come out to 0.249 which is gonna be equal to 2 24.9% And so the maximum efficiency for the first engine is 24.9% and so likewise from Herbie, we can go ahead and find the efficiency of the second engine. So NC two is equal to one minus, and then I'll go ahead and show 150 degrees degree Celsius plus 273.15 that converts it to Kelvin. And then we can add 270 plus 273 +15 on. And then this is gonna come out too. 0.2 to 1 or 22.1%. And so now we can go ahead and find the total efficiency using this equation here. So n total is gonna be equal to one minus one, minus the efficiency of the first engine, uh, multiplied by one minus the efficiency of the second engine. And so now we can go ahead and substitute these values. So this is gonna be, uh, one minus one minus is euro 0.249 and then times one minus 0.2 to 1. Uh, this is gonna come out to 0.415 or 41.5%. Uh, and so the overall efficiency is gonna be 41.5%. This is Patsy. And so now, finally, for D the formula for inefficiency, we can go ahead and use this. So this is gonna be N C three, which will call, uh, the engine three that were comparing it to, uh is one minus. And that's gonna be 1 50 degrees Celsius plus 273 15 uh, over 450 degrees Celsius plus 273.15 on then this is gonna come out too. 41.5% on DSO. Now that you see this, the efficiency of this engine equals the combined efficiency of the first in the second engine. So we actually confirm that

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