Can the Efficiency Be Greater Than One?

This is one of the perennial top search terms for this blog.

Anticlimactic answer: Yes, because input and output are determined also by economics, not only by physics.

Often readers search for the efficiency of a refrigerator. Its efficiency, the ratio of output and input energies, is greater than 1 because the ambient energy is free. System’s operators are interested in the money they pay the utility, in relation to the resulting energy for cooling.

If you use the same thermodynamic machine either as a refrigerator or as a heat pump, efficiencies differ: The same input energy drives the compressor, but the relevant output energy is either the energy released to the ‘hot side’ at the condenser or the energy used for evaporating the refrigerant at the ‘cool side’:

The same machine / cycle is used as a heat pump for heating (left) or a refrigerator or AC for cooling (right). (This should just highlight the principles and does not include any hydraulic details, losses etc. related to detailed differences between refrigerators / ACs and heat pumps.)

For photovoltaic panels the definition has sort of the opposite bias: The sun does not send a bill – as PV installers say in their company’s slogan – but the free solar ambient energy is considered, and thus their efficiency is ‘only’ ~20%.

Half of our generator, now operational for three years: 10 panels, oriented south-east, 265W each, efficiency 16%. (The other 8 panels are oriented south-west).

When systems are combined, you can invent all kinds of efficiencies, depending on system boundaries. If PV panels are ‘included’ in a heat pump system (calculation-wise) the nominal electrical input energy becomes lower. If solar thermal collectors are added to any heating system, the electrical or fossil fuel input decreases.

Output energy may refer to energy measured directly at the outlet of the heat pump or boiler. But it might also mean the energy delivered to the heating circuits – after the thermal losses of a buffer tank have been accounted for. But not 100% of these losses are really lost, if the buffer tank is located in the house.

I’ve seen many different definitions in regulations and related software tools, and you find articles about how to game interpret these guidelines to your advantage. Tools and standards also make arbitrary assumptions about storage tank losses, hysteresis parameter and the like – factors that might be critical for efficiency.

Then there are scaling effects: When the design heat loads of two houses differ by a factor of 2, and the smaller house would use a scaled down heat pump (hypothetically providing 50% output power at the same efficiency), the smaller system’s efficiency is likely to be a bit lower. Auxiliary consumers of electricity – like heating circuit pumps or control systems – will not be perfectly scalable. But the smaller the required output energy is, the better it can be aligned with solar energy usage and storage by a ‘smart’ system – and this might outweigh the additional energy needed for ‘smartness’. Perhaps intermittent negative market prices of electricity could be leveraged.

Definitions of efficiency are also culture-specific, tailored to an academic discipline or industry sector. There are different but remotely related concepts of rating how useful a source of energy is: Gibbs Free Energy is the maximum work a system can deliver, given that pressure and temperature do not change during the process considered – for example in a chemical reaction. On the other hand, Exergy is the useful ‘available’ energy ‘contained’ in a (part of a) system: Sources of energy and heat are rated; e.g. heat energy is only mechanically useful up to the maximum efficiency of an ideal Carnot process. Thus exergy depends on the temperature of the environment where waste heat ends up. The exergy efficiency of a Carnot process is 1, as waste heat is already factored in. On the other hand, the fuel used to drive the process may or may not be included and it may or may not be considered pure exergy – if it is, energy and exergy efficiency would be the same again. If heat energy flows from the hot to the cold part of a system in a heat exchanger, no energy is lost – but exergy is.

You could also extend the system’s boundary spatially and on the time axis: Include investment costs or the cost of harm done to the environment. Consider the primary fuel / energy / exergy to ‘generate’ electricity: If a thermal power plant has 40% efficiency then the heat pump’s efficiency needs to be at least 2,5 to ‘compensate’ for that.

In summary, ‘efficiency’ is the ratio of an output and an input energy, and the definitions may be rather arbitrary as and these energies are determined by a ‘sampling’  time, system boundaries, and additional ‘ratings’.

Einstein and His Patents

No, this is not about Einstein’s achievements as a moonlighting scientific paradigm shifter, while working as a patent examiner in his day job.

Albert Einstein Head

Albert Einstein (Wikimedia)

Einstein is famous for the theories of special and general relativity, and for the correct explanation of the photoelectric effect that has been rewarded with the Nobel prize. It is not so common knowledge that he contributed to the theory of Brownian motion, and found a new way of deducing Max Planck’s famous formula for the intensity of blackbody radiation – a prerequisite for an important invention of the 20th century: the laser.

I had graduated in physics, but I was ignorant about Einstein being an avid inventor himself until I read this biography by Jürgen Neffe and Shelley Frisch.

Einstein had spent part of his youth living near or literally in the electrical engineering company operated by his father and his uncle. As a child Albert Einstein became familiar with resistors, magnets, capacitors, light bulbs, generators, and engines.

Later Einstein worked on improvements of a gyrocompass as a technical expert in a patent dispute,  and together with his student Leo Szilard he filed 17 patents in German and some internationall patents on a refrigerator based on gas absorption. Besides, Einstein invented an automatic camera, and he even tried to optimize air plane wings and torpedos (the latter with not too much success, according to Neffe).

The refrigerator patent has been recovered at the end of the 20st century and prototypes have been built recently.  An excellent account of the history of the invention and its inner workings can be found on  Cocktail Party Physics, and the device is analyzed in all thermodynamic details here.

So Einstein’s early exposure to engineering gadgets might have triggered a live-long interest in “building real stuff”. Nevertheless, one might also trace the inventor’s spirit in his theoretical works: Remember the thought experiments used to explain relativity – a world comprising trains, mirrors, clocks, space ships, and falling elevators. A deterministic thinker’s paradise?

Could we dare to speculate that his life-long qualms with the strange nature of the quantum world is due to his engineering mindset (despite the fact that Einstein laid part of the foundations of quantum mechanics)?

The authors of the biography quoted above assume that Einstein’s informal training in engineering prepared him well for his future job as a patent examiner. He was able to nail down the issues with machine defying the laws of physics . If this is true, the process of issuance of patents has changed considerably. I have learned that patents are not granted primarily for the fact that devices do indeed work but for novelty; so it is OK to invent an alleged perpetuum mobile as long as its design is novel. But I digress.

I am finally happy to add Einstein to the list of physicists who contributed to fundamentals of physics and who worked hands-on. I remember Enrico Fermi being often called the last physicist who was both a theorist and an experimental physicist.  And there is of course Richard Feynman – remember the O-ring demo.

Back to the famous fridge:

It always takes me some time to wrap my head around cooling machines that do not use mechanical compressors. Fortunately, I found this annotated drawing of the Einstein-Szilard machine:

Einstein Refrigerator pat1781541 clarified

… which is explained in detail here. [Edit 2017: Linked to archive.org as original server / site does not work anymore.] How does it work?

Generally, any fridge or heat pump (regardless of process details) is based on closed cycle process including

  • condensation of a refrigerant at high temperature and high pressure – at the rear side of the fridge or when a heat pump transfers energy to the water used for heating
  • evaporation at low temperatures and low pressures – inside the fridge or when a heat draws energy from the heat source.

The pressure difference can be due to a mechanical compressor driven by electrical power. But instead of utilizing the total pressure of a single refrigerant, the partial pressure of a gas in a mixture of gases will do. Ammonia can be use used as a refrigerant, and its partial pressure is increased by evaporating ammonia, heating a mixture of ammonia and water. So the energy needed to drive such an absorption process is heat, not electrical energy.

So why is the Einstein-Szilard process even more complicated and based on three substances – ammonia, butane, and water?

Butane is the actual refrigerant, thus cooling is done when the partial pressure of butane is reduced, and butane evaporates at this reduced vapor pressure (in the Evaporator). The total pressure is nearly constant in this process: The partial pressure of butane is reduced by adding ammonia (from the Generator).

In the Condenser the butane partial pressure is increased by removing ammonia from the vapor. This is where water is needed: Water is sprayed into the ammonia-butane vapor, and ammonia is absorbed by the water droplets (due to its large affinity to water). There are two mixtures of liquids in the condenser: butane-ammonia and ammonia-water, and they are separated due to differences in density.

Finally: where is the external energy (heat) fed into the process? Heat is required to expel ammonia from the ammonia-water mixture (in the Generator) that is collected at the bottom of the Condenser.

Why Do Heat Pumps Pump Energy so Easily?

I know my posts are usually walls of text, but I am trying to improve!

In his landmark physics course, the Feynman Lectures on Physics, Richard Feynman tries to explain what an explanation in physics actually is. You can always understand “the math” and follow a proof step-by-step. Nevertheless, this does not necessarily mean that you have really grasped the clues. Real deep, but yet intuitive understanding becomes harder and harder the more abstract the concepts become. Feynman claims that physicists do not really understand on a fundamental level what cannot be explained to freshman students. He himself has invented ingenious twists and lines of reasoning in order to explain seemingly old and dull physics.

I can only second that. Trying to find those brief, intuitive but exact explanations is an obsession of mine, and I plan to report on my attempts as a would-be-Feynman in my blog.

I start with an arbitrary question: A heat pump is often said to work like a refrigerator, but “just the other way round”. I do not like this explanation too much, as “the other way round” is misleading (actually a heat pump is in some sense also a steam power plant, just working “the other way round”) and it does above all not explain how a refrigerator works. Heat pumps sort of enhance the energy retrieved from the “cold” ambient environment, typically you gain about 4 times as much heating energy – available to you internal heating devices – as you put in electric energy. A super optimized machine would be able to enhance energy by about a factor of 8. Where does this come from? Is this sort of a perpetuum mobile?

No, this is actually due to a fundamental natural constant: The marvellous enhancement factor is due to the fact that also the so-called cold environment is still rather hot compared to the absolute zero temperature of -273,15°C. There is some amount of internal energy connected to virtually anything and this energy reaches a zero point when all of this energy has been extracted.

If we would try to enhance heat from a cold asteroid in outer space – an asteroid without an atmosphere and far away from shining stars, with a temperature near the temperature of cold space, about 3K – then the enhancement factor is very close to 1. In winter (on earth) the heat pump transfers heat from hot 273K (0°C) to a little bit more hot 298K (25°C). If the heat pump would operate on the surface of the sun (5000K) and increase the temperature to 5020K, the multiplication factor would be 251!

As a sidenote a perfectly operating heat pump could be used to explain what temperature actually is or better: why it is just natural to measure the temperature in Kelvin. The limit of -273,15°C has originally been discovered on studying the thermal expansion of gases (extrapolating backwards to low temperatures). But you can start the line of reasoning from describing a  perfectly operating heat pump – and you would define the temperature scale as natural that puts an end to the multiplication capabilities of the heat pump.

Now the question is: Why is there a limited amount of internal energy? Haven’t we learned that energy is something that makes only sense if measured in relation to another “state” anyway (such as measuring a change in gravitational energy)?

It seems I have again failed to keep my post short anyway, so I will leave this question to a future post!

3-ton Slinky Loop

Slinky-shaped ground loops, for a geothermal heat pump (Wikimedia, User Mark Johnson – Marktj)