A Mini-Fridge Investigation

Friends, on 7th June I will be giving a talk at the Cheltenham Science Festival with the illustrious Andrea Sella on the topic of Heat Pumps. And I am growing increasingly anxious.

Click on the image for a larger version. Sadly the publicity for our talk at Cheltenham has used the wrong title! But which title have Andrea and I chosen?

I’ve been working on the talk with Andrea for weeks, but there are still parts that we haven’t finalised, including exactly which demonstrations to include. In my search for demonstrations, I thought I would take a look at a mini-fridge that would be easy to position on a table-top. Forty pounds later, a mini-fridge arrived.

Click on Image for a larger version

I managed to get some nice thermal images, and while I was playing around, I thought I would see if I could work out the actual cooling power of such a tiny refrigerator.

Click on image for a larger version

Measurements

My idea was to measure the rate at which a glass of water cooled inside the fridge. From the definition of the heat capacity of water, I knew that to cool one gramme of water by 1 °C required the removal of 4.18 joules of energy. If I measured the time taken to cool a glass of water, I could work out the rate at which heat energy was being removed in joules per second i.e. watts.

So I poured 300.7 grams of water into a glass weighing 371.9. grams. I then placed the glass inside the refrigerator. I deployed four thermocouples to monitor the progress of the cooling.

• I put one thermocouple in the water.
• I put one thermocouple on the cold plate at the back of the inside of the fridge.
• I put one thermocouple on the hot plate at the back of the outside of the fridge
• I used one thermocouple to monitor the temperature of the ambient air.

I then used a datalogger to record the temperatures every 30 seconds. The results from the first 8 hours are shown below.

Click on image for a larger version. Graph showing the temperature versus time of the four thermocouples positioned around the fridge. See text for details of their location.

The data looked pretty much as I expected, with a couple of anomalies. I do not know the cause of the ‘spike’ at 5.2 hours, but the change in cooling rate just before six elapsed hours was caused by leaving open the door of the kitchen laboratory.

The graph below is the same as the previous graph, but has arrows to show the heat flows.

Click on image for a larger version. Heat flows naturally from the water to the cold sink. The heat pump then moves that heat from the cold sink to the hot sink at the rear of the refrigerator. Heat then flows naturally from the hot sink to the air in the room.

Cooling Power

To work out the cooling power, I started with the data of temperature versus time for the water.

Click on image for a larger version. Graph showing the temperature (°C) of the water versus time.

I then worked out the slope of graph above. The cooling rate is only a few thousandths of a degree per second, so in order to measure this with thermocouples with a resolution of only 0.1 °C, I averaged the cooling data over ±10 minutes.

Click on image for a larger version. Graph showing the cooling rate (°C/s) of the water versus time.

Just after the fridge is switched on, the cooling rate peaks at around 1.5 thousandths of a degree per second, or 5.4 °C/hour. The oscillations in the cooling rate are probably due to convection of the water in the glass.

To convert this cooling rate into an estimate of cooling power, one needs to multiply the data above by the heat capacity of the water and the glass, in this case, 1575 J/°C.

Click on image for a larger version. Graph showing the cooling power of the refrigerator.

The data suggests that the cooling power peaks at around 2.5 watts and then falls to just a fraction of a watt. The electrical power drawn was 40 W so overall the efficiency was around 6%.

Lowest Temperature

The cold plate in the back of the fridge gets impressively cold, cooling to below -5 °C. So why doesn’t the water cool to this temperature?

To investigate this, I plotted the rate of heat flow out of the water (in watts) not versus time, but versus the temperature difference between the water and the cold plate.

Click on image for a larger version. Graph showing the cooling power of the refrigerator as measured by the cooling rate of a glass of water, versus the difference in temperature (Delta T) between the water and the cold back plate.

The data show that the cooling rate is roughly proportional to the temperature difference between the water and the back plate. But the cooling rate falls to zero when there is around 9 °C of temperature difference. Why doesn’t the water keep on cooling?

The reason is that heat is flowing into the water through the walls of the refrigerator. The cooling power of the heat pump probably remains at around 2 or 3 W, but as the internal temperature of the fridge falls, heat ‘leaks’ through the insulating the walls.

Click on image for a larger version. A more elaborate version of the graph above. showing additionally heat flowing into the cold sink from the hot sink, and the room. The fridge uses insulation to try to minimise these flows, but they are not zero.

An analogy would be using a water pump to bail out a leaking vessel. When the rate of bailing is equal to rate at which water is leaking in, then the water level doesn’t change. Similarly, the colder the inside of the refrigerator becomes, the more significant heat leaks become. The lowest temperature occurs when the rate at which the heat pump removes heat is equal to the rate at which heat leaks from the environment.

Reflections

Friends, the mini fridge was a little more powerful than I expected, and I was pleased that I managed to remember how to operate the data logger. But none of this is relevant to the talk I am trying to finish with Andrea – somehow these simple measurement exercises seem very attractive when there is proper work to be done.

A Short Talk about my Low-Carbon Home

Friends, today I abandoned my usual Saturday morning ritual of doing a quiz and a crossword with my wife at our local café.

Instead I travelled far beyond the borders of Teddington to give a  “A Short Talk about my Low-Carbon Home” at the Kingston Efficient Homes Show. There were many celebrities there include Ed Davey, the leader of the Liberal Democrats, and the Green Man, whom Wikipedia informs me is not in fact a pagan mythological figure.

The Green Man visited the Kingston Efficient Homes Show.

I was a little discombobulated at the start of the talk because nobody turned up to introduce me, and the clock in the lecture room was slow. And so while I was just waiting to start, I should already have started. And at the end I was being told to wind up, when in fact I still had many minutes left. Hey Ho.

I promised the audience I would put the Powerpoint slides from the show here, and below is a belated re-recording of the 20 minute talk for those who missed it. Somehow, it is 30 minutes long :-(.

Reflection 

I found the event very moving: it was full of people trying to make the world a better place.

• There were slightly bewildered members of the public prepared to spend money on heat pumps and insulation and solar PV and batteries.
• There were installers – the shock troops on the front line of combating climate change.
• There were architects – including the designers of the fantastic Bale House in Hastings.
• There were members of the local Council and politicians.

But one thing annoyed me: the endless request for estimates of ‘payback time’ or ‘return on investment’.

As Bill Nye, the mild-mannered American science communicator so eloquently put it, “The planet is on fire“. And people still want to find out whether it’s worth their while to put out the fire? He was actually rather more pithy than that.

If you found that video amusing, here’s another more upbeat version.

Breville HotCup: Thermodynamic Reflections

Friends, you may recall my long-standing fascination with boiling water efficiently: see for example:

• Kettle versus Quooker (2023)
• A Watched Pan (2022)
• Gas or Electric Kettle? (2012)

So ‘boiling water’ was a topic on which I thought I had written my last word. But visiting some sophisticated neighbours, I saw that they had a Breville HotCup – a kettle that held a reservoir of water, but which then dispensed just a single cup of boiled water at the push of a button. Wow!

Remember that in a conventional kettle one almost always boils too much water, thus wasting energy. And in a Quooker one keeps several litres of water at 100 °C so it is ready when you require hot water. Could the HotCup be the clever device that boils exactly the right amount of water just when you need it, without wasting energy on ‘standby’?

Well, after reflecting on the future rubbish I was creating, I bought one and tested it. It is a perfectly pleasant item, and does indeed dispense individual cups of water quickly – and since this is how I generally consume tea – I must confess to being pleased.

But after assessing its performance from an energy efficiency standpoint, I found myself disappointed. At best, it is ~ 80% efficient, but at its worst it is only 25% efficient! It took me some time to work out how it could be so bad, but I did eventually figure it out. Allow me to explain.

What is a Breville HotCup?

There a number of HotCup models, but the key idea behind them all is that it is a kettle which holds a reservoir of water, and then boils and dispenses just a single ‘cup-full’ – variable between 150 ml and 320 ml – at a time.

The video below shows the HotCup in operation along with the equipment I used to make measurements.

Measurements

I couldn’t see immediately how the device worked, but I set out to measure its performance using the standard techniques of ‘kitchen calorimetry’.

• I weighed the HotCup when empty and then filled it with about 1.5 lites of water. I then weighed the amount of water dispensed (g).
• I measured the temperature of the water reservoir, and the maximum temperature of the dispensed water (°C).
• I recorded the electricity used on a plug-in electricity meter (in kWh).
• I timed the boiling process using the timer on my phone (s).

From the mass of water dispensed and its rise in temperature, I could work out how much heat energy had been given to the water. I could then compare this with the measured amount of electrical energy consumed. Comparing these two figures I could work out the efficiency with which the consumed energy had been converted into hot water.

Remembering the Golden Rule of Experimental Physics, I repeated the experiment multiple times to assess more or less what was going on. Then when I had practiced a couple times, I made one set of readings with the HotCup set to dispense small cups of water (~150 ml) and one with it set to dispense large cups of water (~330 ml). For each setting I repeated the measurements until the reservoir appeared to be empty. The results are shown below.

Click on image for a larger version. Results for successive SMALL cups of water dispensed. Top Left: The temperature of the reservoir was observed to rise as cups of water were dispensed reaching nearly 80 °C. Top Right: The time taken to dispense a cup of water decreased from about 50 s to about 20 s. Bottom Left: The maximum temperature recorded in the cup into which the water was dispensed. Bottom Right: The estimated efficiency of water heating. The average efficiency is only 25%.

Click on image for a larger version. Results for successive LARGE cups of water dispensed. Top Left: The temperature of the reservoir was observed to rise as cups of water were dispensed. Top Right: The time taken to dispense a cup of water was about 40 s. Bottom Left: The maximum temperature recorded in the cup into which the water was dispensed. Bottom Right: The estimated efficiency of water heating. The average efficiency is about 80%.

Conclusions

Having observed the device in operation and measured its performance, I think I can now see how it works.

I think that the HotCup always boils the same amount of water in a boiling chamber – a mini kettle-within-the-kettle. The device uses the pressure built up within the chamber to push out the boiled fluid, and then discharges the unused hot liquid back into the reservoir.

By analysing the inefficiency of the device as a function of the amount of water dispensed, I estimated the volume of the boiling chamber to be approximately 400 ml.

Click on image for a larger version. Plotting the inefficiency as a function of dispensed volume, I estimate that the ‘boiling chamber’ within the HotCup has a volume of approximately 400 ml.

The TOP LEFT graphs in the two panels above show that the reservoir temperature rises after each cupful has been dispensed. It is clear that this rise is larger for the small cup (150 ml) dispensation in which most of the hot water (400 ml – 150 ml = 250 ml) is put into the reservoir. I made a model of this – shown as a dotted red line – and this seems to roughly describe the data.

Thermodynamic Reflections

Friends, I have had this device in my house now for a couple of months, and since my wife and I generally boil the kettle for individual cups of tea, it is quite convenient.

But as a calorimetric thermodynamicist, I must confess that after making these measurements I was at first disappointed. But on reflection, the performance is actually not so bad.

The inefficiency is the exactly the same as if one used 400 ml of water in a kettle to prepare a single beverage. This volume is close to or below the minimum fill level for many kettles. And so although these results look bad, they are probably no worse than using a kettle.

However, if my wife and I both wanted to drink tea at the same time, or if I wanted to boil larger volumes of water for cooking, a conventional kettle would be more efficient.

Another Heat Pump Spreadsheet: Beyond the Rule of Thumb

Friends, around a year ago I wrote an article and made a YouTube video about using a ‘Rule of Thumb’ for estimating the size of heat pump required to replace a gas boiler in a dwelling.

The ‘Rule of Thumb’ is splendidly simple: one just divides the previous year’s gas consumption by 2,900 to give the heat pump size in kilowatts. So if a dwelling used 10,000 kWh of gas the previous year, then one would estimate that it needed a 3.4 kW heat pump. The YouTube video explaining why the rule works has been watched an astonishing 37,000 times, and many people have left comments telling me they found the rule helpful and accurate.

The basic reason the rule works is because (a) most gas consumption is spent heating homes (rather than heating hot water or food) and (b) the climate of the southern half of the UK does not vary that much. The rule of thumb uses gas consumption as an indicator of the amount heat which enters a dwelling and uses climate data – in the form of heating degree days – to estimate how cold it gets in a particular locale. You can find a detailed description here, here, here and here!

But one or two people have told me that it gave them answers they thought were quite wrong. It turned out that these people often only put their gas boilers on for an hour or two per day, and so most of the time their dwellings were unheated. Alternatively, some people – particularly with families – used a lot of hot water every day – and so this formed an unusually large fraction of their gas consumption.

So I thought it would be nice to develop something just a little more sophisticated than the ‘Rule of Thumb’ that would take account of some of these factors. I did this last summer and sent it to an academic expert for feedback. The feedback was devastating: they basically told me that everything was wrong. And despite trying to modify the spreadsheet to meet their criticism, they seemed unmollified. So, shaken, I abandoned the idea for a while.

But recently I have been thinking about the idea again and decided that in fact I thought the spreadsheet was useful after all, and that it could also help with one other problem: sizing of radiators.

The reason I think this endeavour is important is that people who are thinking about installing heat pumps have faced a campaign by the fossil fuel industry and their knowing (and unknowing) shills, a campaign designed to instil fear, uncertainty and doubt (FUD). Every year of delay in installing heat pumps keeps the profits of fossil fuel companies healthy, and impoverishes the world in which our children will have to live.

This is not to say that there are not legitimate questions and uncertainties about installing a heat pump. So this spreadsheet is a transparent tool that can help people make rational choices and – I hope – help them to overcome the FUD.

• You can download the spreadsheet here:Link
• Spreadsheet updated to version 6.01 on 3/3/23

I have tried to make the spreadsheet Good For Nothing™ 🙂 . But mistakes will have slipped through: if you find one, please accept my apology in advance and let me know in the comments.

Spreadsheets Galore

The ‘spreadsheet’ is actually six spreadsheets linked together in an Excel™ Workbook. Each Spreadsheet has its own ‘tab’. Six spreadsheets may sound daunting, but really this could all be on one spreadsheet. Using several sheets actually makes things simpler.

Click on image for a larger version. The introductory ‘tab’ of the Excel™ Workbook showing the other 6 tabs. Users are recommended to save the downloaded copy and experiment with a ‘working copy’.

• The first spreadsheet helps people estimate the average temperature in their dwelling, and also the maximum temperature they like.
• The second spreadsheet helps people estimate the amount of hot water they use.
• The third spreadsheet uses the ideas behind the Rule of Thumb, but modified to take account of the estimates on the first two spreadsheets. It suggests a likely required size of heat pump and a few other building parameters that specialists might find interesting.
• The fourth spreadsheet allows people to see how the area of radiators and the type of radiators affects how hot the water flowing through the radiators needs to be in order to keep their home at the maximum temperature they desire.
• The fifth spreadsheet allows people to make more detailed calculations based on the number, size and type of radiators in their own dwelling.
• Finally, the sixth spreadsheet summarises the results from the previous spreadsheets and estimates the likely savings in cost and carbon dioxide emissions.

Let me show you each spreadsheet works in a little more detail.

Sheet 1: Household Temperature

Click on image for a larger version. Spreadsheet designed to allow a user to indicate the temperature changes in their home throughout a typical winter day.

Click on image for a larger version. As above, but showing a different temperature profile.

On this tab of the workbook, one can specify how the temperature varies inside a dwelling on a typical winter day. There are four times periods and each one can be set to one of three user-chosen temperatures.

The spreadsheet then calculates:

• The average temperature in the dwelling which is useful for calculating the average heat loss and hence energy consumption.
• The maximum temperature required which determines the required power of a heat pump able to heat the dwelling.

Sheet 2: Domestic Hot Water

Click on image for a larger version. Do you know how much hot water your dwelling uses each day.

I have been told that – in the absence of any other information – a good guess for the amount of gas used to heat hot water in a household is 3 kWh per person per day. This tab uses this figure to estimate how much of the annual gas usage is for domestic hot water.

If a user somehow has a better estimate, they can use their own estimate instead.

Sheet 3: Main Calculation

Click on image for a larger version. This ‘tab’ carries out the main heat pump size calculation.

This tab carries out the same calculation as the Rule of Thumb but now with a little more information about a particular user’s dwelling. It incorporates the data from the first two tabs on average and maximum temperatures and domestic hot water usage. It asks the user for the annual gas consumption and their approximate location (within around 100 miles). The location is used to estimate how cold the weather is likely to have been based on analysis of the heating degree-day records from 21 locations in the UK and Ireland.

Click on image for a larger version. This tab carries out the main heat pump size calculation.

The spreadsheet then estimates several parameters that characterise the level of thermal insulation of the dwelling and – most importantly from the perspective of this article – the heat pump size required for the dwelling.

Sheet 4: Radiators

Click on image for a larger version. This tab allows users to see how the area of radiators, and the type of radiators affect the performance of the heating system.

This tab allows users to see how – in general – the area of radiators, and the type of radiators affects the performance of the heating system. First one sets a maximum flow temperature for the system – this is the temperature of the hot water as it enters the radiators.

Heat pumps typically use weather compensation, which means that when the weather is cold, the heat pump increases the temperature of the water flowing in the radiators. For a heat pump the maximum flow temperature required in the coldest weather should ideally be below 50 °C.

Click on image for a larger version. This tab allows users to see how the area of radiators, and the type of radiators affect the performance of the heating system.

The table above shows – for the heat pump size calculated on the previous tab – what combinations of total radiator area and types of radiator will be able to heat the dwelling adequately.

For heat pumps to work at their very best, the temperature of the water flowing in the radiators should be as low as possible while still allowing the dwelling to be adequately heated.

In the example above the heat pump needs to transfer 5,296 watts of heating power to the dwelling.

• The table shows that this would require 9 square metres of single-panel/single-fin (Type 11) radiators, but the same heating could be done with just 5 square metres of double-panel/double-fin (Type 22) radiators.
• Alternatively one might use 9 square metres of double-panel/double-fin (Type 22) radiators because this would require a flow temperature in the radiators 39. 8°C rather than 49.2 °C – and this reduced flow temperature would result in increased heat pump efficiency, and lower running costs.

Sheet 5: More Radiators

Click on image for a larger version. This tab allows users to see how the number, size and type of radiators in their dwelling affect the performance of the heating system.

The previous tab allowed users to see in general terms how the area of radiators, and the type of radiators affect the performance of the heating system. On this tab a user can input the size (width and height) and type of their existing radiators and see whether – for the flow temperature set on the previous tab – they can release enough heat into their dwelling.

Click on image for a larger version. This TAB allows users to see how the area of radiators, and the type of radiators affect the performance of the heating system.

By putting in data on their existing radiators – the radiator type is input via a drop-down menu – the heating power of each radiator is calculated at the maximum allowed flow temperature. The heating power of each radiator is then summed up to see if the assemblage of radiators in the dwelling is capable of providing enough heating power to keep the dwelling warm on a cold day. This is shown as a percentage on a bar chart.

If a figure of 100% cannot be reached with existing radiators, then users can see whether 100% can be achieved by either adding radiators, or replacing radiators with larger ones, or radiators with more panels and fins.

Sheet 6: Summary

Click on image for a larger version. This tab summarises the results from the previous tabs and compares the cost and carbon dioxide emissions of systems using a gas boiler or alternatively, a heat pump.

Nearly finished! This summary tab collects together the conclusions from the previous spreadsheets. If a user enters the cost of their electricity and gas, the spreadsheet will then estimate the likely running costs of a gas boiler and a comparable heat pump.

The annual costs of the gas installation are estimated based on the users estimate of their own gas consumption. The running costs of the heat pump installation are based on an estimated seasonal coefficient of performance (SCOP).

The coefficient of performance (COP) of a heat pump is a measure of the efficiency of a heat pump measured over a period of typically an hour, a day or a week. In mild weather, the COP will be high (perhaps 4) and in cold weather the COP will be low (perhaps 2.5). SCOP measures the efficiency of a heat pump averaged over a whole year.

If a user experiments with different flow temperatures they will find that the lower the maximum flow temperature they plan for, the higher the achievable SCOP and the lower will be their running costs. Typically users will find that with the relative costs of electricity and gas as they are now (April 2023) at a ratio of roughly 3 to 1, a heat pump installation will commonly be a little bit cheaper to run than a gas boiler, but the difference is not very large compared with the capital cost of the installation.

Click on image for a larger version. This tab summarises the results from the previous tabs and compares the cost and carbon dioxide emissions of systems using a gas boiler or alternatively, a heat pump.

And finally – and this is the point of the entire endeavour – the spreadsheet makes a comparison of the carbon dioxide emissions from a dwelling heated either with a heat pump or a gas boiler. It is here that the entire point of running a heat pump becomes clear: carbon dioxide emissions from a heat pump installation are generally around 75% lower than an equivalent gas boiler. And that’s why this matters.

Click on image for a larger version. Graph showing the annual emissions of carbon dioxide from a gas boiler and an equivalent heat pump installations.

I love Kathy Joseph

Kathy Joseph

Friends, on this blog I have previously declared my love of James Hansen and Greta Thunberg. Today I would like to add Kathy Joseph to that list.

I encountered Kathy Joseph on YouTube where she has a channel focussed on what might broadly be described as the “History of Physics”. But it is so much more than that: it is a channel that celebrates the essential humanity of the endeavour that is Physics. And she glories in the way that a little insight can enhance the joy we feel at comprehending even a small fraction of the wonders in the world around us.

I am currently reading her delightful book, The Lightning Tamers about the pre-history of electrical science and our understanding of the basic physics of matter. And as I read more,  I have had the sinking feeling that she has written a book that I would love to have written. But she has done it better than I ever could have.

In her writing she combines plain language, with careful historical annotations. Let me give an example from page 34 of the book in which we learn the origin of the word “battery”: have you ever wondered why we use the same word for an electrical storage device and an arrangement of artillery?

 [Benjamin] Franklin was having so much fun that on April 29, 1749, he decided to have an electrical “party of pleasure” where he killed a turkey by electric shock, roasted it electrically, toasted to electricity with drinks that would give small electric shocks, then ended the party with “the discharge of guns from an electrical battery.

This party was to have a lasting impact, as after the story of the party was recalled in a book, Leyden Jars [early electrical storage devices] started to sometimes be referred to as batteries even though Franklin was just referring to a battery of toy electric guns powered by Leyden Jars.

And in her YouTube videos you can hear the enthusiasm that pervades her writing. She has what seems like hundreds of videos reaching back over the last 5 years, and they all feature the same direct style of narration combined with historical illustrations. I’ve embedded an early video on electricity below, but she covers much more complex subjects in later videos.

If you study or teach Physics at any level from GCSE to degree level, I guarantee your understanding will be enhanced by the historical context that Kathy Joseph offers.

Christmas Break

Friends, it’s been a busy year, and I’m going to take break from blogging for a week or so.

I am really grateful to everyone who reads this blog. I would probably be writing it whether or not you were reading it because writing helps me to clarify my thoughts. But I find the idea that anyone reads what I have written very moving.

So, thank you, and best wishes to you and yours for 2023.

Anyway…

The other day I clicked back through the 920 articles I have written since 2008 and came across this slightly chaotic video of a talk I gave at NPL in 2017 on the physics of candles.

And re-watching it I was amused and distracted, and so I thought I would re-post it in case you too might be amused or distracted!

Candles really are astounding! For example:

• Did you know that wax in bulk is not flammable?
• Did you know the temperature of a candle flame?
• Did you know that a candle stores 8 times more energy than a stick of dynamite?

If would like to access some of the fancy PowerPoint™ animations you can download the PowerPoint file here.

The highlight of the talk is using a candle to power a thermo-electric generator, which in turn powers a USB port, which in turn powers a torch, which is brighter than the candle.

And by the way, here is the slow-motion candle-relighting movie that is embedded in the PowerPoint but which doesn’t show up well in the lecture theatre view.

Thanks again to Brian Madzima for the videography and editing, and Nikita Mezhnyakov for the photograph.

Estimating the heat capacity of my house

Friends, the spell of cold weather at the start of December 2022 has led to me breathlessly examining data on the thermal performance of the heat pump and the house.

During this period, outside temperatures fell as low as -5 °C and average daily temperatures were below 0 °C. In order to try to keep the internal temperature constant, I studied measurements of internal temperature taken every 2 minutes. The data were pretty stable, only rarely falling outside the bound of 19.5 °C ± 0.5 °C.

But looking in detail, I noticed a curious pattern.

Click on image for a larger version. Two graphs from the period 6th to 18th December 2022. The upper graph shows the air temperature in the middle of the house. At around 01:30 each night the temperature fell sharply. The lower graph shows the rate of change of the air temperature versus time (°C/hour). From this graph it is clear that the rate at which the temperature fell was approximately -0.95 °C/hour.

The upper graph shows sharp falls in temperature at 01:30 each night. These were caused by the heat pump switching to its hot water heating cycle. Prior to this, the heat flowing into the house from the heat pump was more-or-less balanced by the heat flowing out. But when the heat pump switches to heating the domestic hot water, there was no heating from the heat pump and the internal temperature fell.

The lower graph shows the rate of change of the air temperature (°C/hour) versus time over the same period. From this graph it is clear that the rate at which the air temperature fell during the domestic hot water cycles was approximately 0.95 °C/hour.

With a little mathematical analysis (which you can read here if you care) this cooling rate can be combined with knowledge of the heat transfer coefficient (which I estimated a couple of weeks ago) to give estimates of (a) the time constant for the house to cool and (b) the effective heat capacity of the house.

Analysis: Time Constant 

The time constant for the house, is the time for the temperature difference between the inside and outside of the house to fall to ~37% of its initial value after the heating is removed.

The time constant is estimated as (the initial temperature difference) divided by (the initial cooling rate). In this case  the initial temperature difference was typically ~20 °C and the initial cooling rate was 0.95 °C/hour, so the time constant of the house is roughly 21 hours. Sometimes it’s useful to express this in seconds: i.e. 21 x 3,600 = 75,600 seconds.

This suggests that if we switched off all the heating when the house was at 20 °C and the external temperature was 0 °C, the house would cool to roughly 7.4 °C after 21 hours. Intuitively this seems right, but for obvious reasons, I don’t want to actually do this experiment!

Note that this time constant is a characteristic of the house and does not vary with internal or external temperature.

Analysis: Thermal Resistance  

A couple of weeks ago I posted an analysis of the heating power required to heat our house as the ‘temperature demand’  increased as the external temperature fell. The summary graph is shown below.

Click on graph for a larger version. Graph of average heating power (in kW) versus temperature demand (°C) for the first 10 days of December 2022.

From this I concluded that Heat Transfer Coefficient (HTC) for the house was around 165 W/°C.

The inverse of the HTC is known as the thermal resistance that connects the inside of the house to the external environment. So the thermal resistance for the house is ~ 1/165 = 0.00606 °C/W.

Analysis: Heat Capacity  

A general feature of simple thermal analyses is that the time constant, thermal resistance and heat capacity are connected by the formula:

Time constant = Thermal resistance x Heat Capacity

Since we have estimates for the time constant (75,600 s) and the thermal resistance (0.00606 °C/W) we can this estimate the heat capacity of the house as 12,474,000 joules per °C.

This extremely large number is difficult to comprehend, but if we change to units more appropriate for building physics we can express the heat capacity as 3.5 kWh/°C. In other words, if the house were perfectly insulated, it would take 3.5 kWh of heat to raise its temperature by 1 °C.

We can check whether the number makes sense by noticing that the main mass of the house is the bricks from which it is built. A single brick weighs ~3 kg and has a heat capacity of ~2,400 J/°C. So thermally it looks like my house consists of 12,474,000/2,400 ~ 5,200 bricks.

However this estimate is too small. Even considering just the 133 square metres of external walls, if these have the equivalent of 120 bricks per square metre that would come to ~16,000 bricks.

So I think this heat capacity estimate just applies the heat capacity of the first internal parts of the house to cool. This refers to all the surfaces in contact with the air. So I think this is the effective heat capacity for cooling just a degree or two below ambient.

Why did I bother with this? The ‘Setback’ Problem

Friends, sometimes I go upstairs and forget why I went. And sometimes I start analysing things and can’t remember why I started! Fortunately, in this case, I had a really good reason for wanting to know the effective heat capacity of my house.

When I analysed the heat flows previously, I have had to assume that the temperature of the house was stable i.e. that there was a balance between the heat flowing in and the heat flowing out. As long as the temperature of the fabric of the house is stable, then it is neither storing or releasing heat.

However this isn’t enough if we want to understand some very common problems in the thermal physics of houses, such as “the setback problem”: this is the question of whether it’s smart to reduce the temperature of a dwelling (say) overnight and then to re-heat it once again in the morning. To answer this question we need to know about the rate at which a house cools down (it’s time constant) which is equivalent to knowing its heat capacity.

And that is why I have done this prolonged and tedious analysis. The next article will be an analysis of ‘The Setback Problem’. And it will be much more exciting!

Annual variation in location of sunrise and sunset

Friends, we are just 16 days away from the winter solstice, and those of us who live in the UK are dismally aware that at this time of year, the Sun only rises above the horizon for about 8 hours.

And not only does the Sun rise above the horizon for a short time, it also fails to rise very high in the sky.

Together these factors make for poor yields from solar panels.

This article is about a third factor which I feel is sometimes overlooked: the location of sunrise and sunset.

The location of sunrise#1

I asked OpenAI Chatbot about this:

How does the location of sunrise vary through the year in the UK?

The location of sunrise in the UK varies significantly throughout the year. In the winter months, sunrise occurs in the southeast, while in the summer months, it appears in the northeast. The amount of daylight hours in the UK also increases as the year progresses, with the summer months having the longest days and the winter months having the shortest.

This is an excellent answer. Accurate and concise. I followed up:

How far north of east does the sun rise at the summer solstice?

At the summer solstice (the longest day of the year), the sun rises as far north as 70° north latitude in the UK, which is around halfway between East Anglia and the Shetland Islands.

In contrast, this answer is utter nonsense! So I guess I will have to write this article myself!

The location of sunrise#2

I was interested in the location of sunrise because of the new panels I am installing will face about 22° north of east – not a very favourable location.

I looked up data for each week of the year from The Time and Date website: the data below are relevant to London, but you can look up data for many other locations worldwide if you are interested.

Click on image for a larger version. This is an extract from tables at the Time and Date web site. It has both the time of sunrise and sunset and the angle of sunrise and sunset measured clockwise from due North.

I then collated the results and plotted them through the weeks of the year.

Click on image for a larger version. This graph shows how the location of sunrise and sunset vary through the year. Angles of sunrise and sunset measured clockwise from due North.

The graph above shows that the phrase: “the Sun rises in the East and sets in the West” is only approximately true. For 6 months of the year, the Sun rises north of East and sets north of West.

My New Solar Panels
This is probably not news to anyone, but I found it interesting, because I am putting solar panels on my home that face north of East.

Click on image for a larger version. Google Maps view of my house showing existing solar panels in blue and the new panels in Yellow. For 6 months of the year between spring and autumn equinoxes, the panels should produce a useful solar yield in the morning.

After plotting these lines on the map, and noting which houses the lines intercepted, I was able to translate them onto a photograph to show the expected location of sunrise through the year.

Click on image for a larger version. Photograph showing the ‘panels-eye’ view of the street-scene at the back of the house For 6 months of the year between spring and autumn equinoxes, the panels should produce a useful solar yield in the morning.

Considering all the panels on the house,- including the 12 installed in November 2020 – in summer the system should generate from early dawn – only just after 4 a.m. in mid-summer, to almost 8:00 p.m. So despite the poor orientation, the Easy-PV calculator suggests the 5 panels will generate 1,338 kWh per year (268 kWh/panel) compared with 3,860 kWh year from the original 12 panels (322 kWh/panel).

Click on image for a larger version. Charts showing the angular extent of daytime through the year. The orientation of the three sets of panels on the different roofs is shown as red arrowed lines.

Along with the 5 panels on the roof, I have installed three panels on the flat roof which are only at 12° to the horizontal. The Easy-PV calculator suggests these 3 panels will generate 919 kWh per year (306 kWh/panel), although I am not sure I properly accounted for shading.

Click on image for a larger version. The same photograph as above but now showing the panels on the flat roof.

Summary
Sadly, although the panels have been installed for more than month, no inverter has been installed and they have not been connected to the grid. Apparently, this will happen “tomorrow”.

But if the output is as I anticipate, then next year the system will generate around 6 MWh. The amount we draw from the grid should be slightly reduced as I hope we will be off-grid for 6 months rather than 4.5 months year. So considered over a year, cumulative generation should be roughly twice as much as we draw from the grid.

Consequently – considered over a year – we should export almost as much as we import, which is getting close to one definition of carbon neutrality. This is my dream!

Click on image for a larger version. Cumulative PV generation for 2022 is just under 4 MWh, in line with the MCS guidance when the system was installed. Cumulative Grid Consumption is expected to be just over 3 MWh this year. The dotted purple line shows anticipated generation next year.

Heating My Home

Friends, I have made a 5-minute video about heating my home with a heat pump.

It’s nonsense, but once I had thought of the idea of using coloured water to represent heat, I felt compelled to make the video: I hope you enjoy it.

How big is that fire?

Click on the image for a larger version. The picture is courtesy of Michael Newbry.

Friends, you may have noticed that we have recently entered a period of what is euphemistically called “enhanced risk of wildfires”.

And reports of wildfires from around the world include some truly apocalyptic images.

But many of these reports fail to communicate clearly one of the key metrics for fires: the size of the fire.

Some reports do mention the area affected in hectares (abbreviated as ha) or acres, but while I can just about grasp the meaning of one acre or one hectare – I struggle to appreciate the size of a fire covering, say, 6,000 hectares.

In order to convert these statistics to something meaningful, I work out the length of one side of a square with the same area.

Areas expressed in hectares.

A hectare is an area of 100 m x 100 m, or 0.1 km x 0.1 km so that there are 100 hectares in a square kilometre.

So to convert an area expressed in hectares to the side of the square of equal area one takes two steps.

• First one takes the square root of the number of hectares.
• One then divides by 10.

So for a fire with an area of 6,000 hectares the calculation looks like this:

• √6,000 = 77.4
• 77.4÷10 = 7.74 km

Since the original area was probably quite uncertain I would express this as being equivalent to a square with a side of 7 or 8 km.

Areas expressed in acres.

An acre is an area of 63.6 m x 6.36 m, or 0.64 km x 0.64 km so that there are roughly 2.5 acres in a hectare.

I can’t think of an easy way to get a good approximation for acres, but a bad approximation is better than no estimate at all. So I recommend, the following 3- or 4-step process:

• First one divides the number of acres by 2
• Then one takes the square root of half the number of acres.
• One then divides by 10.
• This answer will be about 10% too large.

So for a fire with an area of 15,000 acres the calculation looks like this:

• 15,000÷2 = 7,500
• √7,500 = 86.6
• 86.6÷10 = 8.7 km

At this point one can either just bear in mind that this is a slight over-estimate, or correct by 10%. In this context, the overall uncertainty in the estimate means the last step is barely worthwhile.

How bad is the situation in Europe?

Click on Image for larger version. Estimates of the cumulative area (in hectares) burned by wildfires in each of the EU countries. The red bars show data for this year, and the blue bars show the average area burned between 2006 and 2021.

There is a wonderful website (link) which publishes estimates of wildfire prevalence in all the countries of the EU. One output of the website is shown above:

• The blue bars shows the average area burned from 2006 to 2021
• The red bars shows the average area burned so far this year.

You can immediately see that Spain, Romania, and France are having bad years for wildfires.

But how big an area is 244,924 hectares – the area burned in Spain so far? Using the rule above, one can see that it is an area equivalent to a square with a side of 50 km – roughly equivalent to (say) the area of Cheshire.

The area burned in France so far this year is 60,901 hectares. Using the rule above, one can see that it is an area equivalent to a square with a side of 25 km.

Michael, what was the point of this article?

When trying to visualise large areas expressed in hectares (or acres) I find it useful to work out the length of side of a square which would have the same area.