The Heat Source Paradox

It is not a paradox – it is a straight-forward relation between a heat pump system’s key data:

The lower a heat pump’s performance factor is, the smaller the source can be built.

I would not write this post, hadn’t I found a version of this statement with a positive twist  used in an advert!

In this post I consider a heat pump a blackbox that converts input energy into output heat energy – it ‘multiplies’ energy by a performance factor. A traditional mechanical heat pump uses electrical input energy to drive a mechanical compressor. The uncommon Rotation Heat Pump utilizes the pressure gradient created by centrifugal forces and thus again by electrical power.

But a pressure difference can also be maintained by adsorption/desorption processes or by changing the amount of one fluid dissolved in another; Einstein’s famous refrigerator uses a more complex combination of such dissolution/evaporation processes. Evaporation or desorption can be directly driven by heat: A gas heat pump thus ‘multiplies’ the energy from burning natural gas (and in addition, a heat pump and a gas boiler can be combined in one unit).

The overall performance factor of a gas heat pump – kWh heating energy out over kWh gas in – is about 1,5 – 2. This is lower than 4 – 5 available with mechanical compressors. But the assessment depends on the costs of kWh gas versus kWh electrical energy: If gas is four times cheaper (which nearly is the case in Germany) than burning natural gas in a traditional boiler without any ‘heat pump multiplication’, then the classical boiler can be more economical than using a heat pump with an electrical compressor. If gas is ‘only’ two times as cheap, then a gas heat pump with an overall performance number of ‘only’ 2 will still beat an electrical heat pump with a performance factor of 4.

While the gas heat pump may have its merits under certain market conditions, its performance number is low: For one kWh of gas you only get two kWh of heating energy. This  means you only need to provide one kWh of ‘ambient’ energy from your source – geothermal, water, or air. If the performance factor of an electrical heat pump is 4, you multiply each kWh of input energy by 4. But the heat source has to be able to supply the required 3 kWh. This is the whole ‘paradox’: The better the heat pump’s performance is in terms of heating energy over input energy, the more energy has to be released by a properly designed heat source, like ground loops sufficiently large, a ground-water well providing sufficient flow-rate, an air heat pump’s ventilator powerful enough, or our combination of a big enough solar/air collector plus water tank.

Illustration of the ‘heat source paradox’: The lower the performance number (ratio of output and input energy), the lower is the required ambient energy that has to be provided by ‘the environment’. The output heating energy in red is the target number that has to be met – it is tied to the building’s design heat load.

If you wish to state it that way, a heat pump with inferior performance characteristics has the ‘advantage’ that the source can be smaller – less pipes to be buried in the ground or a smaller water tank. And in an advert for a gas heat pump I found it spelled out exactly in this way, as a pro argument compared to other heat pumps:

The heat source can be built much smaller – investment costs are lower!

It is not wrong, but it is highly misleading. It is like saying that heating electrically with a resistive heating element – and thus a performance number of 1 – is superior because you do not need to invest in building any source of ambient energy at all.

The Collector Size Paradox

Recently I presented the usual update of our system’s and measurement data documentation.The PDF document contains consolidated numbers for each year and month of operations:

Total output heating energy (incl. hot tap water), electrical input energy (incl. brine pump) and its ratio – the performance factor. Seasons always start at Sept.1, except the first season that started at Nov. 2011. For ‘special experiments’ that had an impact on the results see the text and the PDF linked above.

It is finally time to tackle the fundamental questions:

What id the impact of the size of the solar/air collector?


What is the typical output power of the collector?

In 2014 the Chief Engineer had rebuilt the collector so that you can toggle between 12m2 instead of 24m

TOP: Full collector – hydraulics as in seasons 2012, 2013. Active again since Sept. 2017. BOTTOM: Half of the collector, used in seasons 201414, 15, and 16.

Do we have data for seasons we can compare in a reasonable way – seasons that (mainly) differ by collector area?

We disregard seasons 2014 and 2016 – we had to get rid of a nearly 100 years old roof truss and only heated the ground floor with the heat pump.

Attic rebuild project – point of maximum destruction – generation of fuel for the wood stove.

Season 2014 was atypical anyway because of the Ice Storage Challenge experiment.

Then seasonal heating energy should be comparable – so we don’t consider the cold seasons 2012 and 2016.

Remaining warm seasons: 2013 – where the full collector was used – and 2015 (half collector). The whole house was heated with the heat pump; heating and energies and ambient energies were similar – and performance factors were basically identical. So we checked the numbers for the ice months Dec/Feb/Jan. Here a difference can be spotted, but it is far less dramatic than expected. For half the collector:

  • Collector harvest is about 10% lower
  • Performance factor is lower by about 0,2
  • Brine inlet temperature for the heat pump is about 1,5K lower

The upper half of the collector is used, as indicated by hoarfrost.

It was counter-intuitive, and I scrutinized Data Kraken to check it for bugs.

But actually we forgot that we had predicted that years ago: Simulations show the trend correctly, and it suffices to do some basic theoretical calculations. You only need to know how to represent a heat exchanger’s power in two different ways:

Power is either determined by the temperature of the fluid when it enters and exits the exchanger tubes …

[1]   T_brine_outlet – T_brine_inlet * flow_rate * specific_heat

… but power can also be calculated from the heat energy flow from brine to air – over the surface area of the tubes:

[2]   delta_T_brine_air * Exchange_area * some_coefficient

Delta T is an average over the whole exchanger length (actually a logarithmic average but using an arithmetic average is good enough for typical parameters). Some_coefficient is a parameter that characterized heat transfer for area or per length of a tube, so Exchange_area * Some_coefficient could also be called the total heat transfer coefficient.

If several heat exchangers are connected in series their powers are not independent as they share common temperatures of the fluid at the intersection points:

The brine circuit connecting heat pump, collector and the underground water/ice storage tank. The three ‘interesting’ temperatures before/after the heat pump, collector and tank can be calculated from the current power of the heat pump, ambient air temperature, and tank temperature.

When the heat pump is off in ‘collector regeneration mode’ the collector and the heat exchanger in the tank necessarily transfer heat at the same power  per equation [1] – as one’s brine inlet temperature is the other one’s outlet temperature, the flow rate is the same, and also specific heat (whose temperature dependence can be ignored).

But powers can also be expressed by [2]: Each exchanger has a different area, a different heat transfer coefficient, and different mean temperature difference to the ambient medium.

So there are three equations…

  • Power for each exchanger as defined by [1]
  • 2 equations of type [2], one with specific parameters for collector and air, the other for the heat exchanger in the tank.

… and from those the three unknowns can be calculated: Brine inlet temperatures, brine outlet temperature, and harvesting power. All is simple and linear, it is not a big surprise that collector harvesting power is proportional temperature difference between air and tank. The warmer the air, the more you harvest.

The combination of coefficient factors is the ratio of the product of total coefficients and their sum, like: \frac{f_1 * f_2}{f_1 + f_2} – the inverse of the sum of inverses.

This formula shows what one might you have guessed intuitively: If one of the factors is much bigger than the other – if one of the heat exchangers is already much ‘better’ than the others, then it does not help to make the better one even better. In the denominator, the smaller number in the sum can be neglected before and after optimization, the superior properties always cancel out, and the ‘bad’ component fully determines performance. (If one of the ‘factors’ is zero, total power is zero.) Examples for ‘bad’ exchangers: If the heat exchanger tubes in the tank are much too short or if a flat plat collector is used instead of an unglazed collector.

On the other hand, if you make a formerly ‘worse’ exchanger much better, the ratio will change significantly. If both exchangers have properties of the same order of magnitude – which is what we deign our systems for – optimizing one will change things for the better, but never linearly, as effects always cancel out to some extent (You increase numbers in both parts if the fraction).

So there is no ‘rated performance’ in kW or kW per area you could attach to a collector. Its effective performance also depends on the properties of the heat exchanger in the tank.

But there is a subtle consequence to consider: The smaller collector can deliver the same energy and thus ‘has’ twice the power per area. However, air temperature is given, and [2] must hold: In order to achieve this, the delta T between brine and air necessarily has to increase. So brine will be a bit colder and thus the heat pump’s Coefficient of Performance will be a bit lower. Over a full season including the warm periods of heating hot water only the effect is less pronounced – but we see a more significant change in performance data and brine inlet temperature for the ice months in the respective seasons.

Where to Find What?

I have confessed on this blog that I have Mr. Monk DVDs for a reason. We like to categorize, tag, painstakingly re-organize, and re-use. This is reflected in our Innovations in Agriculture …

The Seedbank: Left-over squared timber met the chopsaw.

The Nursery: Rebirth of copper tubes and newspapers.

… as well as in my periodical Raking The Virtual Zen Garden: Updating collections of web resources, especially those related to the heat pump system.

Here is a list of lists, sorted by increasing order of compactification:

But thanks to algorithms, we get helpful advice on presentation from social media platforms: Facebook, for example, encouraged me to tag products in the following photo, so here we go:

“Hand-crafted, artisanal, mobile nursery from recycled metal and wood, for holding biodegradable nursery pots.” Produced without crowd-funding and not submitted to contests concerned with The Intersection of Science, Art, and Innovation.

Frozen Herbs and Latent Energy Storage

… having studied one subject, we immediately have a great deal of direct and precise knowledge … of another.

Richard Feynman

Feynman referred to different phenomena that can be described by equations of the same appearance: Learning how to calculate the distribution of electrical charges gives you the skills to simulate also the flow of heat.

But I extend this to even more down-to-earth analogies – such as the design of a carton of frozen herbs resembling our water-tight underground tank.

(This is not a product placement.)

No, just being a container for frozen stuff is too obvious a connection!

Maybe it is the reclosable lid covering part of the top surface?

Lid of underground water/ice storage tank.

No, too obvious again!

Or it is the intriguing ice structures that grow on the surface: in opened frozen herb boxes long forgotten in the refrigerator – or on a gigantic ice cube in your tank:

Ice Storage Challenge of 2015 - freezing 15m3 of water after having turned off the solar/air collector.

The box of herbs only reveals its secret when dismantled carefully. The Chief Engineer minimizes its volume as a dedicated waste separating citizen:

… not just tramping it down (… although that sometimes helps if some sensors do not co-operate).

He removes the flaps glued to the corners:

And there is was, plain plane and simple:

The Chief Engineer had used exactly this folding technique to cover the walls and floor of the former root cellar with a single piece of pond liner – avoiding to cut and glue the plastic sheet.

How Does It Work? (The Heat Pump System, That Is)

Over the holidays I stayed away from social media, read quantum physics textbooks instead, and The Chief Engineer and I mulled over the fundamental questions of life, the universe and everything. Such as: How to explain our heat pump system?

Many blog postings were actually answers to questions, and am consolidating all these answers to frequently asked questions again in a list of such answers. However, this list has grown quickly.

An astute reader suggested to create an ‘animation’ of the gradual evolution of the system’s state. As I learned from discussions, one major confusion was related to the role of the solar collector and the fact that you have to factor in the history of the heat source: This is true for every heat pump system that uses a heat source that can be ‘depleted’, in contrast to a flow of ground water at a constant temperature for example. With the latter, the ‘state’ of the system only depends on the current ambient temperature, and you can explain it in a way not too different from pontificating on a wood or gas boiler.

One thing you have to accept though is how a heat pump as such works: I have given up to go into thermodynamic details, and I also think that the refrigerator analogy is not helpful. So for this pragmatic introduction a heat pump is just a device that generates heating energy as an output, the input energy being electrical energy and heat energy extracted from a rather cold heat source somewhere near the building. For 8kW heating power you need about 2kW electrical energy and 6kW ambient energy. The ratio of 8kW and 2kW is called the coefficient of performance.

What the typical intro to heat pumps in physics textbooks does not point out is that the ambient heat source actually has to be able to deliver that input energyduring a whole heating season. There is no such thing as the infinite reservoir of energy usually depicted as a large box. Actually, the worse the performance of a heat pump is – the ratio of output heat energy and input electrical energy, the smaller are the demands on the heat source. The Chief Engineer has coined the term The Heat Source Paradox for this!

The lower the temperature of the heat source, the smaller the coefficient of performance is: So if you run an air source heat pump in mid-winter (using a big ventilator) then less energy is extracted from that air source than a geothermal heat pump would extract from ground. But if you build a geothermal heat source that’s too small in relation to a building’s heating demands, you see the same effect: Ground freezes, source temperature decreases, performance decreases, and you need more electrical energy and less ambient energy.

I am harping on the role of the heat source as the whole point of our ‘innovation’ is our special heat source that has two components, both of them being essential: An unglazed solar / air collector and an underground water / ice tank plus the surrounding ground. The collector allows to replenish the energy stored in the tank quickly, even in winter: Air temperature just needs to be some degrees warmer than the cold brine. The tank is a buffer: When no energy is harvested by the collector at ambient temperatures below 0°C, water freezes and releases latent heat. So you can call that an air heat pump with a huge, silent and mainentance-free ‘absorber’ plus a buffer that provides energy for periods of frost and that allows for storing all the energy you don’t need immediately. Ground does provide some energy as well, and I am planning to post about my related simulations.(*) It can be visualized as an extension of the ice / water energy storage into the surroundings. But the active volume or area of ground is smaller than for geothermal systems as most of the ambient energy actually comes from the solar / air collector: The critical months in our climate are Dec-Jan-Feb: Before and after, the collector would be sufficient as the only heat source. In the three ‘ice months’ water is typically frozen in the tank, but even then the collector provides for 75-80% of the ambient energy needed to drive the heat pump.(*)

(*) Edit: This post written in 2017 shows how much energy is stored / exchanged by each component. An overview of essential numbers is given here; emphasis on the volume of ice – which is compared to simulations here.

Components are off-the-shelf products, actually rather simple and cheap ones, such as the most stupid, non-smart brine-water heat pump. What is special is 1) the arrangement of the heat exchanger in the water tank and 2) the custom control logic, that is programming of the control unit.

So here is finally the series of images of the system’s state, shown in a gallery and with captions: You can scroll down to see the series embedded in the post, or click on the first image to see an enlarged view and then click through the slide-show.

More information on the system (technical data, sizing) and measurement data since 2012 can be found in this documentation – updated every few months.

Information for German readers: This post contains the German version of this slide-show.