You Never Know

… when obscure knowledge comes in handy!

You can dismantle an old gutter without efforts, and without any special tools:

Just by gently setting it into twisted motion, effectively applying ~1Hz torsion waves that would lead to fatigue break within a few minutes.

I knew my stint in steel research in the 1990s would finally be good for something.

If you want to create a meme from this and tag it with Work Smart Not Harder, don’t forget to give me proper credits.

No, You Cannot ‘Power Your Home’ by One Hour of Cycling Daily

In the past days different versions of an article had popped up in my social media streams again and again – claiming that you could power your home for 24 hours by cycling for one hour.

Regular readers know that I craft my statements carefully in articles about energy, nearly as in the old times when submitting a scientific paper to a journal, with lots of phrases like Tentatively, we assume…

But in this case, I cannot say it more politely or less distinctly:

No, you cannot power your home by one hour of cycling unless the only electrical appliance in your home is the equivalent of one energy-efficient small computer. I am excluding heating and cooling anyway.

Yes, I know the original article targeted people without access to the power grid. But this information seems to have been lost in uncritical reshares with catchy headlines. Having seen lots of people – whose ‘Western’ homes will never be powered by a treadmill – discussing and cheering this idea, I want to contribute some numbers [*].

This is all the not-exactly-rocket-science math you need, so authors not adding conclusive numbers to their claims have no excuses:

Energy in kWh = Power in Watts times hours divided by 1000

Then you need to be capable to read off your yearly kWh from your utility bill, divide by 365, and/or spot the power in Watts indicated on appliances or to be googled easily.

A professional athlete can cycle at several 100 Watts for some minutes (only) and he just beats a toaster (which needs a power of 500-1000W):

So an average person cannot cycle at more than 100-200W for one hour, delivering 0,2kWh during that hour at best.

With that energy you can power a 20W notebook or light bulb for 10 hours, and nothing more.

Anything with rotating parts like water well pumps, washing machines, or appliances for cutting or mixing need much more power than that, usually a few 100W. Cycling for one hour can drive one device like that for less than half an hour.

An electric stove or a water heater needs about 2kW peak power, at half of the maximum such appliances would consume 1kWh in one hour. An energy-efficient small fridge needs 0,5kWh per day, a large one up to 4kWh.

A TV set could need 150W[**], so you might just be able to power it while watching. I don’t say that this is a bad idea – but it is just very different from ‘powering your home’.

I’ll not link those click-bait articles but an excellent website instead (for the US): Here you can estimate your daily consumption, by picking all your appliances from a list, and learn about the power each one needs. At least it should give you some feeling for the numbers, to be compared with the utility bill, and to identify the most important suckers for energy.

I have scrutinized our base load consumption in this article: In summer (without space heating) our house needs about 10kWh of electrical energy per day, including 1-2 kWh for heating of hot water by the heat pump. The base load – what the house needs when we are away – is about 4kWh per day.

There are numerous articles with energy statistics for different countries, I pick one at random, stating – in line with many others – that a German household needs about 10kWh per day and one in the US about 30kWh. But even for Nigeria the average value per home is about 1,5kWh, several times the output of one hour of cycling.


[*] I’ve added this paragraph on Feb. 8 for clarification as the point came up in some discussions on my post.

[**] Depends on size, see for example this list for TVs common in Germany. I was rather thinking of a bigger one, in line with the typical values given also by the US Department of Energy (300W for a plasma TV!).

Heat Pump System Data: Three Seasons 2012 – 2015

We have updated the documentation of monthly and seasonal measurement data – now including also the full season September 2014 to August 2015.

The overall Seasonal Performance Factor was 4,4 – despite the slightly lower numbers in February and March, when was the solar collector was off during the Ice Storage Challenge.

Edit: I have learned from a question that the SPF is also calculated in BTU/Wh. ‘Our’ SPF uses the same units in nominator and denominator, so 4,4 is in Wh/Wh. The conversion factor is about 3,4 (note that I use a decimal comma BTW), so our SPF [kWh/kWh] is equivalent to an SPF [BTU/Wh] ~ 15.

Monthly Performance Factor, Heat Pump System

Monthly heating energy provided by the heat pump – total of both space heating and hot water water, related electrical input energy, and the ratio = monthly performance factor. The SPF is in kWh/kWh.

The SPF determines economics of heating with a heat pump.

It’s time to compare costs again, based on current minimum prices of electricity and natural gas in our region in Austria (published by regulator e-control):

  • We need about 20.000 kWh (*) of heating energy per year.
  • Assuming a nearly perfect gas boiler with an efficiency of 95%, we would need about 21.050 kWh of gas.
  • Cost of natural gas incl. taxes, grid fees: ~ 0,0600 € / kWh
  • Yearly energy costs for heating with gas would be: € 1.260
  • Given an SPF of 4,4 for the heat pump, 20.000 kWh heating energy demands translate to 4.545 kWh of electrical energy.
  • Costs of electricity incl. taxes, grid: ~ 0,167 € / kWh
  • Yearly energy costs for heating with the heat pump: € 760
  • Yearly savings with the heat pump: € 500 or 40% of the costs of gas.

(*) As indicated in the PDF, In the past year only the ground floor was heated by the heat pump. So we needed only 13.300 kWh. In the first floor we got rid of the remainders of the old roof truss. The season 2012/2013 was more typical, requiring about 19.700 kWh.

The last winter was not too extreme – we needed 100 kWh maximum heating energy per day. The collector was capable of harvesting about 50 kWh / day:

Daily energy balances, heat pump system, season 2014-2015

Daily energies: 1) Heating energy delivered by the heat pump. Heating energy = electrical energy + ambient energy from the tank. 2) Energy supplied by the collector to the water tank, turned off during the Ice Storage Challenge. Negative collector energies indicate cooling of the water tank by the collector during summer nights. 200 kWh peak in January: due to the warm winter storm ‘Felix’.

Ice formation in this season was mainly triggered by turning off the solar collector deliberately. As soon as we turn the collector on again in March the ice was melted quickly, and the temperature increased to the set value of 8°C – a value picked deliberately to prepare for cooling in summer:

Temperatures and ice formation, heat pump system, season 2014-2015

Daily averages of the air temperature and the temperature in the water tank plus volume of ice created by extracting heat from the heat source (water tank).

Further reading / about the system:
I am maintaining a list of answers to Frequently Asked Questions here.

Lest We Forget the Pioneer: Ottokar Tumlirz and His Early Demo of the Coriolis Effect

Two years ago I wrote an article about The Myth of the Toilet Flush, comparing the angular rotation caused by the earth’s rotation to the typical rotation in experiments with garden hoses that make it easy to observe the Coriolis effect. There are several orders of magnitude in difference, and the effect can only be observed in an experiment done extremely carefully, not in the bathtub sink or toilet flush.

Now two awesome science geeks have finally done such a careful experimenteven a time-synchronized one, observing vortices on either hemisphere!

The effect has been demonstrated in a similarly careful experiment in 1908. It had been done on the Northern hemisphere only, but if it can attributed it to the Coriolis effect by ruling out other disturbances, the different senses of rotations are straight-forward.

Austrian physicist Ottokar Tumlirz had published a German  paper called “New physical evidence on the axis of rotation of the earth”. I had created this ugly sketch of his setup:


Rough sketch based on the abstract of Tumlirz’ paper, not showing the vessel containing these components [*]

A cylindrical vessel (not shown in my drawing) is filled with water, and two glass plates are placed into it. The bottom plate has a hole, as well as the vessel. Both holes are connected by a glass tube that has many small holes. The space between the two plates is filled with water and water slowly flows out – from the bulk of the vessel through the the tiny holes into the tube. These radial red lines are bent very slightly due to the Coriolis force, and the Tumlirz added a die to make them visible. He took a photo 24 hours after starting the experiment, and the water must not flow out faster than 1 mm per minute.

Ernst Mach has given an account of Tumlirz’ experiment, quoted in an article titled Inventors I Have Met – anecdotes by a physicist approached by ‘outsider scientists’, once called paradoxers, today often called crackpots. I learned about Ernst Mach’s article from the reference and re-print of the article on this history of physics website.

Mach refers to Tumlirz’ experiment as an example of an idea that seems to belong in the same category at first glance, but is actually correct:

To be sure, Professor Tumlirz has recently performed an experiment which, while externally similar to this, is correct. By this experiment the rotation of the earth can be imitated, if the utmost care is taken, by the direction of the current of water flowing axially out of a cylindrical vessel. Further details are to be found in an article by Tumlirz in the Sitzungsberichte der Wiener Akademie, Vol. 117, 1908. I happened to know the origin of the thought that gave rise to this invention. Tumlirz noticed that the water flowing somewhat unsymmetrically in a glass funnel assumed a swift rotation in the neck of the funnel so that it formed a whirl of air in the axis of the flowing jet. This put it in his mind to increase the slight angular velocity of the water at rest with reference to the earth, by contraction in the axis.


Comment on the German abstract: It seems one line or sentence got lost or mangled when processing the original as this does not make sense: so bendet sich das Wasser zwischen den beiden Glasscheiben [here something is missing] nach dem Rohrchen durch die kleinen Öffnungen.

I have not managed to find the full version of the old paper and the figures and photos online. I would be grateful for pointers.


Update added August 2016: C. Schiller quotes this historical experiment in vol. 1 of his free physics textbook Motion Mountain (p. 135):

Only in 1962, after several attempts by other researchers, Asher Shapiro was the first to verify that the Coriolis effect has a tiny influence on the direction of the vortex flowing out of the bathtub.

Ref: A. H. SHAPIRO, Bath-tub vortex, Nature 196, pp. 1080-1081, 1962

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 as 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’.

100 1870 (Voitsberg steam power plant)

One of Austria’s last coal power plants – Kraftwerk Voitsberg, retired in 2006 (Florian Probst, Wikimedia). Thermodynamically, this is like ‘a heat pump running in reverse. That’s why I don’t like when a heat pump is said to ‘work like a refrigerator, just in reverse’ (Hinting at: The useful heat provided by the heat pump is equivalent to the waste heat of the refrigerator). If you run the cycle backwards, a heat pump would become sort of a steam power plant.

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). (Image: Kara98, Wikimedia).

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.

Temperature scale according to Kelvin (William Thomson)

Carnot engines with different efficiencies due to different lower temperatures. If one of the temperatures is declared the reference temperature, the other can be determined by / defined by the efficiency of the ideal machine (Image: Olivier Cleynen, Wikimedia.)

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

How to Evaluate a Heat Pump’s Performance?

The straight-forward way is to read off two energy values at the end of a period – day, month, or season:

  1. The electrical energy used by the heat pump
  2. and the heating energy delivered.

The Seasonal Performance Factor (SPF) is the ratio of these – the factor the input electrical energy is ‘multiplied with’ to yield heating energy. The difference between these two energies is supplied by the heat source – the underground water tank / ‘cistern’ plus solar collector in our setup.

But there might not be a separate power meter just for the heat pump’s compressor. Fortunately, performance factors can also be evaluated from vendors’ datasheets and measured brine / heating water temperatures:

Datasheets provide the Coefficient of Performance (COP) – the ‘instantaneous’ ratio of heating power and electrical power. The COP decreases with increasing temperature of the heating water, and with decreasing temperature of the source  – the brine circuit immersed in the cold ice / water tank. E.g when heating the water in floor loops to 35°C the COP is a bit greater than 4 if the water in the underground tank is frozen (0°C). The textbook formula based on Carnot’s ideal process for thermodynamic machines is 8,8 for 0°C/35°; realistic COPs are typically by about factor of 2 lower.

Heat Pump Performance, from Datasheets

COPs, eletrical power (input) and heating power (output) of a ‘7 kW’ brine / water heat pump. Temperatures in the legend are heating water infeed temperatures – 35°C as required by floor loops and 50°C for hot water heating.

If you measure the temperature of the brine and the temperature of the heating water every few minutes, you can determine the COP from these diagrams and take averages for days, months, or seasons.

But should PF and average COP actually be the same?

Average power is total energy divided by time, so (with bars denoting averages):

\text{Performance Factor } = \frac {\text{Total Heating Energy } \mathnormal{E_{H}}} {\text{Total Electrical Energy } \mathnormal{E_{E}}} = \frac {\text{Average Heating Power } \mathnormal{\bar{P}_{H}}} {\text{Average Electrical Power }\mathnormal{\bar{P}_{E}} }

On the other hand the average COP is calculated from data taken at many different times. At any point of time t,

\text{Coefficient of Performance(t)} = \frac {\text{Heating Power }P_{H}(t))} {\text{Electrical Power } P_{E}(t))}

Having measured the COP at N times, the average COP is thus:

\overline{COP}(t) = \frac {1}{N} \sum \frac{P_{H}(t)}{P_{E}(t)} = \overline{\frac{P_{H}(t)}{P_{E}(t)}}

\overline{\frac{P_{H}(t)}{P_{E}(t)}} is not necessarily equal to \frac{\overline{P_{H}}}{\overline{P_{E}}}

When is the average of ratios equal to the ratios of the averages?

If electrical power and heating power would fluctuate wildly we would be in trouble. Consider this hypothetical scenario of odd non-physical power readings:

  • PH = 10, PE = 1
  • PH = 2, PE = 20

The ratio of averages is: (10 + 2) / (1 + 20) = 12 / 21 = 0,57
The average of ratios is: (10/1 + 2/20) / 2 = (10 + 0,1) / 2 = 5,05

Quite a difference. Good that typical powers look like this:

Measured heating power, electrical power, COP (heat pump)

Powers measured on 2015-02-20 . Two space heating periods with a COP between 4 and 5, and one heating hot water cycle: the COP gradually decreases as heating water temperature increases.

Powers change only by a fraction of their absolute values – the heat pump is basically ON or OFF.  When these data were taken in February, average daily ambient temperature was between 0°C and 5°C, and per day about 75kWh were used for space heating and hot water. Since heat pump output is constant, daily run times change with heating demands.

Results for the red hot tap water heating cycle:

  • Performance Factor calculated from energies: 3,68
  • Average COP: 3,76.

I wanted to know how much powers are allowed to change without invalidating the average COP method:

Electrical power and heating power rise / fall about linearly, so they can be described by two parameters: Initial powers when the heat pump is turned on, and the slope of the curve or relative change of power within on cycle. The Performance Factor is determined from energies, the areas of trapezoids under the curves. For calculating the COP the ratio needs to be integrated, which results in a not so nice integral.

The important thing is that COP and PF are proportional to the ratio of inital powers and their relative match only depends on the slopes of the heating power and electrical power curves. As long as the relative increase / decrease of those powers is significantly smaller than 1, the difference in performance indicators is just a few percent. In the example curve, the heating energy decreases by 15%, while electrical energy increases by 52% – performance indicators would differ by less than 2%. This small difference is not too sensitive to changes in slopes.

All is well.

Solar collector, spring 2015.

Happily harvesting ambient energy.


Detailed monthly and seasonal performance data are given in this document.

Ice Storage Challenge: High Score!

Released from ice are brook and river
By the quickening glance of the gracious Spring;
The colors of hope to the valley cling,
And weak old Winter himself must shiver,
Withdrawn to the mountains, a crownless king.

These are the first lines of the English version of a famous German poem on spring, from the drama Faust, by Johann Wolfgang von Goethe. Weird factoid about me: I was once inclined to study literature, rather than physics. But finally physics won, so this is a post about joyful toying with modeling heat transport in ice and water.

After 46 days we had a high score: The ice cube, generated by our heat pump, stopped growing at about 15m3. About 10mof water remained unfrozen. After the volume of ice had been in a steady state for a a while, we turned on the solar collector again to return to standard operations.

Where did the energy for the heat pump come from before?

The lid of the tank is insulated against ambient air, the solar collector was not operational, and no ice had been created: The remaining energy has to be provided by the 5th element that cannot be shut off: 1) water 2) ice, 3) ambient air, 4) solar radiation … 5) ground.

Normally ground supplies about 15 W per m2 surface area – deduced from monitoring the power transported with the brine flow and energy accounting for the tank. The active interface between tank and ground below frost depth is about 35 m2. This results in about 0,5 kW in total, thus just 12 kWh per day, much lower than the ~ 50 kWh ambient energy fed into the heat pump.

After much deliberation and playing with the heat transfer equation we came up with this description of the evolution of the ice cube:

Phase 1: Growth of ice into water.

  • Ice starts to grow from the heat exchanger tubes into the remaining water. These tubes are installed in a meandering pattern, traversing the storage tank.
  • At some point the thick layers of ice covering adjacent parts of the pipes touch each other. The surface of this solid ice cube is smaller than the interface between the meandering ice formations and water before. The power needed by the heat pump has to be pushed through a smaller surface – which is only possible if the temperature gradient within the ice gets larger. As the temperature at the ice-water interface has to be 0°C, the temperature at the heat exchanger has to decrease. This is exactly what we see from monitoring data – brine temperature drops well below 0°C.
  • Side-effect: Due to the lower brine temperature the coefficient of performance  decreases slightly. So more of the total heating energy needs to be provided by the electrical input. We call this the heat source paradox: The worse performance is, the more you spare the energy stored in the heat source. Thanks to this self-protection mechanism, the energy in the tank will not suddenly drop to zero.
Ice in the water tank, 2015-03-30

Evolution of the volume of ice, ambient temperature and brine temperature over time. The ice is now quickly melting again – in March the collector is already harvesting enough energy again for balancing heating demands!

Phase 2: Ice touching ground.

  • As long as there is some water between ice and ground, the water temperature is 0°C. This is the temperature ground ‘sees’ and the temperature which is relevant for the low heat transport from ground to water.
  • Ice touches some surfaces of the cuboid tank – the ones where the heat exchanger tubes are closest to the surface. Now ground is directly connected to ice with its temperatures < 0°C. The temperature gradient between ground and ice provides for a higher flow of energy. This is also indicated by the evolution of the temperature in the ground below the tank: While temperatures of undisturbed ground and the region below the tank had been aligned before, ground temperature beneath the tank still kept getting lower – although a few meters away from the tank ground is already warming up again.
  • If enough heat is delivered by ground, no more heat is needed by freezing the remaining water in the tank. When ground temperature reaches zero, it can even freeze – which happens with geothermal systems, too. We might have extended the ice storage into ground.
Temperatures in ground, beneath the water tank.

The temperature sensor closest to the tank in 30 cm underneath. A few meters from the tank ground is already warming up again (following the standard ‘yearly temperature wave’), but below the tank the temperature is still getting lower – as highlighted by the blue rectangle.

Heat transport within ice is actually more efficient than transport in water: Ice has 4 times the heat conductivity of water, and 10 times the thermal diffusivity. The latter is a measure for the time a deposited ‘lump of heat’ will be spread in space:

Heat eqnSo we have built a very efficient cold bridge between the heat exchanger and ground. Everything is consistent with the poetry of the differential equation of heat transfer.

I marvel at the intriguing and mathematically appealing physics in my backyard!

When we grow up, we'll be eggplants!

For the backyard (‘Office desk farming’). Happy Easter everybody!