# Mr. Bubble Was Confused. A Cliffhanger.

This year we experienced a record-breaking January in Austria – the coldest since 30 years. Our heat pump system produced 14m3 of ice in the underground tank.

The volume of ice is measured by Mr. Bubble, the winner of The Ultimate Level Sensor Casting Show run by the Chief Engineer last year:

The classic, analog level sensor was very robust and simple, but required continuous human intervention:

So a multitude of prototypes had been evaluated …

The challenge was to measure small changes in level as 1 mm corresponds to about 0,15 m3 of ice.

Mr. Bubble uses a flow of bubbling air in a tube; the measured pressure increases linearly with the distance of the liquid level from the nozzle:

Mr. Bubble is fine and sane, as long as ice is growing monotonously: Ice grows from the heat exchanger tubes into the water, and the heat exchanger does not float due to buoyancy, as it is attached to the supporting construction. The design makes sure that not-yet-frozen water can always ‘escape’ to higher levels to make room for growing ice. Finally Mr. Bubble lives inside a hollow cylinder of water inside a block of ice. As long as all the ice is covered by water, Mr. Bubble’s calculation is correct.

But when ambient temperature rises and the collector harvests more energy then needed by the heat pump, melting starts at the heat exchanger tubes. The density of ice is smaller than that of water, so the water level in Mr. Bubble’s hollow cylinder is below the surface level of ice:

Mr. Bubble is utterly confused and literally driven over the edge – having to deal with this cliff of ice:

When ice is melted, the surface level inside the hollow cylinder drops quickly as the diameter of the cylinder is much smaller than the width of the tank. So the alleged volume of ice perceived by Mr. Bubble seems to drop extremely fast and out of proportion: 1m3 of ice is equivalent to 93kWh of energy – the energy our heat pump would need on an extremely cold day. On an ice melting day, the heat pump needs much less, so a drop of more than 1m3 per day is an artefact.

As long as there are ice castles on the surface, Mr. Bubble keeps underestimating the volume of ice. When it gets colder, ice grows again, and its growth is then overestimated via the same effect. Mr. Bubble amplifies the oscillations in growing and shrinking of ice.

In the final stages of melting a slab-with-a-hole-like structure ‘mounted’ above the water surface remains. The actual level of water is lower than it was before the ice period. This is reflected in the raw data – the distance measured. The volume of ice output is calibrated not to show negative values, but the underlying measurement data do:

Only when finally all ice has been melted – slowly and via thermal contact with air – then the water level is back to normal.

In the final stages of melting parts of the suspended slab of ice may break off and then floating small icebergs can confuse Mr. Bubble, too:

So how can we picture the true evolution of ice during melting? I am simulating the volume of ice, based on our measurements of air temperature. To be detailed in a future post – this is my cliffhanger!

# Ice Storage Hierarchy of Needs

Data Kraken – the tentacled tangled pieces of software for data analysis – has a secret theoretical sibling, an older one: Before we built our heat source from a cellar, I developed numerical simulations of the future heat pump system. Today this simulation tool comprises e.g. a model of our control system, real-live weather data, energy balances of all storage tanks, and a solution to the heat equation for the ground surrounding the water/ice tank.

I can model the change of the tank temperature and  ‘peak ice’ in a heating season. But the point of these simulations is rather to find out to which parameters the system’s performance reacts particularly sensitive: In a worst case scenario will the storage tank be large enough?

A seemingly fascinating aspect was how peak ice ‘reacts’ to input parameters: It is quite sensitive to the properties of ground and the solar/air collector. If you made either the ground or the collector just ‘a bit worse’, ice seems to grow out of proportion. Taking a step back I realized that I could have come to that conclusion using simple energy accounting instead of differential equations – once I had long-term data for the average energy harvesting power of the collector and ground. Caveat: The simple calculation only works if these estimates are reliable for a chosen system – and this depends e.g. on hydraulic design, control logic, the shape of the tank, and the heat transfer properties of ground and collector.

For the operations of the combined tank+collector source the critical months are the ice months Dec/Jan/Feb when air temperature does not allow harvesting all energy from air. Before and after that period, the solar/air collector is nearly the only source anyway. As I emphasized on this blog again and again, even during the ice months, the collector is still the main source and delivers most of the ambient energy the heat pump needs (if properly sized) in a typical winter. The rest has to come from energy stored in the ground surrounding the tank or from freezing water.

I am finally succumbing to trends of edutainment and storytelling in science communications – here is an infographic:

Using some typical numbers, I am illustrating 4 scenarios in the figure below, for a  system with these parameters:

• A cuboid tank of about 23 m3
• Required ambient energy for the three ice months is ~7000kWh
(about 9330kWh of heating energy at a performance factor of 4)
• ‘Standard’ scenario: The collector delivers 75% of the ambient energy, ground delivers about 18%.
• Worse’ scenarios: Either collector or/and ground energy is reduced by 25% compared to the standard.

Contributions of the three sources add up to the total ambient energy needed – this is yet another way of combining different energies in one balance.

Ambient energy needed by the heat pump in  Dec+Jan+Feb,  as delivered by the three different sources. Latent ‘ice’ energy is also translated to the percentage of water in the tank that would be frozen.

Neither collector nor ground energy change much in relation to the base line. But latent energy has to fill in the gap: As the total collector energy is much higher than the total latent energy content of the tank, an increase in the gap is large in relation to the base ice energy.

If collector and ground would both ‘underdeliver’ by 25% the tank in this scenario would be frozen completely instead of only 23%.

The ice energy is just the peak of the total ambient energy iceberg.

You could call this system an air-geothermal-ice heat pump then!

# On Photovoltaic Generators and Scattering Cross Sections

Subtitle: Dimensional Analysis again.

Our photovoltaic generator has about 5 kW rated ‘peak’ power – 18 panels with 265W each.

South-east oriented part of our generator – 10 panels. The remaining 8 are oriented south-west.

Peak output power is obtained under so-called standard testing condition – 1 kWp (kilo Watt peak) is equivalent to:

• a panel temperature of 25°C (as efficiency depends on temperature)
• an incident angle of sunlight relative to zenith of about 48°C – equivalent to an air mass of 1,5. This determines the spectrum of the electromagnetic radiation.
• an irradiance of solar energy of 1kW per square meter.

Simulated spectra for different air masses (Wikimedia, User Solar Gate). For AM 1 the path of sunlight is shortest and thus absorption is lowest.

The last condition can be rephrased as: We get 1 kW output per kW/minput. 1 kWp is thus defined as:

1 kWp = 1 kW / (1 kW/m2)

Canceling kW, you end up with 1 kWp being equivalent to an area of 1 m2.

Why is this a useful unit?

Solar radiation generates electron-hole pairs in solar cells, operated as photodiodes in reverse bias. Only if the incoming photon has exactly the right energy, solar energy is used efficiently. If the photon is not energetic enough – too ‘red’ – it is lost and converted to heat. If the photon is too blue  – too ‘ultraviolet’ – it generates electrical charges, but the greater part of its energy is wasted as the probability of two photons hitting at the same time is rare. Thus commercial solar panels have an efficiency of less than 20% today. (This does not yet say anything about economics as the total incoming energy is ‘free’.)

The less efficient solar panels are, the more of them you need to obtain a certain target output power. A perfect generator would deliver 1 kW output with a size of 1 m2 at standard test conditions. The kWp rating is equivalent to the area of an ideal generator that would generate the same output power, and it helps with evaluating if your rooftop area is large enough.

Our 4,77 kW generator uses 18 panels, about 1,61 m2 each – so 29 m2 in total. Panels’ efficiency  is then about 4,77 / 29 = 16,4% – a number you can also find in the datasheet.

There is no rated power comparable to that for solar thermal collectors, so I wonder why the unit has been defined in this way. Speculating wildly: Physicists working on solar cells usually have a background in solid state physics, and the design of the kWp rating is equivalent to a familiar concept: Scattering cross section.

An atom can be modeled as a little oscillator, driven by the incident electromagnetic energy. It re-radiates absorbed energy in all directions. Although this can be fully understood only in quantum mechanical terms, simple classical models are successful in explaining some macroscopic parameters, like the index of refraction. The scattering strength of an atom is expressed as:

[ Power scattered ] / [ Incident power of the beam / m2 ]

… the same sort of ratio as discussed above! Power cancels out and the result is an area, imagined as a ‘cross-section’. The atom acts as if it were an opaque disk of a certain area that ‘cuts out’ a respective part of the incident beam and re-radiates it.

The same concept is used for describing interactions between all kinds of particles (not only photons) – the scattering cross section determines the probability that an interaction will occur:

Particles’ scattering strengths are represented by red disks (area = cross section). The probability of a scattering event going to happen is equal to the ratio of the sum of all red disk areas and the total (blue+red) area. (Wikimedia, User FerdiBf)

# Learning General Relativity

Math blogger Joseph Nebus does another A – Z series of posts, explaining technical terms in mathematics. He asked readers for their favorite pick of things to be covered in this series, and I came up with General Covariance. Which he laid out in this post – in his signature style, using neither equations nor pop-science images like deformed rubber mattresses – but ‘just words’. As so often, he manages to explain things really well!

Actually, I asked for that term as I am in the middle of yet another physics (re-)learning project – in the spirit of my ventures into QFT a while back.

Since a while I have now tried (on this blog) to cover only the physics related to something I have both education in and hands-on experience with. Re General Relativity I have neither: My PhD was in applied condensed-matter physics – lasers, superconductors, optics – and this article by physicist Chad Orzel about What Math Do You Need For Physics? covers well what sort of math you need in that case. Quote:

I moved into the lab, and was concerned more with technical details of vacuum pumps and lasers and electronic circuits and computer data acquisition and analysis.

So I cannot find the remotest way to justify why I would need General Relativity on a daily basis – insider jokes about very peculiarly torus-shaped underground water/ice tanks for heat pumps aside.

My motivation is what I described in this post of mine: Math-heavy physics is – for me, that means a statistical sample of 1 – the best way of brazing myself for any type of tech / IT / engineering work. This positive effect is not even directly related to math/physics aspects of that work.

But I also noticed ‘on the internet’ that there is a community of science and math enthusiasts, who indulge in self-studying theoretical physics seriously as a hobby. Often these are physics majors who ended up in very different industry sectors or in management / ‘non-tech’ jobs and who want to reconnect with what they once learned.

For those fellow learners I’d like to publish links to my favorite learning resources.

There seem to be two ways to start a course or book on GR, and sometimes authors toggle between both modes. You can start from the ‘tangible’ physics of our flat space (spacetime) plus special relativity and then gradually ‘add a bit of curvature’ and related concepts. In this way the introduction sounds familiar, and less daunting. Or you could try to introduce the mathematical concepts at a most rigorous abstract level, and return to the actual physics of our 4D spacetime and matter as late as possible.

The latter makes a lot of sense as you better unlearn some things you took for granted about vector and tensor calculus in flat space. A vector must no longer be visualized as an arrow that can be moved around carelessly in space, and one must be very careful in visualizing what transforming coordinates really means.

For motivation or as an ‘upper level pop-sci intro’…

Richard Feynman’s lecture on curved space might be a very good primer. Feynman explains what curved space and curved spacetime actually mean. Yes, he is using that infamous beetle on a balloon, but he also gives some numbers obtained by back-of-the-envelope calculations that explain important concepts.

For learning about the mathematical foundations …

I cannot praise these Lectures given at the Heraeus International Winter School Gravity and Light 2015 enough. Award-winning lecturer Frederic P. Schuller goes to great lengths to introduce concepts carefully and precisely. His goal is to make all implicit assumptions explicit and avoid allusions to misguided ‘intuitions’ one might got have used to when working with vector analysis, tensors, gradients, derivatives etc. in our tangible 3D world – covered by what he calls ‘undergraduate analysis’. Only in lecture 9 the first connection is made back to Newtonian gravity. Then, back to math only for some more lectures, until finally our 4D spacetime is discussed in lecture 13.

Schuller mentions in passing that Einstein himself struggled with the advanced math of his own theory, e.g. in the sense of not yet distinguishing clearly between the mathematical structure that represents the real world (a topological manifold) and the multi-dimensional chart we project our world onto when using an atlas. It is interesting to pair these lectures with this paper on the history and philosophy of general relativity – a link Joseph Nebus has pointed to in his post on covariance.

Learning physics or math from videos you need to be much more disciplined than with plowing through textbooks – in the sense that you absolutely have to do every single step in a derivation on your own. It is easy to delude oneself that you understood something by following a derivation passively, without calculating anything yourself. So what makes these lectures so useful is that tutorial sessions have been recorded as well: Tutorial sheets and videos can be found here.
(Edit: The Youtube channel of the event has not all the recordings of the tutorial sessions, only this conference website has. It seems the former domain does not work any more, but the content is perserved at gravity-and-light.herokuapp.com)

You also find brief notes for these lectures here.

For a ‘physics-only’ introduction …

… I picked a classical, ‘legendary’ resource: Landau and Lifshitz give an introduction to General Relativity in the last third of the second volume in their Course of Theoretical Physics, The Classical Theory of Fields. Landau and Lifshitz’s text is terse, perhaps similar in style to Dirac’s classical introduction to quantum mechanics. No humor, but sublime and elegant.

Landau and Lifshitz don’t need manifolds nor tangent bundles, and they use the 3D curvature tensor of space a lot in addition to the metric tensor of 4D spacetime. They introduce concepts of differences in space and time right from the start, plus the notion of simultaneity. Mathematicians might be shocked by a somewhat handwaving, ‘typical physicist’s’ way to deal with differentials, the way vectors on different points in space are related, etc. – neglecting (at first sight, explore every footnote in detail!) the tower of mathematical structures you actually need to do this precisely.

But I would regard Lev Landau sort of a Richard Feynman of The East, so it takes his genius not make any silly mistakes by taking the seemingly intuitive notions too literally. And I recommend this book only when combined with a most rigorous introduction.

I recommend Sean Carroll’s  Lecture Notes on General Relativity from 1997 (precursor of his textbook), together with his short No-Nonsense Introduction to GR as a summary. Carroll switches between more intuitive physics and very formal math. He keeps his conversational tone – well known to readers of his popular physics books – which makes his lecture notes a pleasure to read.

__________________________________

So this was a long-winded way to present just a bunch of links. This post should also serve as sort of an excuse that I haven’t been really active on social media or followed up closely on other blogs recently. It seems in winter I am secluding myself from the world in order to catch up on theoretical physics.

# And Now for Something Completely Different: Rotation Heat Pump!

Heat pumps for space heating are all very similar: Refrigerant evaporates, pressure is increased by a scroll compressor, refrigerant condenses, pressure is reduced in an expansion value. *yawn*

The question is:

Can a compression heat pump be built in a completely different way?

Austrian start-up ECOP did it: They  invented the so-called Rotation Heat Pump.

It does not have a classical compressor, and the ‘refrigerant’ does not undergo a phase transition. A pressure gradient is created by centrifugal forces: The whole system rotates, including the high-pressure (heat sink) and low-pressure (source) heat exchanger. The low pressure part of the system is positioned closer to the center of the rotation axis, and heat sink and heat source are connected at the axis (using heating water). The system rotates at up to 1800 rounds per minute.

A mixture of noble gases is used in a Joule (Brayton) process, driven in a cycle by a ventilator. Gas is compressed and thus heated up; then it is cooled at constant pressure and energy is released to the heat sink. After expanding the gas, it is heated up again at low pressure by the heat source.

In the textbook Joule cycle, a turbine and a compressor share a common axis: The energy released by the turbine is used to drive the compressor. This is essential, as compression and expansion energies are of the same order of magnitude, and both are considerably larger than the net energy difference – the actual input energy.

In contrast to that, a classical compression heat pump uses a refrigerant that is condensed while releasing heat and then evaporated again at low pressure. There is no mini-turbine to reduce the pressure but only an expansion valve, as there is not much energy to gain.

This explains why the Rotation Heat Pumps absolutely have to have compression efficiencies of nearly 100%, compared to, say, 85% efficiency of a scroll compressor in heat pump used for space heating:

Some numbers for a Joule process (from this German ECOP paper): On expansion of the gas 1200kW are gained, but 1300kW are needed for compression – if there would be no losses at all. So the net input power is 100kW. But if the efficiency of the compression is reduced from 100% to 80% about 1600kW are needed and thus a net input power of 500kW – five times the power compared to the ideal compressor! The coefficient of performance would plummet from 10 to 2,3.

I believe these challenging requirements are why Rotation Heat Pumps are ‘large’ and built for industrial processes. In addition to the high COP, this heat pump is also very versatile: Since there are no phase transitions, you can pick your favorite corner of the thermodynamic state diagram at will: This heat pump works for very different combinations temperatures of the hot target and the cold source.

# Re-Visiting Carnot’s Theorem

The proof by contradiction used in physics textbooks is one of those arguments that appear surprising, then self-evident, then deceptive in its simplicity. You – or maybe only: I – cannot resist turning it over and over in your head again, viewing it from different angles.

tl;dr: I just wanted to introduce the time-honored tradition of ASCII text art images to illustrate Carnot’s Theorem, but this post got out of hand when I mulled about how to  refute an erroneous counter-argument. As there are still research papers being written about Carnot’s efficiency I feel vindicated for writing a really long post though.

Carnot‘s arguments prove that there is a maximum efficiency of a thermodynamic heat engine – a machine that turns heat into mechanical energy. He gives the maximum value by evaluating one specific, idealized process, and then proves that a machine with higher efficiency would give rise to a paradox. The engine uses part of the heat available in a large, hot reservoir of heat and turns it into mechanical work and waste heat – the latter dumped to a colder ‘environment’ in a 4-step process. (Note that while our modern reformulation of the proof by contradiction refers to the Second Law of Thermodynamics, Carnot’s initial version was based on the caloric theory.)

The efficiency of such an engine η – mechanical energy per cycle over input heat energy – only depends on the two temperatures (More details and references here):

$\eta_\text{carnot} = \frac {T_1-T_2}{T_1}$

These are absolute temperatures in Kelvin; this universal efficiency can be used to define what we mean by absolute temperature.

I am going to use ‘nice’ numbers. To make ηcarnot equal to 1/2, the hot temperature
T1 = 273° = 546 K, and the colder ‘environment’ has T2 = 0°C = 273 K.

If this machine is run in reverse, it uses mechanical input energy to ‘pump’ energy from the cold environment to the hot reservoir – it is a heat pump using the ambient reservoir as a heat source. The Coefficient of Performance (COP, ε) of the heat pump is heat output over mechanical input, the inverse of the efficiency of the corresponding engine. εcarnot is 2 for the temperatures given above.

If we combine two such perfect machines – an engine and a heat pump, both connected to the hot space and to the cold environment, their effects cancel out: The mechanical energy released by the engine drives the heat pump which ‘pumps back’ the same amount of energy.

In the ASCII images energies are translated to arrows, and the number of parallel arrows indicates the amount of energy per cycle (or power). For each device, the number or arrows flowing in and out is the same; energy is always conserved. I am viewing this from the heat pump’s perspective, so I call the cold environment the source, and the hot environment room.

Neither of the heat reservoirs are heated or cooled in this ideal case as the same amount of energy flows from and to each of the heat reservoirs:

|----------------------------------------------------------|
|         Hot room at temperature T_1 = 273°C = 546 K      |
|----------------------------------------------------------|
| | | |                         | | | |
v v v v                         ^ ^ ^ ^
| | | |                         | | | |
|------------|                 |---------------|
|   Engine   |->->->->->->->->-|   Heat pump   |
|  Eta = 1/2 |->->->->->->->->-| COP=2 Eta=1/2 |
|------------|                 |---------------|
| |                             | |
v v                             ^ ^
| |                             | |
|----------------------------------------------------------|
|        Cold source at temperature T_2 = 0°C = 273 K      |
|----------------------------------------------------------|

If either of the two machines works less than perfectly and in tandem with a perfect machine, anything is still fine:

If the engine is far less than perfect and has an efficiency of only 1/4 – while the heat pump still works perfectly – more of the engine’s heat energy input is now converted to waste heat and diverted to the environment:

|----------------------------------------------------------|
|         Hot room at temperature T_1 = 273°C = 546 K      |
|----------------------------------------------------------|
| | | |                           | |
v v v v                           ^ ^
| | | |                           | |
|------------|                 |---------------|
|   Engine   |->->->->->->->->-|   Heat pump   |
|  Eta = 1/4 |                 | COP=2 Eta=1/2 |
|------------|                 |---------------|
| | |                             |
v v v                             ^
| | |                             |
|----------------------------------------------------------|
|        Cold source at temperature T_2 = 0°C = 273 K      |
|----------------------------------------------------------|

Now two net units of energy flow from the hot room to the environment (summing up the arrows to and from the devices):

|----------------------------------------------------------|
|         Hot room at temperature T_1 = 273°C = 546 K      |
|----------------------------------------------------------|
| |
v v
| |
|------------------|
|   Combination:   |
| Eta=1/4 COP=1/2  |
|------------------|
| |
v v
| |
|----------------------------------------------------------|
|        Cold source at temperature T_2 = 0°C = 273 K      |
|----------------------------------------------------------|

Using a real-live heat pump with a COP of 3/2 (< 2) together with a perfect engine …

|----------------------------------------------------------|
|         Hot room at temperature T_1 = 273°C = 546 K      |
|----------------------------------------------------------|
| | | |                             | | |
v v v v                             ^ ^ ^
| | | |                             | | |
|------------|                 |-----------------|
|   Engine   |->->->->->->->->-|    Heat pump    |
|  Eta = 1/2 |->->->->->->->->-|     COP=3/2     |
|------------|                 |-----------------|
| |                                 |
v v                                 ^
| |                                 |
|----------------------------------------------------------|
|        Cold source at temperature T_2 = 0°C = 273 K      |
|----------------------------------------------------------|

… causes again a non-paradoxical net flow of one unit of energy from the room to the environment.

In the most extreme case  a poor heat pump (not worth this name) with a COP of 1 just translates mechanical energy into heat energy 1:1. This is a resistive heating element, a heating rod, and net heat fortunately flows from hot to cold without paradoxes:

|----------------------------------------------------------|
|         Hot room at temperature T_1 = 273°C = 546 K      |
|----------------------------------------------------------|
| |                                |
v v                                ^
| |                                |
|------------|                 |-----------------|
|   Engine   |->->->->->->->->-|   'Heat pump'   |
|  Eta = 1/2 |                 |     COP = 1     |
|------------|                 |-----------------|
|
v
|
|----------------------------------------------------------|
|        Cold source at temperature T_2 = 0°C = 273 K      |
|----------------------------------------------------------|

The textbook paradox in encountered, when an ideal heat pump is combined with an allegedly better-than-possible engine, e.g. one with an efficiency:

ηengine = 2/3 (> ηcarnot = 1/2)

|----------------------------------------------------------|
|         Hot room at temperature T_1 = 273°C = 546 K      |
|----------------------------------------------------------|
| | |                           | | | |
v v v                           ^ ^ ^ ^
| | |                           | | | |
|------------|                 |---------------|
|   Engine   |->->->->->->->->-|   Heat pump   |
|  Eta = 2/3 |->->->->->->->->-| COP=2 Eta=1/2 |
|------------|                 |---------------|
|                               | |
v                               ^ ^
|                               | |
|----------------------------------------------------------|
|        Cold source at temperature T_2 = 0°C = 273 K      |
|----------------------------------------------------------|

The net effect / heat flow is then:

|----------------------------------------------------------|
|        Hot room at temperature T_1 = 273°C = 546 K       |
|----------------------------------------------------------|
|
^
|
|------------------|
|   Combination:   |
| Eta=3/2; COP=1/2 |
|------------------|
|
^
|
|----------------------------------------------------------|
|       Cold source at temperature T_2 = 0°C = 273 K       |
|----------------------------------------------------------|

One unit of heat would flow from the environment to the room, from the colder to the warmer body without any other change being made to the system. The combination of these machines would violate the Second Law of Thermodynamics; it is a Perpetuum Mobile of the Second Kind.

If the heat pump has a higher COP than the inverse of the perfect engine’s efficiency, a similar paradox arises, and again one unit of heat flows in the forbidden direction:

|----------------------------------------------------------|
|         Hot room at temperature T_1 = 273°C = 546 K      |
|----------------------------------------------------------|
| |                             | | |
v v                             ^ ^ ^
| |                             | | |
|------------|                 |---------------|
|   Engine   |->->->->->->->->-|   Heat pump   |
|  Eta = 1/2 |                 |    COP = 3    |
|------------|                 |---------------|
|                               | |
v                               ^ ^
|                               | |
|----------------------------------------------------------|
|        Cold source at temperature T_2 = 0°C = 273 K      |
|----------------------------------------------------------|

A weird question: Can’t we circumvent the paradox if we pair the impossible superior engine with a poor heat pump?

|----------------------------------------------------------|
|         Hot room at temperature T_1 = 273°C = 546 K      |
|----------------------------------------------------------|
| | |                             | |
v v v                             ^ ^
| | |                             | |
|------------|                 |---------------|
|   Engine   |->->->->->->->->-|   Heat pump   |
|  Eta = 2/3 |->->->->->->->->-|    COP = 1    |
|------------|                 |---------------|
|
v
|
|----------------------------------------------------------|
|        Cold source at temperature T_2 = 0°C = 273 K      |
|----------------------------------------------------------

Indeed: If the COP of the heat pump (= 1) is smaller than the inverse of the engine’s efficiency (3/2), there will be no apparent violation of the Second Law – one unit of net heat flows from hot to cold.

An engine with low efficiency 1/4 would ‘fix’ the second paradox involving the better-than-perfect heat pump:

|----------------------------------------------------------|
|         Hot room at temperature T_1 = 273°C = 546 K      |
|----------------------------------------------------------|
| | | |                          | | |
v v v v                          ^ ^ ^
| | | |                          | | |
|------------|                 |---------------|
|   Engine   |->->->->->->->->-|   Heat pump   |
|  Eta = 1/4 |                 |     COP=3     |
|------------|                 |---------------|
| | |                            | |
v v v                            ^ ^
| | |                            | |
|----------------------------------------------------------|
|        Cold source at temperature T_2 = 0°C = 273 K      |
|----------------------------------------------------------|

But we cannot combine heat pumps and engines at will, just to circumvent the paradox – one counter-example is sufficient: Any realistic engine combined with any realistic heat pump – plus all combinations of those machines with ‘worse’ ones – have to result in net flow from hot to cold …

The Second Law identifies such ‘sets’ of engines and heat pumps that will all work together nicely. It’s easier to see this when all examples are condensed into one formula:

The heat extracted in total from the hot room – Q1 –  is the difference of heat used by the engine and heat delivered by the heat pump, both of which are defined in relation to the same mechanical work W:

$Q_1 = W\left (\frac{1}{\eta_\text{engine}}-\varepsilon_\text{heatpump}\right)$

This is also automatically equal to Qas another quick calculation shows or by just considering that energy is conserved: Some heat goes into the combination of the two machines, part of it – W – flows internally from the engine to the heat pump. But no part of the input Q1 can be lost, so the output of the combined machine has to match the input. Energy ‘losses’ such as energy due to friction will flow to either of the heat reservoirs: If an engine is less-then-perfect, more heat will be wasted to the environment; and if the heat pump is less-than-perfect a greater part of mechanical energy will be translated to heat only 1:1. You might be even lucky: Some part of heat generated by friction might end up in the hot room.

As Q1 has to be > 0 according to the Second Low, the performance numbers have to related by this inequality:

$\frac{1}{\eta_\text{engine}}\geq\varepsilon_\text{heatpump}$

The equal sign is true if the effects of the two machines just cancel each other.

If we start from a combination of two perfect machines (ηengine = 1/2 = 1/εheatpump) and increase either ηengine or εheatpump, this condition would be violated and heat would flow from cold to hot without efforts.

But also an engine with efficiency = 1 would work happily with the worst heat pump with COP = 1. No paradox would arise at first glance  – as 1/1 >= 1:

|----------------------------------------------------------|
|         Hot room at temperature T_1 = 273°C = 546 K      |
|----------------------------------------------------------|
|                                |
v                                ^
|                                |
|------------|                 |-----------------|
|   Engine   |->->->->->->->->-|   'Heat pump'   |
|   Eta = 1  |                 |      COP=1      |
|------------|                 |-----------------|

|----------------------------------------------------------|
|        Cold source at temperature T_2 = 0°C = 273 K      |
|----------------------------------------------------------|

What’s wrong here?

Because of conservation of energy ε is always greater equal 1; so the set of valid combinations of machines all consistent with each other is defined by:

$\frac{1}{\eta_\text{engine}}\geq\varepsilon_\text{heatpump}\geq1$

… for all efficiencies η and COPs / ε of machines in a valid set. The combination η = ε = 1 is still not ruled out immediately.

But if the alleged best engine (in a ‘set’) would have an efficiency of 1, then the alleged best heat pump would have an Coefficient of Performance of only 1 – and this is actually the only heat pump possible as ε has to be both lower equal and greater equal than 1. It cannot get better without creating paradoxes!

If one real-live heat pump is found that is just slightly better than a heating rod – say
ε = 1,1 – then performance numbers for the set of consisent, non-paradoxical machines need to fulfill:

$\eta_\text{engine}\leq\eta_\text{best engine}$

and

$\varepsilon_\text{heatpump}\leq\varepsilon_\text{best heatpump}$

… in addition to the inequality relating η and ε.

If ε = 1,1 is a candidate for the best heat pump, a set of valid machines would comprise:

• All heat pumps with ε between 1 and 1,1 (as per limits on ε)
• All engines with η between 0 and 0,9 (as per inequality following the Second Law plus limit on η).

Consistent sets of machines are thus given by a stronger condition – by adding a limit for both efficiency and COP ‘in between’:

$\frac{1}{\eta_\text{engine}}\geq\text{Some Number}\geq\varepsilon_\text{heatpump}\geq1$

Carnot has designed a hypothetical ideal heat pump that could have a COP of εcarnot = 1/ηcarnot. It is a limiting case of a reversible machine, but feasible on principle. εcarnot  is thus a valid upper limit for heat pumps, a candidate for Some Number. In order to make this inequality true for all sets of machines (ideal ones plus all worse ones) then 1/ηcarnot = εcarnot also constitutes a limit for engines:

$\frac{1}{\eta_\text{engine}}\geq\frac{1}{\eta_\text{carnot}}\geq\varepsilon_\text{heatpump}\geq1$

So in order to rule out all paradoxes, Some Number in Between has to be provided for each set of machines. But what defines a set? As machines of totally different making have to work with each other without violating this equality, this number can only be a function of the only parameters characterizing the system – the two temperatures

Carnot’s efficiency is only a function of the temperatures. His hypothetical process is reversible, the machine can work either as a heat pump or an engine. If we could come up with a better process for a reversible heat pump (ε > εcarnot), the machine run in reverse would be an engine with η less than ηcarnot, whereas a ‘better’ engine would lower the upper bound for heat pumps.

If you have found one truly reversible process, both η and ε associated with it are necessarily the upper bounds of performance of the respective machines, so you cannot push Some Number in one direction or the other, and the efficiencies of all reversible engines have to be equal – and thus equal to ηcarnot. The ‘resistive heater’ with ε = 1 is the iconic irreversible device. It will not turn into a perfect engine with η = 1 when ‘run in reverse’.

The seemingly odd thing is that 1/ηcarnot appears like a lower bound for ε at first glance if you just declare ηcarnot an upper bound for corresponding engines and take the inverse, while in practice and according to common sense it is the maximum value for all heat pumps, including irreversible ones. (As a rule of thumb a typical heat pump for space heating has a COP only 50% of 1/ηcarnot.)

But this ‘contradiction’ is yet another way of stating that there is one universal performance indicator of all reversible machines making use of two heat reservoirs: The COP of a hypothetical ‘superior’ reversible heat pump would be at least 1/ηcarnot  … as good as Carnot’s reversible machine, maybe better. But the same is true for the hypothetical superior engine with an efficiency of at least ηcarnot. So the performance numbers of all reversible machines (all in one set, characterized by the two temperatures) have to be exactly the same.

Historical piston compressor (from the time when engines with pistons looked like the ones in textbooks), installed 1878 in the salt mine of Bex, Switzerland. 1943 it was still in operation. Such machines used in salt processing were considered the first heat pumps.

# Rowboats, Laser Pulses, and Heat Energy (Boring Title: Dimensional Analysis)

Dimensional analysis means to understand the essentials of a phenomenon in physics and to calculate characteristic numbers – without solving the underlying, often complex, differential equation. The theory of fluid dynamics is full of interesting dimensionless numbers –  Reynolds Number is perhaps most famous.

In the previous post on temperature waves I solved the Heat Equation for a very simple case, in order to answer the question How far does solar energy get into ground in a year? Reason: I have been working on simulations of our heat pump system since a few years. This also involves heat transport between the water/ice tank and ground. If you set out to simulate a complex phenomenon you have to make lots of assumptions about materials’ parameters, and you have to simplify the system and equations you use for modelling the real world. You need a way of cross-checking if your results sound plausible in terms of orders of magnitude. So my goal has been to find yet another method to confirm assumptions I have made about the thermal properties of ground elsewhere.

Before I am going to revisit heat transport, I’ll try to explain what dimensional analysis is – using the best example I’ve ever seen. I borrow it from theoretical physicist – and awesome lecturer – David Tong:

How does the speed of a rowing boat depend in the number of rowers?

References: Tong’s lecture called Dynamics and Relativity (Chapter 3), This is the original paper from 1971 Tong quotes: Rowing: A similarity analysis.

The boat experiences a force of friction in water. As for a car impeded by the friction of the surrounding air, the force of friction depends on velocity.

Force is the change of momentum, momentum is proportional to mass times velocity. Every small ‘parcel’ of water carries a momentum proportional to speed – so force should at least be proportional to one factor of v. But these parcel move at a speed v, so the faster they move the more momentum is exchanged with the boat; so there has to be a second factor of v, and force is proportional to the square of the speed of the boat.

The larger the cross-section of the submerged part of the boat, A, the higher is the number of collisions between parcels of water and the boat, so putting it together:

$F \sim v^{2}A$

Rowers need to put in power to compensate for friction. Power is energy per time, and Energy is force times distance. Since distance over time is velocity, thus power is also force times velocity.

So there is one more factor of v to be included in power:

$P \sim v^{3}A$

For the same reason wind power harvested by wind turbines is proportional to the third power of wind speed.

A boat does not sink because downward gravity and upward buoyancy just compensate each other; buoyancy is the weight of the volume of water displaced. The heavier the load, the more water needs to be displaced. The submerged volume of the boat V is proportional to the weight of the rowers, and thus to their number N if the mass of the boat itself is negligible:

$V \sim N$

The volume of something scales with the third power of its linear dimensions – think of a cube or a sphere; so the surface area scales with the square of the length, and the cross-section A scales with V – and thus with N:

$A \sim N^{\frac{2}{3}}$

Each rower contributes the same share to the total rowing power, so:

$P \sim N$

Inserting for A in the first expression for P:

$P \sim v^{3} N^{\frac{2}{3}}$

Eliminating P as it has been shown to be proportional to N:

$N \sim v^{3} N^{\frac{2}{3}}$
$v^{3} \sim N^{\frac{1}{3}}$
$v \sim N^{\frac{1}{9}}$

… which is in good agreement with measurements according to Tong.

Heat Transport and Characteristic Lengths

In the last post I’ve calculated characteristic lengths, describing how heat is slowly dissipated in ground: 1) The wavelength of the damped oscillation and 2) the run-out length of the enveloping exponential function.

Both are proportional to the square root of a simple number:

$l \sim \sqrt{D \tau}$

… the factor of proportionality being ‘small’ on a logarithmic scale, like π or 2 or their inverse. τ is the period, and D was a number expressing how well the material carries away heat energy.

There is another ‘simple’ scenario that also results in a length scale described by
$\sqrt{D \tau}$ times a small number: If you deposit a confined ‘lump of heat’, a ‘point heat’ it will peter out and the average width of the lump after some time τ is about this length as well.

Using very short laser pulse to heat solid material is very close to depositing ‘point heat’. Decades ago I worked with pulsed excimer lasers, used for ablation (‘shooting off) material from ceramic targets.This type of lasers is used in eye surgery today:

Heat is deposited in nanosecond pulses, and the run-out length of the heat peak in the material is about $\sqrt{D \tau}$ with tau being equal to the very short laser’s pulse length of several nanoseconds. As the pulse duration is short, the penetration depth is short as well, and tissue is ‘cut’ precisely without heating much of the underlying material.

So this type of $\sqrt{D \tau}$ length is not just a result of a calculation for a specific scenario, but it rather seems to encompass important characteristics of heat conduction as such.

The unit of D is area over time, m2/s. If you accept the heat equation as a starting point, analysing the dimensions involved by counting x and t you see that D has to contain two powers of x and one of t. Half of applied physics and engineering is about getting units right.

But I pretend I don’t even know the heat equation and ‘visualize’ heat transport in this way: ‘Something’ – like heat energy – is concentrated in space and closely peters out. The spreading out is faster, the more concentrated it is. A thin needle-like peak quickly becomes a rounded hill, and then is flattened gradually. Concentration in space means curvature. The smaller the space occupied by the lump of heat is, the smaller its radius, the higher its curvature as curvature is the inverse of the radius of a tangential circular path.

I want to relate curvature to the change with time. Change in time has to be measured in units including the inverse of time, curvature is the inverse of space. Equating those, you have to come with something including the square of spatial dimension and one temporal dimension – something like D [m2/s].

How to get a characteristic length from this? D has to be multiplied by a characteristic time, and then we need to take a the square root. So we need to put in some characteristic time, that’s a property of the specific system investigated and not of the equation – like the yearly period or the laser pulse. And the resulting length is exactly that $l \sim \sqrt{D \tau}$ that shows up in any of of the solutions for specific scenarios.

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The characteristic width of the spreading lump of heat is visible in the so-called Green’s functions. These functions described a system’s response to a ‘source’ which resemble a needle-like peak in time. In this case it is a Gaussian functions with a ‘width’ $\sim \sqrt{D \tau}$. See e.g. equation (14) on PDF-page 14 of these lecture notes.