An Efficiency Greater Than 1?

No, my next project is not building a Perpetuum Mobile.

Sometimes I mull upon definitions of performance indicators. It seems straight-forward that the efficiency of a wood log or oil burner is smaller than 1 – if combustion is not perfect you will never be able to turn the caloric value into heat, due to various losses and incomplete combustion.

Our solar panels have an ‘efficiency’ or power ratio of about 16,5%. So 16.5% of solar energy are converted to electrical energy which does not seem a lot. However, that number is meaningless without adding economic context as solar energy is free. Higher efficiency would allow for much smaller panels. If efficiency were only 1% and panels were incredibly cheap and I had ample roof spaces I might not care though.

The coefficient of performance of a heat pump is 4-5 which sometimes leaves you with this weird feeling of using odd definitions. Electrical power is ‘multiplied’ by a factor always greater than one. Is that based on crackpottery?

Heat pump.
Our heat pump. (5 connections: 2x heat source – brine, 3x heating water hot water / heating water supply, joint return).

Actually, we are cheating here when considering the ‘input’ – in contrast to the way we view photovoltaic panels: If 1 kW of electrical power is magically converted to 4 kW of heating power, the remaining 3 kW are provided by a cold or lukewarm heat source. Since those are (economically) free, they don’t count. But you might still wonder, why the number is so much higher than 1.

My favorite answer:

There is an absolute minimum temperature, and our typical refrigerators and heat pumps operate well above it.

The efficiency of thermodynamic machines is most often explained by starting with an ideal process using an ideal substance – using a perfect gas as a refrigerant that runs in a closed circuit. (For more details see pointers in the Further Reading section below). The gas would be expanded at a low temperature. This low temperature is constant during heat is transferred from the heat source to the gas. At a higher temperature the gas is compressed and releases heat. The heat released is the sum of the heat taken in at lower temperatures plus the electrical energy fed in to the compressor – so there is no violation of energy conservation. In order to ‘jump’ from the lower to the higher temperature, the gas is compressed – by a compressor run on electrical power – without exchanging heat with the environment. This process is repeating itself again and again, and with every cycle the same heat energy is released at the higher temperature.

In defining the coefficient of performance the energy from the heat source is omitted, in contrast to the electrical energy:

COP = \frac {\text{Heat released at higher temperature per cycle}}{\text{Electrical energy fed into the compressor per cycle}}

The efficiency of a heat pump is the inverse of the efficiency of an ideal engine – the same machine, running in reverse. The engine has an efficiency lower than 1 as expected. Just as the ambient energy fed into the heat pump is ‘free’, the related heat released by the engine to the environment is useless and thus not included in the engine’s ‘output’.

The calculation (see below) results in a simple expression as the efficiency only depends on temperatures. Naming the higher temperature (heating water) T1 and the temperature of the heat source (‘environment’, our water tank for example) T….

COP = \frac {T_1}{T_1-T_2}

The important thing here is that temperatures have to be calculated in absolute values: 0°C is equal to 273,15 Kelvin, so for a typical heat pump and floor loops the nominator is about 307 K (35°C) whereas the denominator is the difference between both temperature levels – 35°C and 0°C, so 35 K. Thus the theoretical COP is as high as 8,8!

Two silly examples:

  • Would the heat pump operate close to absolute zero, say, trying to pump heat from 5 K to 40 K, the COP would only be
    40 / 35 = 1,14.
  • On the other hand, using the sun as a heat source (6000 K) the COP would be
    6035 / 35 = 172.

So, as heat pump owners we are lucky to live in an environment rather hot compared to absolute zero, on a planet where temperatures don’t vary that much in different places, compared to how far away we are from absolute zero.


Further reading:

Richard Feynman has often used unusual approaches and new perspectives when explaining the basics in his legendary Physics Lectures. He introduces (potential) energy at the very beginning of the course drawing on Carnot’s argument, even before he defines force, acceleration, velocity etc. (!) In deriving the efficiency of an ideal thermodynamic engine many chapters later he pictured a funny machine made from rubber bands, but otherwise he follows the classical arguments:

Chapter 44 of Feynman’s Physics Lectures Vol 1, The Laws of Thermodynamics.

For an ideal gas heat energies and mechanical energies are calculated for the four steps of Carnot’s ideal process – based on the Ideal Gas Law. The result is the much more universal efficiency given above. There can’t be any better machine as combining an ideal engine with an ideal heat pump / refrigerator (the same type of machine running in reverse) would violate the second law of thermodynamics – stated as a principle: Heat cannot flow from a colder to a warmer body and be turned into mechanical energy, with the remaining system staying the same.

Pressure over Volume for Carnot’s process, when using the machine as an engine (running it counter-clockwise it describes a heat pump): AB: Expansion at constant high temperature, BC: Expansion without heat exchange (cooling), CD: Compression at constant low temperature, DA: Compression without heat exhange (gas heats up). [Wikimedia, public domain]
Feynman stated several times in his lectures that he does not want to teach history of physics or downplayed the importance of learning about history of science a bit (though it seems he was well versed in – as e.g. his efforts to follow Newton’s geometrical prove of Kepler’s Laws showed). For historical background of the evolution of Carnot’s ideas and his legacy see the the definitive resource on classical thermodynamics and its history – Peter Mander’s blog

What had puzzled me is once why we accidentally latched onto such a universal law, using just the Ideal Gas Law.The reason is that the Gas Law has the absolute temperature already included. Historically, it did take quite a while until pressure, volume and temperature had been combined in a single equation – see Peter Mander’s excellent article on the historical background of this equation.

Having explained Carnot’s Cycle and efficiency, every course in thermodynamics reveals a deeper explanation: The efficiency of an ideal engine could actually be used as a starting point defining the new scale of temperature! If Carnot engines with different efficiencies work ‘off’ the same higher reference temperature, the lower temperature can be determined by / defined by the efficiency of the ideal machine.

However, according to the following paper, Carnot did not rigorously prove that his ideal cycle would be the optimum one. But it can be done, applying variational principles – optimizing the process for maximum work done or maximum efficiency:

Carnot Theory: Derivation and Extension, paper by Liqiu Wang

12 Comments Add yours

  1. WHAT–ANOTHER one I missed. Could I have been in a coma or something, I wonder?
    Just yesterday Jake (one of my students, who also happens to be a physics major) and I were trying to explain to another non-science student why perpetual machines cannot work. We were trying to do a good job and found that in order to do it we had to backtrack way back and reintroduce the laws of thermodynamics. We based it on the KMT in a simple version. I think we got through to some extent. Of course, none of that stuff comes into our heads easily. To us, now, it seems pretty basic but that’s because we’ve been applying it for decades.

    1. I wonder if you’ve been in touch with Dave? He hasn’t been active online lately.

      1. elkement says:

        No, I haven’t – I had checked his blog explicitly in case notifications had problems.

        1. No worries. I emailed him and found that all is well but changes are afoot.

    2. elkement says:

      (I published more posts than usual in the first half of June, perhaps unexpected :-))
      I think it is always a challenge as you have to introduce some fundamentals as you go, and you have to decide which ‘laws’ you introduce as taken for granted… and you don’t want to start with the elementary forces :-) The problem in discussing with …. uhm…. motivated outsider physicists (= would-be perpetuum mobile builders) is that you might get across as the typical dogmatic scientist then.

  2. Michelle H says:

    (WordPress did drop you from my reader; I had to go back in and add you again.) I wonder, when you write a post like this, or think along the line of these questions, who is more present: the physicist or the engineer? Could you even define each role, or do they integrate too much to separate?

    1. elkement says:

      This is a great question, Michelle, especially in relation to this post. In the chapter from Feynman’ book I quoted, he notes that this research done by engineer Sadi Carnot was one of the few times in history, when hands-on research in engineering contributed to fundamental insights in physics.
      If you define ‘physics’ as just dealing with new insights into the laws of nature, then I am an engineer all the time (plus a dilettante historian of science at times, on this blog).
      But I’d say I consider myself more of a physicist, when I think about phenomena like the details of the growth of your ice cube – especially when I try to explain something beyond improving a system or process by overdoing the math and theory a bit. Most of the time I am engineer, trying to evaluate existing things, and make them work together reliably … researching devices built by humans, not nature… and above all: Consider legal and economic context in parallel to the technical solution.
      Thermodynamics is also the only subjects I had the chance to attend lectures both in a physics and engineering degree program – despite the same name and much overlap, the underlying philosophy was different. My favorite example (though a bit hyperbolic) is the definition of entropy. In physics this was all about explaining the deep connections between arrow of time, underlying statistical mechanics etc. In the engineering thermodynamics a colleague asked the professor what entropy really is – the answer was: Just consider it a calculated property that comes in handy on evaluating processes and machines (compressors for example). The former is intriguing, but the latter makes a lot of sense if you need to get things done.

      1. Michelle H says:

        There is an interesting combination of thought processes, then, which would suggest your curiosity lands you in that marginal space between and overlapping of things (I am thinking of a Venn diagram). I have seen you move around these combinations of culture, nature and technology for as long as I’ve been reading your blog, and it is interesting now to see you working in new applications which overlap all these areas yet again.

        I have a post in my reader that alludes to my interest in an architecture program that I was investigating this winter; I was drawn to it because of the combined emphasis of culture, technical skill, and sustainable environmental consciousness. It seemed to be the “arts” comparative to what you are doing. The program is a bit unusual, I thought, because the primary consideration was not the so-called aesthetics of building, but more practical in the way a building needs to equally meet both green practices and human practical requirements. When I was about to register for some classes here that would help me meet the entrance requirements of this architecture program, I was redirected by the Fine Arts advisor to reconsider maybe an engineering program instead (FA holds the belief that aesthetics should be the priority for architects, and they grasp firmly to their ownership of the pre-architecture students).

        Aside from being slightly annoyed by the myopic scope of this view, it did leave me actively thinking about the fine balances of priorities when people work on complex projects impacting multiple areas of focus, ranging from humans to environment. I’ve also been pondering the deep ethical concerns that are involved (and what is implied) when one focus is subtly given priority over the other.

        The encounter described above gives me an even greater appreciation for how very rare it is to find someone who can exist in various modes of thought almost at once (or, at least, to rapidly move among them with some consistency). There are very few people who choose to publicly model this type of thinking. This is why I ask you about your process, because you explore many different concerns with a balance of curiosity and concern.

        1. elkement says:

          An interesting comment, Michelle!! It seems we really need more Leonardo da Vincis!
          Professional life is still built upon clearly defined categories and tags. One reason I can relate to is that, unfortunately, the generalist’s label is much too often claimed by people who don’t want to deal with any details (in either discipline) – which is totally annoying if you are the so-called narrow-minded specialist who actually has to work on the problem.
          There are many new interdisciplinary degree programs now (as our European systems have been changed a lot, after bachelors’ degrees have been introduced only recently), but I feel ambiguous not to be trained in anything in depth. My pet peeve is replacing math / theory / STEM classes by something like ‘project management’ (real-live anecdote, as – quote – ‘the math-y stuff would turn applicants off’). No doubt knowledge of project management is useful in complex building projects for example – but I think such subjects should be left to post-graduate education, when you already have some experience on some job. Learned only theoretically it makes people following checklists and looking for ‘tools’ (software) too much.
          In an ideal world, however, interdisciplinary programs would encourage students to explore any of the subjects in depth on their own – also including some rigorous math and theory you cannot cover in the curriculum. I just don’t see this happening. In every degree program the most math-y / technical stuff is the weed-out class (math for physicists, chemistry for medical doctors…), and students are more than happy after they have ticked that off.

          My personal approach was not ideal, more pragmatic: Specializing on different things in a serial fashion so I could claim true expertise in the traditional way at any point of time. I admit, one of the reasons I completed another engineering degree was that I could bring the credentials to the table in case somebody asked for them (Nobody ever did – seems credibility comes from demonstrating a pilot system, not from certificates) I would not have needed it as a formal requirement, as my business license was issued based on my physics education before, and we were already knee-deep in building our system before I started this education. My professional title is Consulting Engineer in Applied Physics, as in Austria you can become a self-employed consulting engineer (maybe somewhat comparable to Professional Engineer) in any STEM discipline if you can prove you have additional know-how and experience in economics and entrepreneurship. Perhaps, this also reflects the traditional Austrian / German culture of the value of vocational training, of the university being also training for a job (we had no ‘general ed’ classes, we specialized in one subject, no major/minor), and physics being considered ‘closer’ to engineering.

          But as you noticed, I have changed, and I don’t hide my dilettante / wannabe renaissance approach anymore. After all, I had no formal education in IT at all, and did that for such a long time (including academic teaching). It was good, that computer geeks hardly ask for credentials at all as they only believe in demonstrated skills, so maybe many people believed my degrees were in computer science.
          I think I just got tired of all that show-off – the more it is recommended to polish your self-presentation, the weirder I find it and the more I like to challenge it.

          1. Michelle H says:

            Wow; lots to think over–thanks for sharing all that. I know all too well the trouble over certificates and credentials. This seems especially poignant as I’ve watched the value of my education drop from something meaningful in the workforce system to something of very little value (even a hindrance). This can be demeaning, especially given that this signifies very little about the projects I’ve done or how I can do them. But I suppose this is part of the subtle biases that form around pay grades and incomes, and that exhausting self-marketing game you have already described.

            I guess that thinking of myself in generalist terms comes from deliberately avoiding a specific priority upon approaching a project (and I suppose I need a better word now to describe this!). Often, the priorities of a problem will show up in a self-declared kind of way (i.e. teaching young kids, each is different; or, responding to a fundraising need and building a campaign to fit the situation). I’ve always poured hours of research and thinking into how to develop precise and specific skills to meet those needs; I think this is what made me very good at doing these projects. (And, all too often, I am the first one in and the last one out on almost any project I touch… so I certainly do need to distance from those label associations.) Even my undergrad studies showed this tendency. When I resolved, finally, to stay in Arts I settled into studying literary theory–Deconstruction, specifically–and took only the classes that would help build that framework. Sometimes those classes were in classical studies, or history, but they all fit a larger structure that I was creating in my mind. Yet, this was uncommon. Most of my classmates were tangled up quoting poems and racing to read more novels than the others in the program, perhaps for demonstration purposes only, but all quite generalized. I usually didn’t even finish reading all the novels in a class, as there was a lot of new theory I was learning on the side. But also, most of the faculty who taught me gave similar feedback, that I was working on the level of a doctoral candidate already. I suspect this might be true, since looking at grad studies now I found there is little of anything new the program can offer me except access to a mentor to guide self-study through the last of the hard stuff I haven’t quite done alone. (But then what?)

            But setting all that aside, I also see how this type of coding plays out in STEM. I see in my husband’s work that certificates and professional accreditations are more often used to side-step accountability, rather than be held to it; this is done by saying, ‘someone else’s problem, I’m just the [whatever you want]”. Perhaps all this only demonstrates that people are people, no matter their education or career path. I can see how the system has evolved to suit the majority. And, I suppose that being part of a minority group of thinkers, the best course is to probably hack the system without stagnating within it (as you’ve done). I suspect this is what first caught my interest when I was looking at other programs and decided to consider mathematics: interdisciplinary applications, but also allowing for a highly specialized skill set at the same time. I mostly want to dig into a problem, solve it, and hope that the next problem will be a little bit different with something new to discover. Sadly, there is a game that has to be played out in order to get to those problems… one has to look smart. And these days, too may people think literature majors aren’t smart.

  3. Efficiency has not any strict definition, it’s fundamentally a subjective concept. I often cite the example of a car, asking to students what is its efficiency. They often relevantly answer in defining it as the ratio of the mechanical work produced, over the chemical energy consumed as fuel. But in cold countries as our owns, it’s not enough because a part of the heat released by the engine is often used to warm up the car’s compartment. We can even talk about electricity used for the radio, and so on. We define efficiency as the ratio of what’s useful for us, over what we need to pay for it. As far as heat stolen from surrounding air, underground or a nearby river, is free, it is not taken into account into the expression of efficiency and the resulting number could be greater than one. No reason to worry about that. It’s the reason why I usually clearly make the difference between efficiency and effectiveness which is on the contrary an objective parameter with a deep physical meaning.

    1. elkement says:

      I fully agree! It just found it interesting as I have read / heard debates about those efficiencies – like: physicists blaming engineers not using meaningful definitions or vice versa, or especially the ‘low’ efficiency of PV panels used in political arguments ‘against solar energy’.

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