# Simulating Life-Forms (2): Cooling Energy

I found this comprehensive research report:
Energy Use in the Australian Residential Sector 1986–2020 (June 2008)

There are many interesting results – and the level of detail is impressive: The authors modelled the energy used per appliance type, by e.g. factoring in how building types change slowly over time or by modelling the development of TV sets and their usage. Occupancy factors for buildings are determined from assumptions about typical usage profiles called Stay At Home, At Work or Night Owl.

I zoom in on simulating and predicting usage of air conditioning and thus cooling energy:

They went to great lengths to simulate the behavior of home owners to model operations of air conditioning and thus total cooling energy for a season, for a state or the whole country.

The authors investigated the official simulation software used for rating buildings (from …part2.pdf):

In the AccuRate software, once cooling is invoked the
program continues to assume that the occupant is willing to
tolerate less than optimal comfort conditions and will therefore terminate cooling if in the absence of such cooling the internal temperature would not rise above the summer neutral temperature noted in Table 57, + 2.5oC plus allowances for humidity and air movement as applicable. While this may be appropriate for rating purposes, it is considered to be an unlikely form of behaviour to be adopted by householders in the field and as such this assumption is likely to underestimate the potential space cooling demand. This theory is supported by the survey work undertaken by McGreggor in South Australia.

This confirms what I am saying all the time: The more modern a building is, or generally nowadays given ‘modern’ home owners’ requirements, the more important would it be to actually simulate humans’ behavior, on top of the physics and the control logic.

The research study also points out e.g. that AC usage has been on the rise, because units got affordable, modern houses are built with less focus on shading, and home owners demand higher standards of comfort. Ducted cooling systems that cover the cooling load of the whole house are being implemented, and they replace systems for cooling single zones only. Those ducted systems have a rated output cooling power greater than 10kW – so the authors (and it seems Australian governmental decision makers) are worried about the impact on the stability of the power grid on hot days [*].

Once AC had been turned on for the first time in the hot season, home owners don’t switch it off again when the theoretical ‘neutral’ summer temperature would be reached again, but they keep it on and try to maintain a lower temperature (22-23°C) that is about constant irrespective of temperature outside. So small differences in actual behavior cause huge error bars in total cooling energy for a season:

The impact of this resetting of the cooling thermostat operation was found to be significant. A comparison was undertaken between cooling loads determined using the AccuRate default thermostat settings and the modified settings as described above. A single-storey brick veneer detached dwelling with concrete slab on ground floor and ceiling insulation was used for the comparison. The comparison was undertaken in both the Adelaide and the Darwin climate zones. In Adelaide the modified settings produced an increased annual cooling load 64% higher than that using the AccuRate default settings.

The report also confirms my anecdotal evidence: In winter (colder regions) people heat rooms to higher temperatures than ‘expected’; in summer (warmer regions) people want to cool to a lower temperature:

This is perhaps not surprising, de Dear notes that: “preferred temperature for a particular building did not necessarily coincide with thermal neutrality, and this semantic discrepancy was most evident in HVAC buildings where preference was depressed below neutrality in warm climates and elevated above neutrality in cold climates (ie people preferred to feel cooler than neutral in warm climates, and warmer than neutral in cold climates)” (Richard de Dear et al 1997, P xi).

I noticed that the same people who (over-)heat their rooms to 24°C in winter might want to cool to 20°C in summer. In middle Europe AC in private homes has been uncommon, but I believe it is on the rise, too, also because home owners got accustomed to a certain level of cooling when they work in typical office buildings.

My conclusion is (yet again) that you cannot reliably ‘predict’ cooling energy. It’s already hard to do so for heating energy for low energy houses, but nearly impossible for cooling energy. All you can do – from a practical / system’s design perspective – is to make sure that there is an ‘infinite’ source of cooling energy available.

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[*] Edit: And it actually happenend in February 2017.

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?

or

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.

# Simulations: Levels of Consciousness

In a recent post I showed these results of simulations for our heat pump system:

I focused on the technical details – this post will be more philosophical.

What is a ‘simulation’ – opposed to simplified calculations of monthly or yearly average temperatures or energies? The latter are provided by tools used by governmental agencies or standardization bodies – allowing for a comparison of different systems.

In a true simulation the time intervals so small that you catch all ‘relevant’ changes of a system. If a heating system is turned on for one hour, then turned off again, he time slot needs to be smaller than one hour. I argued before that calculating meaningful monthly numbers requires to incorporate data that had been obtained before by measurements – or by true simulations.

For our system, the heat flow between ground and the water/ice tank is important. In our simplified sizing tool – which is not a simulation – I use average numbers. I validated them by comparing with measurements: The contribution of ground can be determined indirectly; by tallying all the other energies involved. In the detailed simulation I calculate the temperature in ground as a function of time and of distance from the tank, by solving the Heat Equation numerically. Energy flow is then proportional to the temperature gradient at the walls of the tank. You need to make assumptions about the thermal properties of ground, and a simplified geometry of the tank is considered.

Engineering / applied physics in my opinion is about applying a good-enough-approach in order to solve one specific problem. It’s about knowing your numbers and their limits. It is tempting to get carried away by nerdy physics details, and focus on simulating what you know exactly – forgetting that there are huge error bars because of unknowns.

This is the hierarchy I keep in mind:

On the lowest level is the simulation physics, that is: about modelling how ‘nature’ and system’s components react – to changes in the previous time slot. Temperatures change because energies flows, and energy flows because of temperature differences. The heat pump’s output power depends on heating water temperature and brine temperature. Energy of the building is ‘lost’ to the environment via heat conduction; heat exchangers immersed in tanks deposit energy there or retrieve it. I found that getting the serial connection of heat exchangers right in the model was crucial, and it required a self-consistent calculation for three temperatures at the same point of time, rather than trying to ‘follow round the brine’. I used the information on average brine temperatures obtained by these method to run a simplified version of the simulation using daily averages only – for estimating the maximum volume of ice for two decades.

So this means you need to model your exact hydraulic setup, or at least you need to know which features of your setup are critical and worthy to model in detail. But the same also holds for the second level, the simulation of control logic. I try to mirror production control logic as far as possible: This code determines how pumps and valves will react, depending on the system’s prior status before. Both in real life and in the simulation threshold values and ‘hystereses’ are critical: You start to heat if some temperature falls below X, but you only stop heating if it has risen above X plus some Delta. Typical brine-water heat pumps always provide approximately the same output power, so you control operations time and buffer heating energy. If Delta for heating the hot water buffer tank is too large, the heat pump’s performance will suffer. The Coefficient of Performance of the heat pump decreases with increasing heating water temperature. Changing an innocuous parameter will change results a lot, and the ‘control model’ should be given the same vigilance as the ‘physics model’.

Control units can be tweaked at different levels: ‘Experts’ can change the logic, but end users can change non-critical parameters, such as set point temperatures.We don’t restrict expert access in systems we provide the control unit for. But it make sense to require extra input for the expert level though – to prevent accidental changes.

And here we enter level 3 – users’ behavior. We humans are bad at trying to outsmart the controller.

[Life-form in my home] always sets the controller to ‘Sun’. [little sun icon indicating manually set parameters]. Can’t you program something so that nothing actually changes when you pick ‘Sun’?

With heat pumps utilizing ground or water sources – ‘built’ storage repositories with limited capacity – unexpected and irregular system changes are critical: You have to size your source in advance. You cannot simply order one more lorry load of wood pellets or oil if you ‘run out of fuel’. If the source of ambient energy is depleted, the heat pump finally will refuse to work below a certain source temperature. The heat pump’s rated power has match the heating demands and the size of the source exactly. It also must not be oversized in order to avoid turning on and off the compressor too often.

Thus you need good estimates for peak heat load and yearly energy needs, and models should include extreme weather (‘physics’) but also erratic users’ behaviour. The more modern the building, the more important spikes in hot tap water usage get in relation to space heating. A vendor of wood pellet stoves told me that delivering peak energy for hot water – used in private bathrooms that match spas – is a greater challenge today than delivering space heating energy. Energy certificates of modern buildings take into account huge estimated solar and internal energy gains – calculated according to standards. But the true heating power needed on a certain day will depend on the strategy or automation home owners use when managing their shades.

Typical gas boilers are oversized (in terms of kW rated power) by a factor of 2 or more in Germany, but with heat pumps you need to be more careful. However, this also means that heat pump systems cannot and should not be planned for rare peak demands, such as: 10 overnight guests want to shower in the morning one after the other, on an extremely cold day, or for heating up the building quickly after temperature had been decreased during a leave of absence.

The nerdy answer is that a smart home would know when your vacation ends and start heating up well in advance. Not sure what to do about the showering guests as in this case ‘missing’ power cannot be compensated by more time. Perhaps a gamified approach will work: An app will do something funny / provide incentives and notifications so that people wait for the water to heat up again. But what about planning for renting a part of the house out someday? Maybe a very good AI will predict what your grandchildren are likely to do, based on automated genetics monitoring.

The challenge of simulating human behaviour is ultimately governed by constraints on resources – such as the size of the heat source: Future heating demands and energy usage is unknown but the heat source has to be sized today. If the system is ‘open’ and connected to a ‘grid’ in a convenient way problems seem to go away: You order whatever you need, including energy, any time. The opposite is planning for true self-sufficiency: I once developed a simulation for an off-grid system using photovoltaic generators and wind power – for a mountain shelter. They had to meet tough regulations and hygienic standards like any other restaurant, e.g.: to use ‘industry-grade’ dishwashers needing 10kW of power. In order to provide that by solar power (plus battery) you needed to make an estimate on the number of guests likely to visit … and thus on how many people would go hiking on a specific day … and thus maybe on the weather forecast. I tried to factor in the ‘visiting probability’ based on the current weather.

I think many of these problem can be ‘resolved’ by recognizing that they are first world problems. It takes tremendous efforts – in terms of energy use or systems’ complexity – to obtain 100% availability and to cover all exceptional use cases. You would need the design heat load only for a few days every decade. On most winter days a properly sized heat pump is operating for only 12 hours. The simple, low tech solution would be to accept the very very rare intermittent 18,5°C room temperature mitigated by proper clothing. Accepting a 20-minute delay of your shower solves the hot water issue. An economical analysis can reveal the (most likely very small) trade-off of providing exceptional peak energy by a ‘backup’ electrical heating element – or by using that wood stove that you installed ‘as a backup’ but mostly for ornamental reasons because it is dreadful to fetch the wood logs when it is really cold.

But our ‘modern’ expectations and convenience needs are also reflected in regulations. Contractors are afraid of being sued by malicious clients who (quote) sit next their heat pump and count its operating cycles – to compare the numbers with the ones to be ‘guaranteed. In a weather-challenged region at more than 2.000 meters altitude people need to steam clean dishes and use stainless steel instead of wood – where wooden plates have been used for centuries. I believe that regulators are as prone as anybody else to fall into the nerdy trap described above: You monitor, measure, calculate, and regulate the things in detail that you can measure and because you can measure them – not because these things were top priorities or had the most profound impact.

# Heat Transport: What I Wrote So Far.

Don’t worry, The Subversive Elkement will publish the usual silly summer posting soon! Now am just tying up loose ends.

In the next months I will keep writing about heat transport: Detailed simulations versus maverick’s rules of thumb, numerical solutions versus insights from the few things you can solve analytically, and applications to our heat pump system.

So I checked what I have already written – and I discovered a series which does not show up as such in various lists, tags, categories:

[2014-12-14] Cistern-Based Heat Pump – Research Done in 1993 in Iowa. Pioneering work, but the authors dismissed a solar collector for economic reasons. They used a steady-state estimate of the heat flow from ground to the tank, and did not test the setup in winter.

[2015-01-28] More Ice? Exploring Spacetime of Climate and Weather. A simplified simulation based on historical weather data – only using daily averages. Focus: Estimate of the maximum volume of ice per season, demonstration of yearly variations. As explained later (2017) in more detail I had to use information from detailed simulations though – to calculate the energy harvested by the collector correctly in such a simple model.

[2015-04-01] Ice Storage Challenge: High Score! Our heat pump created an ice cube of about 15m3 because we had turned the collector off. About 10m3 of water remained unfrozen, most likely when / because the ice cube touched ground. Some qualitative discussions of heat transport phenomena involved and of relevant thermal parameters.

[2016-01-07] How Does It Work? (The Heat Pump System, That Is) Our system, in a slide-show of operating statuses throughput a typical year. For each period typical temperatures are given and the ‘typical’ direction of heat flow.

[2016-01-22] Temperature Waves and Geothermal Energy. ‘Geothermal’ energy used by heat pumps is mainly stored solar energy. A simple model: The temperature at the surface of the earth varies sinusoidally throughout the year – this the boundary condition for the heat equation. This differential equation links the temporal change of temperature to its spatial variation. I solve the equation, show some figures, and check how results compare to the thermal diffusivity of ground obtained from measurements.

[2016-03-01] Rowboats, Laser Pulses, and Heat Energy (Boring Title: Dimensional Analysis). Re-visiting heat transport and heat diffusion length, this time only analyzing dimensional relationships. By looking at the heat equation (without the need to solve it) a characteristic length can be calculated: ‘How far does heat get in a certain time?’

[2017-02-05] Earth, Air, Water, and Ice. Data analysis of the heating season 2014/15 (when we turned off the solar/air collector to simulate a harsher winter) – and an attempt to show energy storages, heat exchangers, and heat flows in one schematic. From the net energy ‘in the tank’ the contribution of ground can be calculated.

[2017-02-22] Ice Storage Hierarchy of Needs. Continued from the previous post – bird’s eye view: How much energy comes from which sources, and which input parameters are critical? I try to answer when and if simple energy accounting makes sense in comparison to detailed simulations.

[2017-05-02] Simulating Peak Ice. I compare measurements of the level in the tank with simulations of the evolution of the volume of ice. Simulations (1-minute intervals) comprise a model of the control logic, the varying performance factor of the heat pump, heat transport in ground, energy balances for the hot and cold tanks, and the heat exchangers connected in series.

(Adding the following after having published this post. However, there is no guarantee I will update this post forever ;-))

[2017-08-17] Simulations: Levels of Consciousness. Bird’s Eye View: How does simulating heat transport fit into my big picture of simulating the heat pump system or buildings or heating systems in general? I consider it part of the ‘physics’ layer of a hierarchy of levels.

Planned episodes? Later this year (2017) or next year I might write about the error made when considering simplified geometry – like modeling a linear 1D flow when the actualy symmetry is e.g. spherical.

# Simulating Peak Ice

This year ice in the tank was finally melted between March 5 to March 10 – as ‘visual inspection’ showed. Level sensor Mr. Bubble was confused during the melting phase; thus it was an interesting exercise to compare simulations to measurements.

Simulations use the measured ambient temperature and solar radiation as an input, data points are taken every minute. Air temperature determines the heating energy needed by the house: Simulated heat load is increasing linearly until a maximum ‘cut off’ temperature.

The control logic of the real controller (UVR1611 / UVR16x2) is mirrored in the simulation: The controller’s heating curve determines the set temperature for the heating water, and it switches the virtual 3-way valves: Diverting heating water either to the hygienic storage or the buffer tank for space heating, and including the collector in the brine circuit if air temperature is high enough compared to brine temperature. In the brine circuit, three heat exchangers are connected in series: Three temperatures at different points are determined self-consistently from three equations that use underground tank temperature, air temperature, and the heat pump evaporator’s power as input parameters.

The hydraulic schematic for reference, as displayed in the controller’s visualization (See this article for details on operations.)

The Coefficient of Performance of the heat pump, its heating power, and its electrical input power are determined by heating water temperature and brine temperature – from polynomial fit curves to vendors’ data sheet.

So for every minute, the temperatures of tanks – hot and cold – and the volume of ice can be calculated from energy balances. The heating circuits and tap water consume energy, the heat pump delivers energy. The heat exchanger in the tank releases energy or harvests energy, and the collector exchanges energy with the environment. The heat flow between tank and ground is calculated by numerically solving the Heat Equation, using the nearly constant temperature in about 10 meters depth as a boundary condition.

For validating the simulation and for fine-tuning input parameters – like the thermal properties of ground or the building – I cross-check calculated versus measured daily / monthly energies and average temperatures.

Measurements for this winter show the artificial oscillations during the melting phase because Mr. Bubble faces the cliff of ice:

Simulations show growing of ice and the evolution of the tank temperature in agreement with measurements. The melting of ice is in line with observations. The ‘plateau’ shows the oscillations that Mr. Bubble notices, but the true amplitude is smaller:

Simulated peak ice is about 0,7m3 greater than the measured value. This can be explained by my neglecting temperature gradients within water or ice in the tank:

When there is only a bit of ice yet (small peak in December), tank temperature is underestimated: In reality, the density anomaly of water causes a zone of 4°C at the bottom, below the ice.

When the ice block is more massive (end of January), I overestimate brine temperature as ice has less than 0°C, at least intermittently when the heat pump is turned on. Thus the temperature difference between ambient air and brine is underestimated, and so is the simulated energy harvested from the collector – and more energy needs to be provided by freezing water.

However, a difference in volume of less than 10% is uncritical for system’s sizing, especially if you err on the size of caution. Temperature gradients in ice and convection in water should be less critical if heat exchanger tubes traverse the volume of tank evenly – our prime design principle.

I have got questions about the efficiency of immersed heat exchangers in the tank – will heat transfer deteriorate if the layer of ice becomes too thick? No, according also to this very detailed research report on simulations of ‘ice storage heat pump systems’ (p.5). We grow so-called ‘ice on coil’ which is compared to flat-plate heat exchangers:

… for the coil, the total heat transfer (UA), accounting for the growing ice surface, shows only a small decrease with growing ice thickness. The heat transfer resistance of the growing ice layer is partially compensated by the increased heat transfer area around the coil. In the case of the flat plate, on the contrary, also the UA-value decreases rapidly with growing ice thickness.

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For system’s configuration data see the last chapter of this documentation.

# 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!

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Continued: Here are some details on simulations.

# Temperature Waves and Geothermal Energy

Nearly all of renewable energy exploited today is, in a sense, solar energy. Photovoltaic cells convert solar radiation into electricity, solar thermal collectors heat hot water. Plants need solar power for photosynthesis, for ‘creating biomass’. The motion of water and air is influenced by the fictitious forces caused by the earth’s rotation, but by temperature gradients imposed by the distribution of solar energy as well.

Also geothermal heat pumps with ground loops near the surface actually use solar energy deposited in summer and stored for winter – that’s why I think that ‘geothermal heat pumps’ is a bit of a misnomer.

Collector (heat exchanger) for brine-water heat pumps.

Within the first ~10 meters below the surface, temperature fluctuates throughout the year; at 10m the temperature remains about constant and equal to 10-15°C for the whole year.

Only at higher depths the flow of ‘real’ geothermal energy can be spotted: In the top layer of the earth’s crust the temperatures rises about linearly, at about 3°C (3K) per 100m. The details depend on geological peculiarities, it can be higher in active regions. This is the energy utilized by geothermal power plants delivering electricity and/or heat.

Geothermal gradient adapted from Boehler, R. (1996). Melting temperature of the Earth’s mantle and core: Earth’s thermal structure. Annual Review of Earth and Planetary Sciences, 24(1), 15–40. (Wikimedia, user Bkilli1). Geothermal power plants use boreholes a few kilometers deep.

This geothermal energy originates from radioactive decays and from the violent past of the primordial earth: when the kinetic energy of celestial objects colliding with each other turned into heat.

The flow of geothermal energy per area directed to the surface, associated with this gradient is about 65 mW/m2 on continents:

Global map of the flow of heat, in mW/m2, from Earth’s interior to the surface. Davies, J. H., & Davies, D. R. (2010). Earth’s surface heat flux. Solid Earth, 1(1), 5-24. (Wikimedia user Bkilli1)

Some comparisons:

• It is small compared to the energy from the sun: In middle Europe, the sun provides about 1.000 kWh per m2 and year, thus 1.000.000Wh / 8.760h = 144W/m2 on average.
• It also much lower than the rule-of-thumb power of ‘flat’ ground loop collectors – about 20W/m2
• The total ‘cooling power’ of the earth is several 1010kW: Would the energy not be replenished by radioactive decay, the earth would lose a some seemingly impressive 1014kWh per year, yet this would result only in a temperature difference of ~10-7°C (This is just a back-of-the-envelope check of orders of magnitude, based on earth’s mass and surface area, see links at the bottom for detailed values).

The constant energy in 10m depth – the ‘neutral zone’ – is about the same as the average temperature of the earth (averaged over one year over the surface of the earth): About 14°C. I will show below that this is not a coincidence: The temperature right below the fluctuating temperature wave ‘driven’ by the sun has to be equal to the average value at the surface. It is misleading to attribute the 10°C in 10m depths to the ‘hot inner earth’ only.

In this post I am toying with theoretical calculations, but in order not so scare readers off too much I show the figures first, and add the derivation as an appendix. My goal is to compare these results with our measurements, to cross-check assumptions for the thermal properties of ground I use in numerical simulations of our heat pump system (which I need for modeling e.g. the expected maximum volume of ice)

1. The surface temperature varies periodically in a year, and I use maximum, minimum and average temperature from our measurements, (corrected a bit for the mild last seasons). These are daily averages as I am not interested in the daily temperature changes between and night.
2. A constant geothermal flow of 65 mW/m2 is superimposed to that.
3. The slow transport of solar energy into ground is governed by a thermal property of ground, called the thermal diffusivity. It describes ‘how quickly’ a lump of heat deposited will spread; its unit is area per time. I use an assumption for this number based on values for soil in the literature.

I am determining the temperature as a function of depth and of time by solving the differential equation that governs heat conduction. This equation tells us how a spatial distribution of heat energy or ‘temperature field’ will slowly evolve with time, given the temperature at the boundary of the interesting part of space in question – in this case the surface of the earth. Fortunately, the yearly oscillation of air temperature is about the simplest boundary condition one could have, so you can calculate the solution analytically.
Another nice feature of the underlying equation is that it allows for adding different solutions: I can just add the effect of the real geothermal flow of energy to the fluctuations caused by solar energy.

The result is a  ‘damped temperature wave’; the temperature varies periodically with time and space: The spatial maximum of temperature moves from the surface to a point below and back: In summer (beginning of August) the measured temperature is maximum at the surface, but in autumn the maximum is found some meters below – heat flows back from ground to the surface then:

Calculated ground temperature, based on measurements of the yearly variation of the temperature at the surface and an assumption of the thermal properties of ground. Calculated for typical middle European maximum and minimum temperatures.

This figure is in line with the images shown in every textbook of geothermal energy. Since the wave is symmetrical about the yearly average, the temperature in about 10m depth, when the wave has ‘run out’, has to be equal to the yearly average at the surface. The wave does not have much chance to oscillate as it is damped down in the middle of the first period, so the length of decay is much shorter than the wavelength.

The geothermal flow just adds a small distortion, an asymmetry of the ‘wave’. It is seen only when switching to a larger scale.

Some data as in previous plot, just extended to greater depths. The geothermal gradient is about 3°C/100m, the detailed value being calculated from the value of thermal conductivity also used to model the fluctuations.

Now varying time instead of space: The higher the depth, the more time it takes for ground to reach maximum temperature. The lag of the maximum temperature is proportional to depth: For 1m difference in depth it is less than a month.

Temporal change of ground temperature at different depths. The wave is damped, but other simply ‘moving into the earth’ at a constant speed.

Measuring the time difference between the maxima for different depths lets us determine the ‘speed of propagation’ of this wave – its wavelength divided by its period. Actually, the speed depends in a simple way on the thermal diffusivity and the period as I show below.

But this gives me an opportunity to cross-check my assumption for diffusivity: I  need to compare the calculations with the experimentally determined delay of the maximum. We measure ground temperature at different depths, below our ice/water tank but also in undisturbed ground:

Temperature measured with Pt1000 sensors – comparing ground temperature at different depths, and the related ‘lag’. Indicated by vertical dotted lines, the approximate positions of maxima and minima. The lag is about 10-15 days.

The lag derived from the figure is in the same order as the lag derived from the calculation and thus in accordance with my assumed thermal diffusivity: In 70cm depth, the temperature peak is delayed by about two weeks.

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Appendix: Calculations and background.

I am trying to give an outline of my solution, plus some ‘motivation’ of where the differential equation comes from.

Heat transfer is governed by the same type of equation that describes also the diffusion of gas molecules or similar phenomena. Something lumped together in space slowly peters out, spatial irregularities are flattened. Or: The temporal change – the first derivative with respect to time – is ‘driven’ by a spatial curvature, the second derivative with respect to space.

$\frac{\partial T}{\partial t} = D\frac{\partial^{2} T}{\partial x^{2}}$

This is the heat transfer equation for a region of space that does not have any sources or sinks of heat – places where heat energy would be created from ‘nothing’ or vanish – like an underground nuclear reaction (or freezing of ice). All we know about the material is covered by the constant D, called thermal diffusivity.

The equation is based on local conservation of energy: The energy stored in a small volume of space can only change if something is created or removed within that volume (‘sources’) or if it flows out of the volume through its surface. This is a very general principles applicable to almost anything in physics. Without sources or sinks, this translates to:

$\frac{\partial [energy\,density]}{\partial t} = -\frac{\partial \overrightarrow{[energy\,flow]}}{\partial x}$

The energy density [J/m3] stored in a volume of material by heating it up from some start temperature is proportional to temperature, proportionality factors being the mass density ρ [kg/m3] and the specific heat cp [J/kg] of this material. The energy flow per area [W/m2] is typically nearly proportional to the temperature gradient, the constant being heat conductivity κ [W/mK]. The gradient is the first-order derivative in space, so inserting all this we end with the second derivative in space.

All three characteristic constants of the heat conducting material can be combined into one – the diffusivity mentioned before:

$D = \frac{\kappa }{\varrho \, c_{p} }$

So changes in more than one of these parameters can compensate for each other; for example low density can compensate for low conductivity. I hinted at this when writing about heat conduction in our gigantic ice cube: Ice has a higher conductivity and a lower specific heat than water, thus a much higher diffusivity.

I am considering a vast area of ground irradiated by the sun, so heat conduction will be one-dimensional and temperature changes only along the axis perpendicular to the surface. At the surface the temperature varies periodically throughout the year. t=0 is to be associated with beginning of August – our experimentally determined maximum – and the minimum is observed at the beginning of February.

This assumption is just the boundary condition needed to solve this partial differential equation. The real ‘wavy’  variation of temperature is closed to a sine wave, which makes the calculation also very easy. As a physicist I have trained to used a complex exponential function rather than sine or cosine, keeping in mind that only real part describes the real world. This a legitimate choice, thanks to the linearity of the differential equation:

$T(t,x=0) = T_{0} e^{i\omega t}$

with ω being the angular frequency corresponding to one year (2π/ω = 1 year).

It oscillates about 0, with an amplitude of T0. But after all, the definition of 0°C is arbitrary and – again thanks to linearity – we can use this solution and just add a constant function to shift it to the desired value. A constant does neither change with space or time and thus solves the equation trivially.

If you have more complicated sources or sinks, you would represent those mathematically as a composition of simpler ‘sources’, for each of which you can find a quick solution and then add up add the solutions, again thanks to linearity. We are lucky that our boundary condition consist just of one such simple harmonic wave, and we guess at the solution for all of space, adding a spatial wave to the temporal one.

So this is the ansatz – an educated guess for the function that we hope to solve the differential equation:

$T(t,x) = T_{0} e^{i\omega t + \beta x}$

It’s the temperature at the surface, multiplied by an exponential function. x is positive and increasing with depth. β is some number we don’t know yet. For x=0 it’s equal to the boundary temperature. Would it be a real, negative number, temperature would decrease exponentially with depth.

The ansatz is inserted into the heat equation, and every differentiation with respect to either space or time just yields a factor; then the exponential function can be cancelled from the heat transfer equation. We end up with a constraint for the factor β:

$i\omega = D\beta^{2}$

Taking the square root of the complex number, there would be two solutions:

$\beta=\pm \sqrt{\frac{\omega}{2D}}(1+i))$

β has a real and an imaginary part: Using it in T(x,t) the real part corresponds to exponential ‘decay’ while the imaginary part is an oscillation (similar to the temporal one).

Both real and imaginary parts of this function solve the equation (as any linear combination does). So we take the real part and insert β – only the solution for β with negative sign makes sense as the other one would describe temperature increasing to infinity.

$T(t,x) = Re \left(T_{0}e^{i\omega t} e^{-\sqrt{\frac{\omega}{2D}}(1+i)x}\right)$

The thing in the exponent has to be dimension-less, so we can express the combinations of constants as characteristic lengths, and insert the definition of ω=2π/τ):

$T(t,x) = T_{0} e^{-\frac{x}{l}}cos\left(2\pi\left(\frac {t} {\tau} -\frac{x}{\lambda }\right)\right)$

The two lengths are:

• the wavelength of the oscillation $\lambda = \sqrt{4\pi D\tau }$
• and the attenuation length  $l = \frac{\lambda}{2\pi} = \sqrt{\frac{D\tau}{\pi}}$

So the ratio between those lengths does not depend on the properties of the material and the wavelength is always much shorter than the attenuation length. That’s why there is hardly one period visible in the plots.

The plots have been created with this parameters:

• Heat conductivity κ = 0,0019 kW/mK
• Density ρ = 2000 kg/m3
• Specific heat cp = 1,3 kJ/kgK
• tau = 1 year = 8760 hours

Thus:

• Diffusivity D = 0,002631 m2/h
• Wavelength λ = 17 m
• Attenuation length l = 2,7 m

The wave (any wave) propagates with a speed v equivalent to wavelength over period: v = λ / tau.

$v = \frac{\lambda}{\tau} = \frac{\sqrt{4\pi D\tau}}{\tau} = \sqrt{\frac{4\pi D}{\tau}}$

The speed depends only on the period and the diffusivity.

The maximum of the temperature as observed in a certain depth x is delayed by a time equal x over v. Cross-checking our measurements of the temperature T(30cm) and T(100cm), I would thus expect a delay by 0,7m / (17m/8760h) = 360 h = 15 days which is approximately in agreement with experiments (taking orders of magnitude). Note one thing though: Only the square root of D is needed in calculations, so any error I make in assumptions for D will be generously reduced.

I have not yet included the geothermal linear temperature gradient in the calculation. Again we are grateful for linearity: A linear – zero-curvature – temperature profile that does not change with time is also a trivial solution of the equation that can be added to our special exponential solution.

So the full solution shown in the plot is the sum of:

• The damped oscillation (oscillating about 0°C)
• Plus a constant denoting the true yearly average temperature
• Plus a linear decrease with depth, the linear correction being 0 at the surface to meet the boundary condition.

If there would be no geothermal gradient (thus no flow from beneath) the temperature at infinite distance (practically in 20m) would be the same as the average temperature of the surface.

Daily changes could be taken into account by adding yet another solution that satisfies an amendment to the boundary condition: Daily fluctuations of temperatures would be superimposed to the yearly oscillations. The derivation would be exactly the same, just the period is different by a factor of 365. Since the characteristic lengths go with the square root of the period, yearly and daily lengths differ only by a factor of about 19.

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