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# Ideal Heat Engine w/ Non-Zero Power

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According to the second law of thermodynamics, it is impossible for:
heat energy to flow from a colder body to a hotter body
an ideal heat engine to have the efficiency of 99%
an ideal heat engine to have non-zero power.
a physical process to yield more energy than what is put in

^As taken from a homework problem, the explanation:

The ideal engine follows a reversible cycle--therefore, an infinitely slow one. If the work is being done at the infinitely slow rate, the power of such an engine is zero.

An alternative way to state the second law of thermodynamics is as follows:

It is impossible to construct a cyclical heat engine whose sole effect is the continuous transfer of heat energy from a colder object to a hotter one.

This statement is known as Clausius statement of the second law. Note the word "sole." Of course, it is possible to construct a machine in which a heat flow from a colder to a hotter object is accompanied by another process, such as work input.

Is anyone able to explain this in layman terms, i.e. to an first-level college physics student...

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It is not clear to me what your question is.

Where exactly are you having difficulty with the material you have posted?

The part in upright text cannot be complete by itself, was it a multiple choice question, with the underlined part the 'correct' answer?

And whose explanation is in italics?

Is this the explanation of the 'correct' answer and do you understand why the others are incorrect?

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Some assertions don't belong to the second law, others not to this qualitative law. Neither do I understand clearly te question.

However, I'll take with much skepticism any claim like "hence infinitely slow". Many systems show no loss at all with finite speed (nanomachines consisting of aggregates of few atoms, superfluids, superconductors...) and I wouldn't be surprised if someone had already built a thermal engine with them. Not necessarily an engine that moves a train, but one that ruins the previous general assertion.

Also, beware with "efficiency". If one takes it as output over input, it's always 1. A better idea would be energy benefit over energy cost, but then the efficiency of a heat pump is >1. And especially in thermal engines, one defines an isentropic efficiency by the output power over the power allowed by the temperature ratio; still different, and some engines achieve an excellent isentropic efficiency.

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It is not clear to me what your question is.

Where exactly are you having difficulty with the material you have posted?

The part in upright text cannot be complete by itself, was it a multiple choice question, with the underlined part the 'correct' answer?

And whose explanation is in italics?

Is this the explanation of the 'correct' answer and do you understand why the others are incorrect?

The difficulty I'm having is with understanding the explanation for the answer they provided.

It was a multiple choice question, basically: of the four choices, which one would be impossible? The underlined answer is the 'correct' answer. The explanation is provided by the text the problem was selected from so it is the explanation of the correct answer, given by the book.

I do understand why the other answers are incorrect - it is possible for 1) heat to flow from a colder body to a hotter body, as in a heat pump, 2) 99% efficiency, just not 100%, 3) a physical process to yield more energy than what is put in - this is kind of a misleading statement but I believe heat pumps satisfy this - that is the work input (typically electrical work) is typically smaller than the heat transfer to the hot/cold reservoir (thus COP's tend to be larger than 1).

My question is why can an ideal engine NOT have non-zero power? If the engine isn't running, does it not have non-zero power? Or as they put it: if the work is being done at the infinitely slow rate, the power of such an engine is zero. The "explanation" seems to contradict itself.

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which one would be impossible?

The situation would seem to be as I thought, but you have not read the question correctly.

It did not ask which statement was impossible full stop.

It asked which statement was forbidden (made impossible) by the second law.

For example (4) is a correct statement, ie it is impossible for a physical process to yield more energy that is put in, but just nothing to do with the second law

So look again at all 4 statements and see why they are accepted or rejected. Look particularly at the definition of an ideal heat engine.

If you were to say that all four statements are poorly worded and somewhat ambiguous, I would agree with you.

For example the first statement "it is impossible for heat energy to flow from a colder body to a hotter body" is incomplete so can be

taken two ways which make it either possible or impossible,depending upopn how you complete it.

However neither interpretation is against the second law.

Edited by studiot
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Thank you for your pointers, yes the fourth answer is indeed with relation to the 1st law rather than the 2nd law. However in trying to prove my misreading of the question you didnt really answer the question of why it is impossible for an ideal engine to have a nonzero power, according to the 2nd law. It did not specifically ask for a scenario in which the engine was at full stop, but in order to consider something impossible we must more or less consider all possibilities no?

I think I'm going to bury this problem as a poorly worded one in the back of my mind somewhere. Thank you both for your input.

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No I didn't yet answer (3) because I was trying to lead the discussion in that direction. It does worry me that several of your other statements were incorrect and I thought you would appreciate the insight gained by finding for yourself the correct answers. That is why I have only given hints so far. Here are a few more.

There are four laws in classical thermodynamics.

There are four statements to compare with these laws. Each statement is either correct or incorrect and to do with one or more of the four laws.

To reinforce what was said about (3) - How long does it take to complete one single cycle< and what does the second law say about cycles?

There is much to be gained from all the answers to this question.

We have now dealt with (4), what about (1) and (2)?

Does heat flow from a cold body to a hot one in a heat pump?

How do you think that happens?

Can you offer a heat pump design that achieves this remarkable feat?

I do understand why the other answers are incorrect - it is possible for 1) heat to flow from a colder body to a hotter body, as in a heat pump,

Edited by studiot
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Does heat flow from a cold body to a hot one in a heat pump?

How do you think that happens?

Can you offer a heat pump design that achieves this remarkable feat?

Okay, I'll follow along with you.

My answers to your questions (elaborated further from my previous post):

1) Heat can flow from a cold body to a hot one in a cooling process. This particular heat pump would be a refrigerator. If we were to look at an energy diagram for a refrigerator, we see that heat is "pumped" from the cold reservoir (e.g. inside of a refrigerator) to the hot reservoir (e.g. your kitchen). The cold reservoir is simply a 'sub-system' that has a lower temperature than the hot reservoir. The reason you can't have 100% efficiency is because you'd need a cold reservoir with 0K, and to my knowledge that is not possible.

2) Heat flows from a colder region to a hotter region via work input. Otherwise, heat flow from a colder to hotter would violate the second law (that entropy of a system must increase or remain the same).

3) I suppose the answer to #1 gives the general idea here.

Also...

To reinforce what was said about (3) - How long does it take to complete one single cycle< and what does the second law say about cycles?

The second law doesn't say anything specific about cycles. My confusion stems from how they relate power (energy/time) to this. I understand how a "zero power" would arise, that is making time infinitely large would give you a zero power. But what makes an infinitely large time impossible, is what I don't understand.

Edited by ahyaa
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Ahyaa, are you reading the question correctly, or am I misunderstanding your confusion? It looks like you're confused about why "zero power" is impossible, but the question and explanation state that nonzero power is impossible. Or, put another way, the power must be zero and anything else is impossible.

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What strikes me is that I have made several statements that you haven't asked me about once.

For instance I said there are four laws of classical thermodynamics.

Each can be stated in several equivalent ways.

The most fundamental was introduced after the first law and called the zeroth law.

It is the most fundamental because all the other three refer to equilibrium states and there is no point doing this unless you can identify and define an equilibrium state. That is what the zeroth law is about.

One version states that heat cannot flow from a colder to a hotter body.

So (1) is true but because of the zeroth law, not the second.

Any heat pump works like this simple model (there are more complicated ones)

There is a working fluid which is cooled by expansion to below the environment temperature.

Heat flows from the environment into the working fluid by the zeroth law, cooling the environment and warming the fluid.

Because of the zeroth law the fluid cannot become warmer than the environment.

The working fluid is then compressed to raise its temperature and pumped to a heat exchanger where it is hotter than the output medium so heat flows from the working fluid to the output medium, again by the zeroth law.

The working fluid is pumped back to the environment heat exchanger and expanded to cool it below the environment temperature.

This is a cyclic process.

At no point is heat transferred directly from a colder to a hotter body.

Edited by studiot
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Ahyaa, are you reading the question correctly, or am I misunderstanding your confusion? It looks like you're confused about why "zero power" is impossible, but the question and explanation state that nonzero power is impossible. Or, put another way, the power must be zero and anything else is impossible.

You're misunderstanding here, I said because I understand how zero power is possible, the answer that "nonzero power is impossible" seems wrong.

One version states that heat cannot flow from a colder to a hotter body.

I have not heard of the zeroth law being re-stated in such a way. I wouldn't disagree that it may be out of my scope, so if you would like to enlighten me I would be highly interested in how this restatement is arrived at.

(Actually, looking back and thinking about this I agree completely, but I just haven't seen the law restated in this way, but it is good to know).

So (1) is true but because of the zeroth law, not the second.

Your heat pump model does make sense and I find it very interesting. But would you not say that heat flows from the compressed working fluid to output medium, and from the environment to the expanded working fluid, due to an increase in entropy?

Edited by ahyaa
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There is a certain sense in analysing (1) then (2) then (3) then (4).

Note that (4) refers to energy transfer whereas (1) refers to heat only.

Failure to make this distinction is a common cause of failure to correctly analyse a thermodynamics problem.

Entropy is the ratio of heat transferred to temperature, not the energy.

But would you not say that heat flows from the compressed working fluid to output medium, and from the environment to the expanded working fluid, due to an increase in entropy?

Entropy of what? But no, I would not say that because classical entropy change is defined in terms of cyclic processes and equilibrium states.

Can we set entropy aside until the end since it was not mentioned in the question?.

The heat engine needs further development to cope with (2) and (3).

If you bring bodies A and B (at different temperatures with Ta > Tb) into thermal contact heat will flow directly from A to B.

You cannot make this a heat engine ie you cannot extract work from this you have to have an intermediate stage which is why you have to have a working fluid.

Incidentally for your issue with (3)

Power is the rate of doing work = work/time Note (3) says power, not work.

For something that happens infinitely slowly, time = infinity so you are dividing by infinity, which is equivalent to multiplying by zero (in engineering maths not pure maths).

So for any finite value of work the power is exactly zero.

Edited by studiot
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You're misunderstanding here, I said because I understand how zero power is possible, the answer that "nonzero power is impossible" seems wrong.

I'm getting tripped up by the "because" part of that. If "nonzero power is impossible" then the only possible power is zero. So understanding how zero power is possible wouldn't contradict nonzero power being impossible.

Unless you mean that because of the way you understand zero power being possible, it seems like you should be able to get more than zero power?

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