# First Year of Rooftop Solar Power and Heat Pump: Re-Visiting Economics

After I presented details for selected days, I am going to review overall performance in the first year. From June 2015 to May 2016 …

• … we needed 6.600 kWh of electrical energy in total.
• The heat pump consumed about 3.600 kWh of that …
• … in order to ‘pump it up to’ 16.800 kWh of heating energy (incl. hot tap water heating). This was a mild season! .
• The remaining 3.000kWh were used by household and office appliances, control, and circulation pumps.

(Disclaimer: I am from Austria –> decimal commas and dot thousands separator 🙂

The photovoltaic generator …

• … harvested about 5.600kWh / year – not too bad for our 4,8kW system with panels oriented partly south-east and partly south-west.
• 2.000 kWh of that were used directly and the rest was fed into the grid.
• So 30% of our consumption was provided directly by the PV generator (self-sufficiency quota) and
• 35% of PV energy produced was utilized immediately (self-consumption quota).

Monthly energy balances show the striking difference between summer and winter: In summer the small energy needed to heat hot water can easily be provided by solar power. But in winter only a fraction of the energy needed can be harvested, even on perfectly sunny days.

Figures below show…

• … the total energy consumed in the house as the sum of the energy for the heat pump and the rest used by appliances …
• … and as the sum of energy consumed immediately and the rest provided by the utility.
• The total energy ‘generated’ by the solar panels, as a sum of the energy consumed directly (same aqua bar as in the sum of consumption) and the rest fed into the grid.

In June we needed only 300kWh (10kWh per day). The PV total output was more then 700kWh, and 200kW of that was directly delivered by the PV system – so the PV generator covered 65%. It would be rather easy to become autonomous by using a small, <10kWh battery and ‘shifting’ the missing 3,3kWh per day from sunny to dark hours.

But in January we needed 1100kWh and PV provided less than 200kWh in total. So a battery would not help as there is no energy left to be ‘shifted’.

Daily PV energy balances show that this is true for every single day in January:

We harvest typically less than 10 kWh per day, but we need  more than 30kWh. On the coldest days in January, the heat pump needed about 33kWh – thus heating energy was about 130kWh:

Our house’s heat consumption is typical for a well-renovated old building. If we would re-build our house from scratch, according to low energy standards, we might need only 50-60% energy at best. Then heat pump’s input energy could be cut in half (violet bar). But even then, daily total energy consumption would exceed PV production.

Economics

I have covered economics of the system without battery here and our system has lived up to the expectations: Profits were € 575, the sum of energy sales at market price  (€ 0,06 / kWh) and by not having to pay € 0,18 for power consumed directly.

In Austria turn-key PV systems (without batteries) cost about € 2.000 / kW rated power – so we earned about 6% of the costs. Not bad – given political discussions about negative interest rates. (All numbers are market prices, no subsidies included.)

But it is also interesting to compare profits to heating costs: In this season electrical energy needed for the heat pump translates to € 650. So our profits from the PV generator nearly amounts to the total heating costs.

Economics of batteries

Last year’s assessment of the economics of a system with battery is still valid: We could increase self-sufficiency from 30% to 55% using a battery and ‘shift’ additional 2.000 kWh to the dark hours. This would result in additional € 240 profits of per year.

If a battery has a life time of 20 years (optimistic estimate!) it must not cost more than € 5.000 to ever pay itself off. This is less than prices I have seen in quotes so far.

Off-grid living and autonomy

Energy autonomy might be valued more than economical profits. Some things to consider:

Planning a true off-grid system is planning for a few days in a row without sunshine. Increasing the size of the battery would not help: The larger the battery the larger the losses, and in winter the battery would never be full. It is hard to store thermal energy for another season, but it is even harder to store electrical energy. Theoretically, the area of panels could be massively oversized (by a factor – not a small investment), but then even more surplus has to be ‘wasted’ in summer.

The system has to provide enough energy per day and required peak load in every moment (see spikes in the previous post), but power needs also to be distributed to the 3 phases of electrical power in the right proportion: In Austria energy meters calculate a sum over 3 phases: A system might seem ‘autonomous’ when connected to the grid, but it would not be able to operate off-grid. Example: The PV generator produces 1kW per phase = 3kW in total, while 2kW are used by a water cooker on phase 1. The meter says you feed in 1kW to the grid, but technically you need 1kW extra from the grid for the water cooker and feed in 1kW on phase 2 and 3 each; so there is a surplus of 1kW in total. Disconnected from the grid, the water cooker would not work as 1kW is missing.

A battery does not provide off-grid capabilities automatically, nor do PV panels provide backup power when the sun is shining but the grid is down: During a power outage the PV system’s inverter has to turn off the whole system – otherwise people working on the power lines outside could be hurt by the power fed into the grid. True backup systems need to disconnect from the power grid safely first. Backup capabilities need to be compliant with local safety regulations and come with additional (potentially clunky / expensive) gadgets.

# Half a Year of Solar Power and Smart Metering

Our PV generator and new metering setup is now operational for half a year; this is my next wall of figures. For the first time I am combining data from all our loggers (PV inverter, smart meter for consumption, and heat pump system’s monitoring), and I give a summary on our scrutinizing the building’s electrical power base load.

For comparison: These are data for Eastern Austria (in sunny Burgenland). Our PV generator has 4.77kWp, 10 panels oriented south-east and 8 south-west. Typical yearly energy production in our place, about 48° latitude: ~ 5.300 kWh. In the first 6 months – May to November 2015 – we harvested about 4.000kWh.
Our house (private home and office) meets the statistical average of an Austrian private home, that is about 3.500 kWh/year for appliances (excl. heating, and cooling is negligible here). We heat with a heat pump and need about 7.200kWh electrical energy per year in total.

In the following plots daily and monthly energy balances are presented in three ways:

1. Total consumption of the building as the sum of the PV energy used immediately, and the energy from the utility.
2. The same total consumption as the sum of the heat pump compressor’s input energy and the remaining energy for appliances, computers, control etc.
3. Total energy generated by PV panels as the sum of energy used (same amount as contributing to 1) and the energy sold to the utility.

In summer there is more PV  energy available than needed and – even with a battery – the rest would needed to be fed into the grid. In October, heating season starts and more energy is needed by the heat pump that can be provided by solar energy.

This is maybe demonstrated best by comparing the self-sufficiency quota (ratio of PV energy and energy consumed) and the self-consumption quota (ratio of PV energy consumed and PV production). Number ‘flip’ in October:

In November we had some unusually hot record-breaking days while the weather became more typical at the end of the month:

This is reflected in energy consumption: November 10 was nearly like a summer day, when the heat pump only had to heat hot water, but on the colder day it needed about 20kWh (resulting in 80-100kWh heating energy).

In July, we had the chance to measure what the building without life-forms needs per day – the absolute minimum baseline. On July 10, 11, and 12 we were away and about 4kWh were consumed per day160W on average.

Note that the 4kWh baseline is 2-3 times the energy the heat pump’s compressor needs for hot water heating every day:

We catalogued all devices, googled for data sheets and measured power consumption, flipped a lot of switches, and watched the smart meter tracking the current consumption of each device.

Consumption minus production: Current values when I started to write this post, the sun was about to set. In order to measure the consumption of individual devices they have been switched an of off one after the other, after sunset.

We abandoned some gadgets and re-considered usage. But in this post I want to focus on the base load only – on all devices that contribute to the 160W baseline.

As we know from quantum physics, the observing changes the result of the measurement. It was not a surprise that the devices used for measuring, monitoring and metering plus required IT infrastructure make up the main part of the base load.

Control & IT base load – 79W

• Network infrastructure, telephone, and data loggers – 35W: Internet provider’s DSL modem / router, our router + WLAN access point, switch, ISDN phone network termination, data loggers / ethernet gateways for our control unit, Uninterruptible Power Supply (UPS).
• Control and monitoring unit for the heat pump system, controlling various valves and pumps: 12W.
• The heat pump’s internal control: 10W
• Three different power meters: 22W: 1) Siemens smart meter of the utility, 2) our own smart meter with data logger and WLAN, 3) dumb meter for overall electrical input energy of the heat pump (compressor plus auxiliary energy). The latter needs 8W despite its dumbness.

Other household base load – 39W

• Unobtrusive small gadgets – 12W: Electrical toothbrush, motion detectors, door bell, water softener, that obnoxious clock at the stove which is always wrong and can’t be turned off either, standby energy of microwave oven and of the PV generator’s inverter.
• Refrigerator – 27W: 0,65 kWh per day.

Non-essential IT networking infrastructure – 10W

• WLAN access point and router for the base floor – for connecting the PV inverter and the smart meter and providing WLAN to all rooms.

These are not required 24/7; you don’t lose data by turning them off. Remembering to turn off daily might be a challenge:

Non-24/7 office devices – 21W. Now turned off with a flip switch every evening, and only turned on when needed.

• Scanner/Printer/Fax: 8W. Surprisingly, there was no difference between ‘standby’ and ‘turned off’ using the soft button – it always needs 8W unless you really disconnect it.
• Server in hibernated state 4W. Note that it took a small hack of the operating system already to hibernate the server operating system at all. Years ago the server was on 24/7 and its energy consumption amounted to 500kWh a year.

Stuff retired after this ‘project’ – 16W.

• Radio alarm clock – 5W. Most useless consumption of energy ever. But this post is not meant as bragging about the smartest use of energy ever, but about providing a realistic account backed up by data.
• Test and backup devices – 7W. Backup notebooks, charging all the time, backup router for playground subnet not really required 24/7, timer switch most likely needing more energy than it saved by switching something else.
• Second old Uninterruptable Power Supply – 4W. used for one connected device only, in addition to the main one. It was purchased in the last century when peculiarities of the local power grid had rebooted  computers every day at 4:00 PM.

In total, we were able to reduce the base load by about 40W, 25% of the original value. This does not sound much – it is equivalent to a small light bulb. But on the other hand, it amounts to 350kWh / year, that is 10% of the yearly energy consumption!

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Logging setup:

• Temperature / compressor’s electrical power: Universal control UVR1611 and C.M.I. as data logger, logging interval 90 seconds. Temperature sensor: PT1000. Power meter:  CAN Energy Meter. Log files are exported daily to CSV files using Winsol. Logging interval: 90 seconds.
• PV output power: Datamanager 2.0 included with PV inverter Fronius Symo 4.5-3-M, logging interval 5 minutes.
• Consumed energy: Smart meter EM-210, logging interval 15 minutes.
• CSV log files are imported into Microsoft SQL Server 2014 for analysis and consolidation. Plots are created with Microsoft Excel as front end to SQL Server, from daily and monthly views on joined UVR1611 / Fronius Symo / EM-210 tables.

# Economics of the Solar Air Collector

In the previous post I gave an overview of our recently compiled data for the heat pump system.

The figure below, showing the seasonal performance factor and daily energy balances, gave rise to an interesting question:

In February the solar collector was off for research purposes, and the performance factor was just a bit lower than in January. Does the small increase in performance – and the related modest decrease in costs of electrical energy – justify the investment of installing a solar collector?

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

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’.

Depending on desired pay-back time, it might not – but this is the ‘wrong question’ to ask. Without the solar collector, the performance factor would not have been higher than 4 before it was turned off; so you must not compare just these two months without taking into account the history of energy storage in the whole season.

Bringing up the schematic again; the components active in space heating mode plus collector are highlighted:

(1) Off-the-shelf heat pump. (2) Energy-efficient brine pump. (3) Underground water tank, can also be used as a cistern. (4) Ribbed pipe unglazed solar collector (5) 3-way valve: Diverting brine to flow through the collector, depending on ambient temperature. (6) Hot water is heated indirectly using a large heat exchanger in the tank. (7) Buffer tank with a heat exchanger for cooling. (8) Heating circuit pump and mixer, for controlling the supply temperature. (9) 3-way valve for switching to cooling mode. (10) 3-way valve for toggling between room heating and hot water heating.

The combination of solar collector and tank is ‘the heat source’, but the primary energy source is ambient air. The unglazed collector allows for extracting energy from it efficiently. Without the tank this system would resemble an air heat pump system – albeit with a quiet heat exchanger instead of a ventilator. You would need the emergency heating element much more often in a typical middle European winter, resulting in a lower seasonal performance factor. We built this system also because it is more economical than a noisy and higher-maintenance air heat pump system in the long run.

Our measurements over three years show that about 75%-80% of the energy extracted from the tank by the heat pump is delivered to it by the solar collector in the same period (see section ‘Ambient Energy’ in monthly and yearly overviews). The remaining energy is from surrounding ground or freezing water. The water tank is a buffer for periods of a few very cold days or weeks. So the solar collector is an essential component – not an option.

In Oct, Nov, and March typically all the energy needed for heating is harvested by the solar collector in the same month. In ‘Ice Months’  Dec, Jan, Feb freezing of water provides for the difference. The ice cube is melted again in the remaining months, by the surplus of solar / air energy – in summer delivered indirectly via ground.

The winter 2014/2015 had been unusually mild, so we had hardly created any ice before February. The collector had managed to replenish the energy quickly, even in December and January. The plot of daily energies over time show that the energy harvested by the collector in these months is only a bit lower than the heating energy consumed by the house! So the energy in the tank was filled to the brim before we turned the collector off on February 1. Had the winter been harsher we might have had 10 m3 of ice already on that day, and we might have needed 140kWh per day of heating energy, rather than 75kWh. We would have encountered  the phenomena noted during the Ice Storage Challenge earlier.

This post has been written by Elke Stangl, on her blog. Just adding this in case the post gets stolen in its entirety again, as it happened to other posts tagged with ‘Solar’ recently.

# 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 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 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:

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).

I am maintaining a list of answers to Frequently Asked Questions here.

# Having Survived the Hottest July Ever (Thanks, Natural Cooling!)

July 2015 was the hottest July ever since meteorological data had been recorded in Austria (since 248 years). We had more than 38°C ambient air temperature at some days; so finally a chance to stress-test our heat pump system’s cooling option.

Heating versus cooling mode

In space heating ‘winter’ mode, the heat pump extracts heat from the heat source – a combination of underground water / ice tank and unglazed solar collector – and heats the bulk volume of the buffer storage tank. We have two heating circuits exchanging heat with this tank – one for the classical old radiators in ground floor, and one for the floor heating loops in the first floor – our repurposed attic.

Space heating mode: The heat pump (1) heats the buffer tank (7), which in turn heats the heating circuits (only one circuit shown, each has its circuit pump and mixer control). Heat source: Solar collector (4) and water / ice storage (3) connected in a single brine circuit. The heat exchanger in the tank is built from the same ribbed pipes as the solar collector. If the ambient temperature is too low too allow for harvesting of energy the 3-way valve (5) makes the brine flow bypass the collector.

The heat pump either heats the buffer tank for space heating, or the hygienic tank for hot tap water. (This posting has a plot with heating power versus time for both modes).

We heat hot tap water indirectly, using a hygienic storage tank with a large internal heat exchanger. Therefore we don’t need to fight legionella by heating to high temperatures, and we only need to heat the bulk volume of the tank to 50°C – which keeps the Coefficient of Performance high.

Hot tap water heating mode: The flow of water heated by the heat pump is diverted to the hygienic storage tank (6). Otherwise, the heat source is used in the same way as for space heating. In this picture, the collector is ‘turned off’ – corresponding to heating water on e.g. a very cold winter evening.

In summer, the still rather cold underground water tank can be used for cooling. Our floor heating loops become cooling loops and we simply use the cool water or ice in the underground tank for natural (‘passive’) cooling. So the heat pump can keep heating water – this is different from systems that turn an air-air heat pump into an air conditioner by reverting the cycle of the refrigerant.

Heating hot water in parallel to cooling is beneficial as the heat pump extracts heat from the underground tank and cools it further!

Cooling mode: Via automated 3-way valve (9) brine is diverted to flow through the heat exchanger in the buffer tank (7). Water in the buffer tank is cooled down so water in the floor ‘heating’ / cooling loops. If the heat pump operates in parallel to heat hot tap water, it cools the brine.

How we optimize cooling power this summer

Water tank temperature. You could tweak the control to keep the large ice cube as long as possible, but there is a the trade-off: The cooler the tank,  the lower the heat pump’s performance factor in heating mode. This year we kept the tank at 8°C after ‘ice season’ as long as possible. To achieve this, the solar collector is bypassed if ambient temperature is ‘too high’. The temperature in the tank rose quickly in April – so our ice is long melted:

The red arrow indicates the end of the ice period; then the set temperature of the tank was 8°C (‘Ice storage tank’ is rather a common term denoting this type of heat source than indicating that it really contains ice all the time.) Green arrows indicate three spells of hot weather. The tank’s temperature increased gradually, being heating by the surrounding ground and by space cooling. At the beginning of August its temperature is close to 20°C, so cooling energy has nearly be used up completely.

At the beginning of July the minimum inlet temperature in the floor loops was 17°C, determined by the dew point (monitored by our control system that controls the mixer accordingly); at the end of the month maximum daily ambient air temperatures were greater than 35°C, and the cooling water had about 21°C.

Room temperature. Cooling was activated only if the room temperature in the 1st floor was higher than 24°C – this allows for keeping as much cooling energy as possible for the really hot periods. We feel that 25°C in the office is absolutely OK as temperatures outside are more then 10°C higher.

Scheduling hot water heating. After the installation of our PV panels we set the hot water heating time slots to periods with high solar radiation – when you have more than 2 kW output power on cloudless days. So we utilized the solar energy generator in the most economic way and the heat pump supports cooling exactly when cooling is needed.

Using the collector for cooling in the night. If the ambient temperature drops to a value lower than the tank temperature, the solar collector can actually cool the tank!

Ventilation. I have been asked if we have forced ventilation, ductwork, and automated awnings etc. No, we haven’t – we just open all the windows during the night and ‘manually operated’ shades attached to the outside of the windows. We call them the Deflector Shields:

Manually operated ventilation – to be shut off at sunrise. We had already 30°C air temperature at 08:00 AM on some days.

South-east deflector shields down. We feel there is still enough light in the (single large) room as we only activate the subset of shields facing the sun directly.

These are details for two typical hot days in July:

The blue line exhibits the cooling power measured for the brine ‘cooling’ circuit. If the heat pump is off, cooling power is about 1 kW; during heat pump operations (blue arrows) 4 kW can be obtained. Night-time ventilation is crucial to keep room temperatures at reasonable levels.

The cooling power is lower than so-called standard cooling load as defined in AC standards – the power required to keep the temperature at about 24°C in steady-state conditions, when ambient temperature would be 30°C and no shades are used. For our attic-office this standard cooling power would amount to more than 10 kW which is higher than the standard (worst case) heating load in winter.

Overall electrical energy balance

I have been asked for a comparison of the energy needed in the house, the heat pump in particular, and the energy delivered by the PV panels and fed in to the grid.

PV numbers in July were not much different from June’s – here is the overview on June and July, maximum PV power on cloudless days has decreased further due to the higher temperatures:

In July, our daily consumption slightly decreased to 9-10 kWh per day, the heat pump needs 1-2 kWh of that. The generator provides for 23 kWh per day,

Currently the weather forecast says, we will have more than 35°C each noon and 20-25° minimum in the night until end of this week. We might experience the utter depletion of our cooling energy storage before it will be replenished again on a rainy next weekend.

# 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?

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.

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’.

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.

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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 carnotcycle.wordpress.com:

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.

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.

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:

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.

Happily harvesting ambient energy.

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