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An ideal refrigerator or ideal heat pump is equivalent to a Carnot engine running in reverse. That is, energy $mid Q_{d}$ is taken in from a cold reservoir and ener...

Question

An ideal refrigerator or ideal heat pump is equivalent to a Carnot engine running in reverse. That is, energy $mid Q_{d}$ is taken in from a cold reservoir and energy $left|Q_{h}ight|$ is rejected to a hot reservoir. (a) Show that the work that must be supplied to run the refrigerator or heat pump is$$W=frac{T_{h}-T_{c}}{T_{c}}left|Q_{c}ight|$$(b) Show that the coefficient of performance of the ideal refrigerator is$$mathrm{COP}=frac{T_{c}}{T_{h}-T_{c}}$$

An ideal refrigerator or ideal heat pump is equivalent to a Carnot engine running in reverse. That is, energy $mid Q_{d}$ is taken in from a cold reservoir and energy $left|Q_{h} ight|$ is rejected to a hot reservoir. (a) Show that the work that must be supplied to run the refrigerator or heat pump is $$ W=frac{T_{h}-T_{c}}{T_{c}}left|Q_{c} ight| $$ (b) Show that the coefficient of performance of the ideal refrigerator is $$ mathrm{COP}=frac{T_{c}}{T_{h}-T_{c}} $$



Answers

(II) $(a)$ Given that the coefficient of performance of a refrigerator is defined (Eq. 4a) as
$$\mathrm{COP}=\frac{Q_{\mathrm{L}}}{W}$$ show that for an ideal (Carnot) refrigerator,
(b) Write the COP in terms of the efficiency e of the reversible heat engine obtained by running the refrigerator backward. (c) What is the coefficient of performance for an ideal refrigerator that maintains a freezer compartment at $-18^{\circ} \mathrm{C}$ when the condenser's temperature is $24^{\circ} \mathrm{C} ?$

So we're trying to heat up a room on the coefficient of performances 3.8. Remember the definition off coefficient of performance? It is que el That is the amount of heat extractor from the lower temperature bath divided by the world with model us. Marty lie, of course. So, no, you remember. Work is simply que edge minus Q. Well, that is the amount off he dumped into the house minus the amount of heat sucked in from outside. Using this become right decides you added by U H minus Cuba. I simply drop the model of signs for now. So rearranging this, we can see that Hey, time skew edge minus key Well is equal to que ele. So using this, we can see that Q edge is equal to I'm sorry. We're supposed to find the work done not think you etch So what will insert do is substitute for Q l you elicit will do you ej minus w we substitute this instant off The other thing It's okay. The coefficient off performance is equal. Do u h minus w by w which means w k is equal. Duke you JJ minus w which means double music will do you a JJ By Keay Bliss One simplest that. So what done is 7.54 killer Jules you wanted by 3.8 less one? Remember that, Q? Which is the amount of heat needed to keep the whom heart or one is equal to Q edge. Not que ele on This turns out to be 1.57 military IDs. I'm sorry, Mega Jules. So another You know this This is the amount off work you need to do for our so power is 1.57 mega jewels. So 1.57 to 10 parsecs. Jules, divided by one hour. Remember that one house has 60 minutes, and each minute has 60 seconds. So that 1.57 I'm Stan par six by 3600 Jules per second. If you remember, June per second is a what on this number turns out to be 440. What? So this is the power with which you need. You need to run the heater to keep your house warm.

All right, so we have this he engine, right? And the energy transfer from the hot reservoirs partly going to go into the cold reservoir and part of it, is it gonna actually do work? And so the efficiency of an engine is gonna be, uh, calculated by just taking the amount of work that gets done and dividing it by the, you know, total initial heat from the hot source. Okay. And so how do we calculate work? Well, as we know, work is gonna be the change in energy. Right? And so the change in energy Well, I was just going to be the hot, minus the cold energy. And so we can then take the efficiency former to make it equal to Q H minus QC Divided by qh. And so from the question we can plug in our numbers, Then we get energy or energy efficiency is gonna be equal to 1.7 kilo jewels minus 1.20 kilo jewels divided by 0.70 killer. Jules and I give us an answer about 0.294 And so there's your answer for party Now. Next we have to figure out the work Well, again as we know, the work is just gonna be the change in energy. And so we can take the 1.70 killer jewels and subtract the 1.20 killer jewels of this cold reservoir. And we'll get about five or not boat. We'll get exactly 500 jewels of work that's being done every cycle. And now we have the power. So how do we calculate the power? Well, the power is just energy over time, changing energy over time, right? And so all we gotta do is take the work and divided by the time. And that's gonna be equal to 500 rules divided by each cycle US 30.3 seconds, giving us an answer of about 1.67 times 10 to the three, Watts said. There's your three answers. That's it.

All right, So for this question, were using the refrigerator coefficient formula. Right? And that's just gonna be equal to K, which is a coefficient performance is gonna be Go to the heat energy. Uh uh, the cold reservoir divide by the amount of work. Okay, so we know that K is five from the question. Another case five. And we know that the Q cold is equal to 120 jewels, right? And we're trying to find work. So if we rearrange the former for work will get que cold divided by K, and that's gonna be equal 220 jewels divided by five. And that's gonna give us 24 drools for the amount of work done in each cycle. Okay, so there is your answer to pour eight. Next, we have to find out the Emma, the energy going to the hot part, the hot reservoir. And so if we look at work, we realized workers changing energy, so we can also be read. It is Q h minus. Que seat. Right, cause, um, due to the second law of thermodynamics, you know, it always go from the energy. Always transferred from hot to cold And so our work is going to be Q hot minus que cold. And so, if you re arrange, we're trying to solve a que hot will get you heart equals to work plus que cold and so will get the answer off. 24 jewels plus 120 jewels close to 144 jewels and, yeah, there we have our to answers, and that's it.

All right to this question. We have our efficiency of a heat engine formula, and we have Accardo engine efficiency for me. Okay, Now, the thing is, this is going to the question is talking about some cartilaginous running in reverse? Writes a refrigerator. So instead of focusing on cue pH for the first form that we're actually gonna use QC and then in a similar matter, the Carnot cycle formula is actually gonna be th one sec about of a T. C. Okay, so we make these changes because it's a refrigerator. Then we can, you know, do some easy, you know, substitution and stuff like that to get our answer If we, uh because this is gonna be a cardinal engine for real. If we replace the E with the teacher when it's d c over TC, you can see the answer becomes evident. Okay, we get th about his TC divided by D. C. Is equal toe work divided like you see, and I'm ago. Q C times teach minus TC over TC is able to work and there's a chance for the first part of the second part. We're looking the Seop that for a Seop a refrigerator. It's always equal to QC over work. Now, something you could notice is that you see over work is the universe of work over QC. Right? And so this is gonna be equal toe one overeat because this is the universe about. And so if we also take the inverse of this, we'll get TC over th minus T c, which is our answer. So there you have it.


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