# Why can the efficiency of a refrigerator (heat pump) be greater than one?

• patrickmoloney
In summary: Miles Per Gallon'. It's a convenient shorthand for comparing different types of vehicles.In summary, the efficiency of a heat engine is greater than the efficiency of a heat pump.
patrickmoloney
In a heat engine thermal energy is converted into mechanic energy.

In a heat pump or refrigeration cycle - fluid is being circulated between hot and cold wells.

But how can the efficiency be greater than 1?

Is it because $COP_{\text{heating}}>COP_{\text{cooling}}$

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Because [primarily] you are just moving energy, not converting it from one kind to another.
Is it because COPheating>COPcooling
No, cooling COP is still greater than 1. Heating COP is a little better because you get to keep the input mechanical energy as heat.

berkeman and CWatters
COP is NOT efficiency.

sophiecentaur
clem said:
COP is NOT efficiency.
No, but it is similar. Efficiency of a heat engine is work out over heat in whereas COP of a heat pump (or air conditioner) is heat out over work in. They are kind of inverses -- but at the same time, they are also both "what do you want" divided by "what do you pay", which makes them sound similar.

Lets say, by any chance the efficiency were one, the amount of heat removed from the inside of the refrigerator would be precisely be the same amount of heat dissipated from the compressor of the fridge. And if the efficiency were more than one you would be removing more heat from the inside than the amount of heat dissipated by the compressor.

Efficiency is a misleading way of looking at COP. Imagine you had a perfect heat engine and you tried to get work out of it from the heat energy you managed to pump into a hot reservoir. The work out from the heat engine would at best be only the work put into the original heat pump. Why? Because the work available from the heat engine depends upon the temperature difference - the 'thermal' Joules in the air out of the heat pump.

You could shift a lot of energy out of a swimming pool into raising the temperature of a nearby pool and it would not cost you much (for a small temperature rise) But how much work could you hope to get out of a heat engine with a temperature difference of, say 5C.

sophiecentaur said:
Efficiency is a misleading way of looking at COP.
I like it because it speaks to what we know and how we use it every day. I have a small electric space heater blowing hot air at me right now. For 1500 W of input, I get 1500 W of heat; 100% efficiency, or 1:1 COP. But if it were a heat pump, I could probably get over 6000 W of heat: 4:1 COP. It's a direct comparison of what heat I get (what I want) for my input power.

And by the same token, I could ask: do I need more air conditioning in the summer than I need heat in the winter? And how much does that cost in input energy? I've found that most people have no idea that an air conditioner has a better than 1:1 COP.

sophiecentaur
If you think ‘efficiency’ is the only word needed then there would be no need for COP. We spend our time telling students (rightly) that efficiency cannot be greater than unity so the solution is surely to avoid the word for heat transfer. ‘People’ are sloppy enough using Science terms, as it is.

sophiecentaur said:
If you think ‘efficiency’ is the only word needed then there would be no need for COP. We spend our time telling students (rightly) that efficiency cannot be greater than unity so the solution is surely to avoid the word for heat transfer. ‘People’ are sloppy enough using Science terms, as it is.
"Efficiency" is not always unitless. Hence miles per gallon.

sophiecentaur
jbriggs444 said:
"Efficiency" is not always unitless. Hence miles per gallon.
I don't know the proper term to describe that particular (mis?)use of the word 'efficiency' but mpg is surely just a convenience for making loose comparisons and doesn't survive too much scrutiny. It takes the term 'efficiency' way outside the range of situations where you could actually rely on using it for valid comparisons. I would have thought that the 'pure' work out / work in should be sacrosanct - if only for the sake of 'students'.

I remember the first time I came across the idea that Heat Pumps could do better than 100% and it left me a bit uneasy - after all that we'd been told in Mechanics classes.

Perspective is everything. COP is (as Russ Waters said) exactly what an engineer applying a heat pump needs to know. It is the 'energy efficiency'. As COP ignores the energy 'collected' at the evaporator, it is not very useful from a thermodynamics perspective. It implicitly assumes that the heat available at the evaporator is infinite - It is, if my wife doesn't pile lawn chairs around the outside unit.

MPG is (more properly, IMHO) 'Fuel Efficiency'. 'Efficiency' (without 'energy' or 'fuel' or 'Thermodynamic') is ambiguous.

russ_watters
I could be happy with that if the definition of Efficiency was actually "Useful Energy Out / Energy In".

sophiecentaur said:
If you think ‘efficiency’ is the only word needed then there would be no need for COP. We spend our time telling students (rightly) that efficiency cannot be greater than unity so the solution is surely to avoid the word for heat transfer. ‘People’ are sloppy enough using Science terms, as it is.
I'll re-frame: I'm sure the term COP was created for the purpose of differentiating it from thermodynamic efficiency. And that's fine, though I'm not sure it is entirely necessary. If it had been called "heat moving efficiency" I'd have been ok with that because while it is important to recognize how they are different, it is also important to recognize how they are the same. But I get it:
I remember the first time I came across the idea that Heat Pumps could do better than 100% and it left me a bit uneasy - after all that we'd been told in Mechanics classes.
Yeah, that comes from starting off not knowing how they are different, and I think that unease keeps you focused on ensuring people recognize how they are different. And that's fine.
sophiecentaur said:
I don't know the proper term to describe that particular (mis?)use of the word 'efficiency' but mpg is surely just a convenience for making loose comparisons and doesn't survive too much scrutiny. It takes the term 'efficiency' way outside the range of situations where you could actually rely on using it for valid comparisons. I would have thought that the 'pure' work out / work in should be sacrosanct - if only for the sake of 'students'.
Well, thermodynamics doesn't own the term "efficiency". Computer efficiency can be measured in FLOPS per Watt, for example. Worker efficiency can be measured in hours expected per hour expended.

Fuel efficiency or economy is a fun one because there's no theoretical limit: the work output of a standard car ride is zero.

Dullard and jbriggs444
russ_watters said:
Well, thermodynamics doesn't own the term "efficiency". Computer efficiency can be measured in FLOPS per Watt, for example. Worker efficiency can be measured in hours expected per hour expended.
I can go along with all of that but then I remember how fussy PF is about terminology in other directions. "You can't use Weight when you mean Mass" "You cannot use Force when you mean Impulse" and many others. In general conversation, many people use all sorts of terms in all sorts of 'bad' ways. I would say that the thread title (and this is PF) demands stricter usage of the terms associated with Efficiency. PF is used as a first port of call for all sorts of people who are after a bit of precision in a very fuzzy world.

sophiecentaur said:
I can go along with all of that but then I remember how fussy PF is about terminology in other directions. "You can't use Weight when you mean Mass" "You cannot use Force when you mean Impulse" and many others.
These are physical quantities which have precise definitions in physics. Efficiency is a general purpose engineering term.

russ_watters
A.T. said:
Efficiency is a general purpose engineering term.
As an oily handed Engineer, I could take offence at that distinction. Nothing general purpose about how we were taught that definition; it was a specific relationship. And, in any case, COP was invented and accepted and is used formally. I sort of get the impression that this thread considers it almost a redundant term; it clearly isn't.
I would be surprised if anyone who was taught Physics in School would not remember being presented with the word 'efficiency' as a pretty well defined formula. At the same time, in School we would have understood that a new kitchen tool made things more efficient and we would happily have discussed the Weight of an object in kg. But both those examples involve wearing different hats from the formal Physicist. I don't see why, on the one hand, it's ok to be very loose about efficiency yet very tight about weight. That's just not consistent. You have begged the question in your 'definitions' of those terms.

A.T. said:
These are physical quantities which have precise definitions in physics. Efficiency is a general purpose engineering term.
sophiecentaur said:
As an oily handed Engineer, I could take offence at that distinction.
I don't. I fully embrace the reduced rigor of engineering vs physics, and use my judgement to decide how much rigor to apply, and when -- and to make up terms if I think they can be useful. Engineers have responsibilities to design and implement things that physics makes possible, but we also have a responsibility to interface with the general public to enable the public to use and make decisions on what we make. So it's not just about use of the terms for engineering purposes, but also for communicating with the public in the way they can understand. That's why mpg is used for cars/trucks (passenger-mile or ton-mile per gallon for trains and planes), and for heating and air conditioning the public is given HSPF and SEER. SEER has the word "efficiency" in it, but it's actually seasonal average COP (in bastardized units of BTU/Watt-hr).
https://energystar.zendesk.com/hc/en-us/articles/212111387-What-is-SEER-EER-HSPF-

A simple reason for the overlap could just be that there's no verb form of "COP". You can ask "How efficient is this air conditioner?" but you can't really ask "How 'COP' is this air conditioner?"

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sophiecentaur said:
I could be happy with that if the definition of Efficiency was actually "Useful Energy Out / Energy In".
Well, let me quote from D. V. Schroeder, An Introduction to Thermal Physics (Addison Wesley Longman, 2000)
Schroeder (p. 123) said:
Let me therefore define the efficiency of an engine, ##e##, as the benefit/cost ratio:
$$e \equiv \frac{\textrm{benefit}}{\textrm{cost}} = \frac{W}{Q_h}. \quad \quad (4.1)$$
Schroeder (p. 128) said:
How should we define the "efficiency" of a refrigerator? Again, the relevant number is the benefit/cost ratio, but this time the benefit is ##Q_c## while the cost is ##W##. To avoid confusion with equation 4.1, this ratio is called the coefficient of performance:
$$\mathrm{COP} = \frac{\textrm{benefit}}{\textrm{cost}} = \frac{Q_c}{W}. \quad \quad (4.6)$$
(bolding in the original).

vanhees71 and russ_watters
DrClaude said:
Well, let me quote from D. V. Schroeder,
At least my idea puts me in good company.

Actually, no. The cite suggests that 'efficiency' is a lot more application-specific (and flexible) than you have been arguing. He's walking with everyone but you.

russ_watters, vanhees71 and DrClaude
sophiecentaur said:
I could be happy with that if the definition of Efficiency was actually "Useful Energy Out / Energy In".
My point was to show that it is common to define efficiency as "Useful Energy Out / Energy In."

russ_watters, sophiecentaur and vanhees71
DrClaude said:
My point was to show that it is common to define efficiency as "Useful Energy Out / Energy In."
Or even more generally as "benefit/cost", where the cost is not necessarily all the energy that goes in.

vanhees71, russ_watters and sophiecentaur
@A.T. and @DrClaude I can see where you're coming from and it makes a certain amount of sense. However, as far as I can see, it comes down to whether Efficiency has dimensions or not. Mechanical Efficiency needs none but the others all need qualifications and specifications. They are not readily interchangeable , either.

Take different systems of levers, springs and gears and you can make useful comparisons of their relative efficiencies. But how can you make that sort of comparison between, for instance, home heating systems where you start with a 'bucket full' of fuel and end with a person sitting in a chair, in a room, 'feeling' (or not) that he is warm enough. This is a wide open comparison system, using different steps all along the energy chain. There is no Percentage Efficiency to be calculated - just cost and satisfaction.

As Physicists, don't you feel the strength of a dimensionless efficiency value? Is that not attractive to the measurer within you? Do you not feel a certain unease in using that measure in the same breath as mpg or the accountant's bottom line for a new factory? (and I am never happy with the vocabulary of accountants). I really cannot see (apart from it being a good, fun topic for discussion) why you would want to use the same term for such different 'quantities'?

Why do people use COP? Surely for the very reason that it isn't straight efficiency. If not, then it would be redundant.

sophiecentaur said:
As Physicists, don't you feel the strength of a dimensionless efficiency value? Is that not attractive to the measurer within you? Do you not feel a certain unease in using that measure in the same breath as mpg or the accountant's bottom line for a new factory? (and I am never happy with the vocabulary of accountants). I really cannot see (apart from it being a good, fun topic for discussion) why you would want to use the same term for such different 'quantities'?
It's all down to language, not physics. There is no disagreement about what formula to use to compute any given result from any given input. Personally, I am happy if an "efficiency" expert comes into my place of work and assesses the results using metrics which are not dimensionless. And I am happy if one uses lines of code per man-month in a coding shop, ton-miles per gallon on the road, and occupancy rate in the hotel business. No skin off my nose regardless of the word used to describe those metrics.

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russ_watters
jbriggs444 said:
It's all down to language, not physics.
I could go along with that except that the terms used in Physics are usually expected to be in their Physics Context when used on PF. It's a matter of context and I think you would have a problem if your 'efficiency expert' told you that your office was working at 110% efficiency (standard pet hate for pedants - so my idea is not restricted to just me). You would need your paid expert to define his efficiency units at first.

sophiecentaur said:
... efficiency cannot be greater than unity ...
Do you have a similar problem with negative efficiency?

I don't know, but isn't this just a matter of definition? In German it's called "Wirkungsgrad", which I'd also translate as "efficiency". For a refrigerator it's the amount of heat transported (per unit time) from the colder reservoir to the hotter devided by the work (per unit time) used to do so. Thus it's for an ideal reversible Carnot process
$$\epsilon_{\text{Carnot,refr.}} = \frac{Q_{\text{cold}}}{W}=\frac{T_{<}}{T_{>}-T_{<}}.$$
The 2nd Law says ##\epsilon_{\text{real,refr.}}<\epsilon_{\text{Carnot,refr.}}##.

For a heat pump it's
$$\epsilon_{\text{Carnot,hp.}}=\frac{Q_{\text{warm}}}{W}=\frac{T_>}{T_{>}-T_<}=\frac{1}{1-T_</T_>}.$$
The latter is ##>1##.

vanhees71 said:
Thus it's for an ideal reversible Carnot process
$$\epsilon_{\text{Carnot,refr.}} = \frac{Q_{\text{cold}}}{W}=\frac{T_{<}}{T_{>}-T_{<}}.$$
The 2nd Law says ##\epsilon_{\text{real,refr.}}<\epsilon_{\text{Carnot,refr.}}##.

For a heat pump it's
$$\epsilon_{\text{Carnot,hp.}}=\frac{Q_{\text{warm}}}{W}=\frac{T_>}{T_{>}-T_<}=\frac{1}{1-T_</T_>}.$$
The latter is ##>1##.
That's a really good point. The common refrain that "efficiency can't exceed 100%" is a first-law (conservation of energy) statement that actually has very limited applicability to thermodynamics. It applies to things like electrical<->mechanical, chemical<->thermal and mechanical<->mechanical conversions, but it doesn't apply to thermal<->mechanical conversions, which are rely on the 2nd law/Carnot's theorem.

For example, we sometimes get people asking why car engines are so inefficient at ~25% based on the incorrect belief that the theoretical limit is 100%. So really, in most thermodynamic contexts the statement that "efficiency can't exceed 100%" is not very useful.

BTW, nobody directly responded to this post that resurrected the thread:
Shubham5421 said:
Lets say, by any chance the efficiency were one, the amount of heat removed from the inside of the refrigerator would be precisely be the same amount of heat dissipated from the compressor of the fridge. And if the efficiency were more than one you would be removing more heat from the inside than the amount of heat dissipated by the compressor.
True. Rather than the commonly cited COP you could use heat absorbed/heat rejected, which can't exceed 100%. It would solve the "problem" being discussed, but makes the metric a little less user-friendly.

russ_watters said:
That's a really good point. The common refrain that "efficiency can't exceed 100%" is a first-law (conservation of energy) statement that actually has very limited applicability to thermodynamics. It applies to things like electrical<->mechanical, chemical<->thermal and mechanical<->mechanical conversions, but it doesn't apply to thermal<->mechanical conversions, which are rely on the 2nd law/Carnot's theorem.

For example, we sometimes get people asking why car engines are so inefficient at ~25% based on the incorrect belief that the theoretical limit is 100%. So really, in most thermodynamic contexts the statement that "efficiency can't exceed 100%" is not very useful.
Of course, for a heat engine the "efficiency" (or however you call it) is defined by
##\frac{W}{Q_{\text{hot}}}## with the Carnot value
$$\epsilon_{\text{Carnot, he.}}=1-\frac{T_{\text{hot}}}{T_{\text{cold}}}<1.$$
"Efficiency" is always a meausure for the "wanted effect" or "benefit" per "efford", i.e., in this case I have to get heat from the hot reservoir ("efford") to get mechanical work ("benefit").

## 1. Why can the efficiency of a refrigerator be greater than one?

The efficiency of a refrigerator, or heat pump, can be greater than one because it does not create energy, but rather transfers it from one place to another. This transfer of energy can result in an efficiency greater than one, as the refrigerator is not limited by the amount of energy it can produce.

## 2. How is the efficiency of a refrigerator calculated?

The efficiency of a refrigerator is calculated by dividing the heat extracted from the cold space by the work done on the system. This ratio can result in a value greater than one if the heat extracted is greater than the work done.

## 3. Can the efficiency of a refrigerator ever be less than one?

No, the efficiency of a refrigerator cannot be less than one. This is because the system is designed to transfer heat from a colder space to a warmer space, which requires work to be done. Therefore, the efficiency will always be greater than or equal to one.

## 4. What factors can affect the efficiency of a refrigerator?

The efficiency of a refrigerator can be affected by various factors such as the design and size of the refrigerator, the type of refrigerant used, and the temperature difference between the hot and cold spaces. Other factors such as the age and maintenance of the refrigerator can also impact its efficiency.

## 5. How can the efficiency of a refrigerator be improved?

The efficiency of a refrigerator can be improved by using more efficient components, such as a better compressor or insulation material. Keeping the refrigerator well-maintained and using it at recommended temperatures can also help improve its efficiency. Additionally, regularly cleaning the condenser coils and ensuring proper air flow can also contribute to a more efficient refrigerator.

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