Problem

(a) What is the best coefficient of performance f…

Problem 41 Hard Difficulty

(a) What is the best coefficient of performance for a heat pump that has a hot reservoir temperature of $50.0^{\circ} \mathrm{C}$ and a cold reservoir temperature of $-20.0^{\circ} \mathrm{C}$ ? (b) How much heat transfer occurs into the warm environment if $3.60 \times 10^{7} \mathrm{J}$ of work $(10.0 \mathrm{kW} \cdot \mathrm{h})$ is put into it? (c) If the cost of this work input is 10.0 cents/k $\mathrm{W} \cdot \mathrm{h}$ , how does its cost compare with the direct heat transfer achieved by burning natural gas at a cost of 85.0 cents per therm. (A therm is a common unit of energy for natural gas and equals $1.055 \times 10^{8} \mathrm{J}$ .)

Answer

A. $C O P_{\mathrm{hp}}=4.61$
B. $3.97 \times 10^{4} \mathrm{kcal}$
C. 3.44

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Physics 101 Mechanics

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Chapter 15

Thermodynamics

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Video Transcript

in the first part of the question. That is hard, eh? We are supposed to find the coefficient of performance, which is best coefficient of performance. So we are given in this case, the cold reserve or temperature on the hopes visible temperature so we can find that we can use that to find efficiency. Let's first right, C O H P. That is the coefficient off hit engine, uh, best proficient off performance off you trained you, which is one divided, but it, er that is efficiency off engine on dhe. This comes from the fact here that we have Q divided by work, which is equal to cure. Divided by eight hour times. Q. Which is efficiency? Times hit, supplied. So in the end, it's just one divided by eat Data, as you can see here on eight off or car knows Indian, which is the best efficiency you can have, is one minus TC divided by th. Now, when you are, you are going to use cheesy and th make sure that you use Kelvin temperatures and negative 20 degrees is 2 53 Kelvin on positive 50 degrees is 23 23 Kelvin. So if you perform this calculation, you'll find that efficiency is 0.2167 Now we can use this to find the coefficient off performance, which is one divided by efficiency. So this is equal to four points. It's one as given in the answer with a question in the part B. We are asked to find how much he transfer occurs into warm environment if 3.6 times 10 power jewels off work is put into it. So we are given the work here, and we're supposed to find this. Q Q is w but eight or, in this case, so 3.6 multiplied by 10. Power seven, divided by 0.2167 gives you 1.66 times 10 power Jews. Answer in the answer below. The question on numerator is given in terms of calories to find in terms of calories. All you have to do is divided by 4.184 so we don't need to do that. Jules is fine on in part. See, we have to compare the cost off it. Engine to Cole. So not cold. Sorry. Natural gas now cost off heat engine in this case is given us 10 cents per kilowatt hour on 3.6 times 10 Power seven Jude's is 10 kilowatt hour, so we can't, which is also given in the question. So this is 10 kilowatt per hour and we have 10 cents Well, one kilowatt hour. So multiplication just cancels this units and we are left with 100 cents, which is $1. So this is the cost off. Ideal Indian now cost for natural gas is 8 85 cents for thought. Now here we have to understand how much heat we are going to use to produce the work off this amount 3.6 times 10 power seven. So to get this this much work done, we need this much off heat. So we need this much off heat from natural gas is so this is hit there in the numerator and but the cost given tow us off Natural gas is 85 cents for third, so we must convert this heat into thirds. How would we do that? Is dividing it by heat In one thing, that is 1.55 times 10 power eight. So cost will be 85 since, so if you could form this calculation, you'll find that answer is 1.337 approximated to three significant digits is 1.34 So in this case, the cost it is $1.34 so you can see heat engine seems much more efficient compared to natural gas. But also make sure to understand that this comes from the efficiency of car knows engine. We will not have this much efficiency, so natural gas in reality can be more efficient, cost efficient than he did in you.

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