Archive for November, 2020

COVID-19: Finally evidence that Lockdown#2 is working.

November 27, 2020

Summary

Friends: Lockdown#2 began 22 days ago on 5th November, and this week’s data finally suggest that it is working!

Last week I wrote that if Lockdown#2 were working, then we would expect to see:

  • The rate of positive tests falling.
  • 5 to 10 days later, hospital admissions would fall.
  • After a further 5 to 10 days, the rate of deaths would fall.

The latest data are discussed below but the headline is this: the rate of positive tests is declining. Consequently, the rate of  hospital admissions and deaths should eventually fall too.

Recent data for the number of daily positive tests and their 7-day retrospective average together with last week’s optimistic projection shown as a dotted line (– – –). Click for a larger version.

Also, the rate at which positive tests are falling is in line what I thought were the most optimistic assumptions conceivable. Basically – and this surprises me – based on these few day’s data, Lockdown#2 appears to be as effective as Lockdown#1.

This is good news:

Restrictive though life is under the current measures, there is much more activity than there was under Lockdown#1. Collectively we seem to have got better at social distancing and wearing masks and hand washing. This achievement offers us the prospect of a workable compromise between lives and livelihoods.

But…

  • with a 7-day-average death rate of 460 people per day this is not a good situation. In fact it is amongst the worst in the world.
  • with viral prevalence still around 1% there is an ongoing risk that the death rate could again increase rapidly if adherence to restrictive measures lapses, such as at Christmas.

Let’s look at the data.

Data#1. Prevalence

Since late April the ONS prevalence survey has been randomly testing people in England each week to look for the virus. They then collate their data into fortnightly periods to increase the sensitivity of their tests. Details of their full results are described methodically in this ‘bulletin‘.

Data from the ONS: Click for a larger version.

The number of people tested and the number of positive tests are given in their table above. ONS estimate that at the end of the measurement period on 21st November 2020 on average 1.1% of the UK population were actively infected – 0.1% down on the previous week’s estimate.

The raw count of positive tests was:

  • 2,040 from 169,333 people tested in the two weeks to 6th November,
  • 1,974 from 158,308 people tested in the preceding two weeks, and
  • 1,876 from 182,184 people tested in the two weeks preceding that.

The data in the table above are graphed below.

ONS Estimated prevalence of COVID-19 in England. Click for a larger version.

Note these estimates come from random survey tests (so-called Pillar 4 tests), not clinical tests.

I have shown two curves on the graph above.

  • The black dotted line (– – –) is the same curve I have plotted for the previous eleven weeks. (Link)
  • It is a fit to the 3 black data points and shows what we might expect if viral prevalence were doubling every 15 days.
  • The blue continuous curve is the ONS model for what is ‘really happening’.
    • I cannot explain how this estimate lies consistently below the data on which the model is based. Clearly there is something I have still not understood. I wrote to ONS 10 days ago requesting an explanation but I have not received an acknowledgement or a reply.

Data#2. Tests and Deaths

The graph below shows three quantities on the same logarithmic scale:

  • the number of positive tests per day
  • the number of people newly admitted to hospital each day
  • the number of deaths per day.

The data were downloaded from the government’s ‘dashboard’ site.

  • Positive tests refer to Pillar 1 (hospital) and Pillar 2 (community) tests combined – not the Pillar 4 tests from the ONS survey.
  • The deaths refer to deaths within 28 days of a test.
  • Hospital admissions for the UK nations combined.

All curves are 7-day retrospective rolling averages of the data since July.

Data for positive casesdaily hospital admissions and daily deaths. Click for a larger version. Recent details for each quantity are given in separate graphs below.

The graph shows the data alongside exponentially decreasing and then increasing trends shown as dotted lines.

  • The declining trends correspond to quantities halving every 21 days – the rate at which the epidemic declined during Lockdown#1.
  • The increasing trends correspond to quantities doubling every 15 days.

Back in July, the three data sets initially fell with similar time-dependencies and then rose through the autumn.

Data#3. Details and Projections

Because we are now in the non-exponential phase of the epidemic, I have re-plotted recent data for cases, admissions and deaths on a linear scale below.

I have also included some projections (as dotted lines) taking the most optimistic view possible. I have imagined that:

  • The effects of Lockdown#2 will begin to show through on Day 324,
    • Looking back from Day 331, this seems to have been about right.
  • The declines will be as swift as in Lockdown#1 (halving every 21 days).
    • Based on limited data, this too seems to be about right.
  • The declines will continue after the 2nd December end of Lockdown#2.
    • The continued extensive and restrictive tiers announced this week means this is possible
  • The decline in hospital admissions is delayed 5 days with respect to cases
    • There is some hint that this may be already visible.
  • The delay in deaths is delayed 5 days with respect to hospital admissions.
    • A 5-day delay is the shortest conceivable and indeed deaths are still rising.

Here are this week’s updates.

Recent data for the number of daily positive tests and their 7-day retrospective average together with last week’s optimistic projection shown as a dotted line. Click for a larger version.

Recent data for the number of daily hospital admissions and their 7-day retrospective average together with last week’s optimistic projection shown as a dotted line. Click for a larger version..

Recent data for the number of daily deaths and their 7-day retrospective average together with an optimistic projection shown as a dotted line. Click for a larger version.

The curved dotted-line projections on the linear graphs above are effectively the same as the straight-line projections on the combined (logarithmic) graph. Please note that:

  • These are guidelines not predictions.
  • Look to see if quantities fall faster than this, or slower than this.

Out of curiosity I have also plotted the rate of hospital admissions relative to start of Lockdown for Lockdowns #1 and #2 below. We see that the 7-day averages rose for a clear two weeks after the start of Lockdown#1, and this has also happened in Lockdown#2.

Data for 7-day retrospective averages of daily hospital admissions relative to start of Lockdowns#1 and #2. Click for a larger version.

So…

I am allowing myself to feel briefly encouraged: We have good vaccine news and seem to have found a way of living which – albeit highly restrictive – allows some economic and social activity while still driving down viral prevalence.

Stay safe.

Acoustic Thermometry in Ancient China

November 23, 2020

Have you ever stared into a well and tried to see the bottom? Even shining a torch down, it can be  hard to see just how far down the water is. But occasionally you can catch a reflection from the surface and appreciate the depth.

Yesterday, I recalled that sense of vertigo as I peered into the depth and darkness of the well of my own ignorance.

I was reading a paper on some weather records in China,

An Introduction to Some Historical Government Weather Records of China

Pao K. Wang and De’er Zhang

Bulletin of the American Meteorological Society Vol 69, No 7 July 1988 pp 753-758

And as the paper described some practices from the 18th Century B.C. i.e. 3800 years ago – a glint of light reflected off the surface of my ignorance and I realised just how colossally deep the well was.

  • I know almost nothing about China.

I paused, caught my breath and continued reading. Here are some of the things I learned.

Shang Dynasty

In the introduction the authors describe the earliest known weather records from the Shang Dynasty, dating as far back as the 18th Century BC. The world was a busy place back then.

The weather records were carved on ox bones or turtle shells by ‘diviners’.

The diviners predicted the weather based on the pattern of cracking of the bone or shell when it was burned. But – astonishingly in my view – they then later carved the actual weather on the same piece of bone or shell!

Zhou Dynasty

In the Zhou dynasty (1111 BC to 246 BC), there were no weather records as such, but people did note extreme weather events such as floods or droughts.

Crucially they recorded the dates at which rivers or lakes froze, and the dates at which particular flowers blossomed. These records have been used to indirectly infer the climate at that time.

West Han Dynasty

In a book, Huai-Nan-Zi, dating to 120 BC a passage states:

“…by hanging feather and charcoal together one can know the dryness or wetness of the air…
When it is dry, the charcoal is light. When it is wet, the charcoal is heavy.

This is clearly a description of a primitive hygrometer operating on the principle that charcoal has a very large internal surface area that can adsorb a relatively large amount of moisture when exposed to moist air. But what is the role of the feather?

Later, in the years leading to 220 AD, it is reported that local governments are required to report the amount of rainfall “…from the beginning of spring to summer, and the beginning of the fall.

There are no records of the rain gauges used, but there must surely have been a standardised procedure.

Descriptions of rain gauges appeared much later in a 1247 AD book Nine Chapters of Mathematics. This describes four techniques for determining rainfall and snowfall from which the shapes of the gauges can be determined. But sadly no actual records remain.

Later Han Dynasty

But perhaps the most fascinating instrument described is a device dating from 1086 AD for measuring the temperature of the soil at different depths.

My visualisation of an early Chinese acoustic ‘thermoscope’ for measuring soil temperature at different depths. Click for a larger version.

The instrument was called  guan (meaning ‘Reed Pipe’ or ‘Scale Tubes’) and was original used as ‘pitch pipes’ for providing standard pitches for tuning musical instruments. Their use is described as follows:

  • Tubes of different lengths – with different natural pitches – were buried so that different lengths were in the ground, but the length above ground was fixed.
  • Ashes were placed in the tube and these were supposed to rise when the tubes were exposed to the ‘proper’ ground temperature.
  • Since ground temperature varied with depth and with the time of year, the ashes would ‘fly’ out of different tubes at different dates giving an idea when a particular temperature had been achieved at a certain depth.

There is a good deal about this explanation that I don’t understand. Did people blow over the tubes? How did they know what was the ‘right’ soil condition? When the tube resonated, how did the ash ‘fly’?

But despite these questions, it is clear that the instrument was using the principle that the speed of sound in air – and hence the resonant frequency of a musical tube – changes with temperature.

This is the same principle that I used when I worked at NPL to build the world’s most accurate thermometer. And I also built several tube like thermometers not-so-dissimilar to the  guan.

But the  guan appears to have been more of a thermoscope than a thermometer i.e. it indicated certain temperature conditions but did not ascribe a number to that condition.

What’s missing?

One of the curiosities of the paper is something which is not there, and whose absence is not mentioned.

Given the centuries of development, and the early prowess of Chinese industry in the manufacture of silk and gunpowder and paper – processes for which temperature and humidity control are critical – I found it surprising that there was no mention of an equivalent to the numerical temperature scales developed in Europe.

Early temperature scales were developed in Europe by Newton and Rømer in 1701 and later perfected by Fahrenheit, Celsius and many others.

Of course reading one paper does very little to amend the vastness of my ignorance of China and its history. And there may indeed have been some kind of alternate scale. But if in fact there was no ‘indigenous’ Chinese temperature scale, this probably speaks to some key difference in the societies at that time.

So?

So those are the drops of knowledge that fell in the deep well of my ignorance.

I enjoyed the wait as the drops fell and I listened for the splash as they reached the surface. But the level did not change very much!

  • If anyone has any information about books I should read, or sources I should consult,  I would be grateful if you could drop me a line.

Addendum

Thanks to Peter G who e-mailed to highlight the work of Joseph Needham who too had asked questions about the anomaly of early Chinese discoveries and inventions, but the relative slowness of industrial innovation. The wikipedia page is fascinating in itself.

COVID-19: No evidence yet that Lockdown#2 is working

November 21, 2020

Summary

Friends: Lockdown#2 began fifteen days ago on 5th November.

If Lockdown#2 were working, then we would expect to see:

  • The rate of positive tests falling.
  • 5 to 10 days later, hospital admissions would fall
  • After a further 5 to 10 days, the rate of deaths would fall

The latest data are discussed below but the headline is this: positive tests are not obviously declining. We thus have no reason to expect hospital admissions and deaths to fall.

If Lockdown#2 is indeed failing then:

  • We have no practical way to reduce the death rate.
  • With viral prevalence at ~1% there will be the permanent ongoing risk that the death rate could again increase rapidly if adherence to restrictive measures lapses.
    • For example if 10 million people meet in groups of 6 at Christmas, then there will be over 100,000 ‘group infection opportunities’ potentially affecting more than half a million people and raising the risk of fresh outbreaks.

At the end of this article I make some projections based on the most optimistic assumptions conceivable. There is no reason to be so optimistic – but I find the pessimistic projections personally unbearable.

But for now, let’s look at the data.

Data#1. Prevalence

Since late April the ONS prevalence survey has been randomly testing people in England each week to look for the virus. They then collate their data into fortnightly periods to increase the sensitivity of their tests. Details of their full results are described methodically in this ‘bulletin‘.

Data from the ONS: Click for a larger version.

The number of people tested and the number of positive tests are given in their table above. ONS estimate that at the end of the measurement period on 14th November 2020 on average 1.22% of the UK population were actively infected – almost the same as the previous two weeks.

The raw count of positive tests was:

  • 2,046 from 160,355 people tested in the two weeks to 6th November,
  • 1,930 from 163,234 people tested in the preceding two weeks, and
  • 1,474 from 182,729 people tested in the two weeks preceding that.

The number of people tested has continued to fall for the second fortnight in a row. The data in the table above are graphed below.

Estimated prevalence of COVID-19 in England. Click for a larger version.

Note these estimates come from random survey tests (so-called Pillar 4 tests), not clinical tests.

I have shown two curves on the graph above.

  • The black dotted line (– – –) is the same curve I have plotted for the previous ten weeks. (Link)
  • It is a fit to the 3 black data points and shows what we might expect if viral prevalence were doubling every 15 days.
    • It looks like the exponential growth phase is truly over. For now. Since these data end on November 14th, this must have been substantially the result of the previous ‘Tiered Measures’ rather than Lockdown#2.
  • The blue continuous curve is the ONS model for what is ‘really happening’.
    • I cannot explain how this estimate lies consistently below the data on which the model is based. Clearly there is something I have still not understood. I wrote to ONS last week requesting an explanation but I have not received an acknowledgement or a reply.

Data#2. Tests and Deaths

The graph below shows three quantities on the same logarithmic scale:

  • the number of positive tests per day
  • the number of people newly admitted to hospital each day
  • the number of deaths per day.

The data were downloaded from the government’s ‘dashboard’ site.

  • Positive tests refer to Pillar 1 (hospital) and Pillar 2 (community) tests combined – not the Pillar 4 tests from the ONS survey.
  • The deaths refer to deaths within 28 days of a test.
  • Hospital admissions for the UK nations combined.

All curves are 7-day retrospective rolling averages of the data since July.

Data for positive casesdaily hospital admissions and daily deaths. Click for a larger version. Recent details for each quantity are given in separate graphs below.

The graph shows the data alongside exponentially decreasing and then increasing trends shown as dotted lines.

  • The declining trends correspond to quantities halving every 21 days – the rate at which the epidemic declined during Lockdown#1.
  • The increasing trends correspond to quantities doubling every 15 days.

Back in July, the three data sets initially fell with similar time-dependencies and then rose through the autumn.

Data#3. Details and Projections

Because we are now in the non-exponential phase of the epidemic, I have re-plotted recent data for cases, admissions and deaths on a linear scale below.

I have also included some projections (as dotted lines) taking the most optimistic view possible. I have imagined that:

  • The effects of Lockdown#2 will begin to show through today,
    • I think this is unlikely.
  • The declines will be as swift as in Lockdown#1 (halving every 21 days).
    • I think this is unlikely.
  • The declines will continue after the 2nd December end of Lockdown#2.
    • I think this is unlikely.
  • The decline in hospital admissions is delayed 5 days with respect to cases
    • A 5-day delay is the shortest conceivable.
  • The delay in deaths is delayed 5 days with respect to hospital admissions.
    • A 5-day delay is the shortest conceivable.

There is no reason to be so optimistic – but I find the pessimistic projections personally unbearable.

Recent data for the number of daily positive tests and their 7-day retrospective average together with an optimistic projection shown as a dotted line. Click for a larger version.

Recent data for the number of daily hospital admissions and their 7-day retrospective average together with an optimistic projection shown as a dotted line. Click for a larger version.

Recent data for the number of daily deaths and their 7-day retrospective average together with an optimistic projection shown as a dotted line. Click for a larger version.

The curved dotted-line projections on the linear graphs above are effectively the same as the straight-line projections on the combined (logarithmic) graph. Please note that:

  • These are guidelines not predictions.
  • Look to see if quantities fall faster than this, or slower than this.

Out of curiosity I have also plotted the rate of hospital admissions relative to start of Lockdown for Lockdowns #1 and #2 below. We see that the 7-day averages rose for a clear two weeks after the start of Lockdown#1, and this has also happened in Lockdown#2.

However in Lockdown#1, the number of cases had probably peaked just after Lockdown: we don’t know because the tests were not available back then.  As we have seen, the number of positive tests is not yet obviously falling even two weeks into Lockdown#2.

Data for 7-day retrospective averages of daily hospital admissions relative to start of Lockdowns#1 and #2. Click for a larger version.

Mortality

As the rate of change of hospital admissions has slowed, we can now revisit the analysis from earlier in the pandemic when the death rate of people who entered hospital was around 20%.

To make this estimate we need to estimate the delay between hospital admission and death.

Looking at the last couple of weeks, Hospital admissions in the range 1500 to 1700 per day seem to result in deaths in range 350 to 420 per day which amounts to a mortality of roughly 24 ± 3%.

So…

A very depressing week: that vaccine cannot arrive soon enough!

Stay safe.

Solar Power in Teddington

November 18, 2020

An Accidental Installation

Slightly to my surprise, I became the owner of a solar power installation last week.

I had planned the works on my house in this order:

  • Triple-glazing
  • External Wall Insulation (EWI)

Wait for a Winter

  • Heat pump
  • Solar Panels
  • Battery

This plan was partly rational. The largest carbon emissions are associated with heating, and so tackling those first – and evaluating their performance overwinter – was sensible.

But I was irrationally averse to getting solar panels because they felt like an indulgence – something I would “allow myself as a treat” after the hard work had been done. However, I changed my mind.

The main reason was that for the EWI work, I needed my neighbour’s permission to put scaffolding in the side passage by their building. My neighbour is an NHS clinic and it took several weeks to locate the person responsible and submit appropriate safety documentation. Although it was all perfectly pleasant – it was long-winded and not a process I wanted to repeat.

With that in mind, once the EWI scaffolding was erected, I called up a local solar power installer, Andy Powell from GreenCap Energy and asked whether he could install a system in the next two weeks using the EWI scaffolding. He visited the next day, and said that he could indeed use the scaffolding and that a 12-panel system would cost £4200. Importantly, he was able to install it the following week.

I had been thinking of all kinds of clever arrangements of panels, but it turns that if you want to put more than 12 panels on your roof, you need a special licence – which takes quite a bit of work – and time.

I reflected that if I installed the panels now, I would save roughly £1000 by using the existing EWI scaffolding. And that £4200 was much less than I had expected. So I put myself in Andy Powell’s capable hands and let him get on with it!

And there was one more piece of serendipity. At that point in the EWI work, it was possible to run the power cable from the panels to the main distribution board by running an armoured cable outside the house, buried underneath the EWI. This saved a lot of mess inside the house.

#1: The Components

The system consists of 12 solar panels, a device called an inverter, some isolation switches and generation meter.

  • The 1.7 m x 1 m panels are from Q-cells. The choice of panels available is bewildering so I just accepted Andy’s recommendation. They look beautiful and seem to work just fine.
  • Each panel generates roughly 40 V and up to 10 A amperes of DC current. They are connected in two banks to the inverter through two cables that poke through a small hole in the roof.
  • The Inverter is a SOLIS 4G 3.6 kW model. It takes the DC voltage and turns it into 220 V AC that can be used around the house or exported to the grid. There is also an add-on that enables the system to be monitored from a phone.
  • In my installation this AC current then goes back outside via an armoured cable buried in the wall  and then comes back inside under the floor, through a power meter, to the distribution board.

The gallery below shows some pictures of the process: click a picture for a larger version.

#2: The Site

Google Maps view of my home showing the shape and orientation of the available roofs. And photographs before and after the installation. Click for a larger image.

There were two roofs available for solar panels on my house – a smaller triangular roof facing 25° east of south, and a larger roof facing 65° west of south.

My initial plan was to cram as many solar panels on the south-facing roof as possible, but being triangular, the large 1.7 m x 1 m panels do not fill up the space very efficiently. I could have squeezed 7 panels on, but in the end I opted for 6 on each roof – it seemed to look a little less ugly.

To my surprise – experimentation with Easy PV software (more details later in the article) seemed to show that the orientation wouldn’t make much difference to the overall energy generated.

The path of the Sun at the summer and winter solstices and the equinoxes. In the summer when most solar energy is generated, the sun sets up to 30° north of west, and so the west-facing panels continue generating later in the day after the south facing panels are in shadow. Having panels on both roofs allows for generation for a longer fraction of the long summer days. Click for a larger version.

I think the reason is that – as Andy Powell pointed out – no panel can generate for more than 12 hours because it doesn’t work when the Sun is behind it!

But in the summer, when most solar energy is generated, the Sun is above the horizon for up to 16 hours and at the solstice it sets more than 30° north of due west.

And so the west-facing panels continue generating later in the day after the south-facing panels are in shadow. Having panels on both roofs allows for generation for a longer fraction of the long summer days.

The path of the Sun at the summer and winter solstices and the equinoxes. The yellow zone shows sun orientations at which only the south-facing panels generate. The red zone shows sun orientations at which only the west-facing panels generate. The purple zone shows sun orientations at which both sets of panels generate.

Of course it is not just the east-west position of the Sun – the so-called azimuthal angle – that affects generation – the height of the sun in the sky – its elevation – is also important.

Sites such as this one will plot maps showing the course of the Sun through the sky on any particular date from your particular location.

The blue line shows the path of the Sun through the sky on November 11th for my location at 53° latitude and 0° longitude. 180° corresponds to due South. Click for a larger version. The shaded yellow boxes indicate the azimuthal angles at which the two banks of panels generate. And the green line shows the optimum elevation of the Sun.

From the figure above we see that the Sun is low in the sky at this time of year (Doh!) – it only rises 20° above the horizon at midday – but that even at this time of year, the generation from the west-facing panels in the afternoon prolongs the useful generation time. From initial observations the power on the two banks of panels is equal at about 1 p.m.

#3: Expected Performance

Frustratingly, working out the expected performance of a solar installation is complicated. One needs to:

  • calculate the Sun path diagram for each day of the year,
  • factor in the weather,
  • consider each bank of panels separately.

Fortunately, approved installers such as Greencap can run standard calculations, or using software like Easy PV you can – after some messing about – come up with your own estimate.

Output from Easy PV software allows one to calculate how much electricity is likely to be generated by each bank of solar panels in a year.

For my installation, both estimates suggested that I should expect to generate roughly 3700 kWh of electricity each year. This figure is roughly how much electricity my house uses each year.

If I could capture each one of those generated kilowatt hours and use it to displace one that I buy from EDF  I would save more than £800/year. However, things are not so simple.

Looking around  one can find a few records (such as this one) of people’s generated power. They seem to indicate that in the UK I should expect roughly 5 times as much daily generation in the summer as in the Winter.

Putting that information together with the fact that I can expect 3700 kWh over the year I concocted a function (sine squared with an offset in case you care) to guide my expectations.

So in the summer I can expect the panels to generate perhaps 15 kWh/day – much more than I need – but in the winter the panels might only generate perhaps 3 kWh/day – much less than I need.

My guess at how many kWh per day I can expect from my solar panels. The average household consumption is shown is shown as a red dotted line (- – -). Click for a larger version.

#4: Actual Performance

I only have 10 days of data for the panels and these are plotted on the graph above and they seem broadly in line with my expectations.

I can already see the effect on my electricity usage. As can be seen on the graph below, my daily average use is 2.1 kWh below the average before the panels were installed.

Daily electricity usage (from a smart meter) before and after solar panel installation. Click for a larger version.

This may not sound much but even if the panels only ever performed at that level, this would prevent the emission of 73 kg of CO2 per year, and (@£0.24/unit) save me £175 per year. For those of you that are interested, that’s a 4.2% return on investment. But I expect the panel performance to be much better than this when averaged over the year.

But what is hard to capture in words is the sheer wonder of the installation. In bright November sunshine the panels generate more than 2 kW of electrical power – so I can boil a kettle and still see the smart meter read zero usage.

#5: What next?

For the next few weeks I intend to let the dust settle, and try to get my head around how the system is working.

But one obvious difficultly – which will become more pressing as we move into Spring – is that when the Sun shines the panels produce kilowatts of electricity whereas the house itself generally consumes just a few hundred watts.

At the moment any excess electricity is exported to the grid – my smart meter says 13 kWh so far – as a gift to the nation!

So in the next few weeks I will sign up with a company to buy this electricity. There are several companies who will buy at between £0.03 and £0.055 per kWh.

In the longer term it may well make sense to get a battery as well. But batteries are expensive, and the more I have thought about it, the main use of a battery in my situation would not be to store solar electricity, but to switch the time at which I bought electricity from the day (when electricity is carbon intensive and expensive) to the night (when electricity is cheap and generally less carbon intensive). But that is a question for another time.

External Wall Insulation: How well does it work?

November 16, 2020

Be Constructive will probably finish my External Wall Insulation (EWI) in just a day or two, but I am already trying to see its effect, even as they apply the last coats of render.

So “How well does it work?”

I won’t have a definitive answer until later in the winter, but this article describes the procedures I am using and you can look at the preliminary data and come to your own – preliminary – conclusions.

The data  

I have written about this before (here, here, here and here!) but please allow me to recap.

To work out how well the EWI is working I measure two things:

  • the difference between external and internal temperatures, and
  • the amount of gas I use each day or each week.

For the last two years I have done this weekly – even I find it too tedious to read the gas meter every day!

But a few weeks ago – in the same week as the EWI work started – I switched to using a ‘smart meter’ and this allows me to download a spreadsheet with daily gas (and electricity) consumption. So now I can work with either daily or weekly averages of gas consumption.

I can then get either the daily or weekly average temperatures from the weather station I have in my back garden. If you don’t have a weather station in your garden then you can use data from nearby stations on the Weather Underground (Link: Zoom in to find weather stations near you).

So how do I use the daily and weekly data to estimate how well the EWI is working?

Weekly data

The graphs below are complex so I will describe each element in turn.

We start with my average daily gas consumption (blue squares) since November 2018. I have averaged the data over 5 weeks to remove anomalously cold or warm weeks. This makes it easier to view the general trend of the data.

My average daily gas consumption (in kWh/day) over the last two years. Click for a larger version and see the text for details.

The data are plotted versus days since the start of 2019, and several events which I think might have affected gas consumption are shown in pink.

  • Triple Glazing of most windows.
  • Installation of a chimney sheep.
  • Triple Glazing of remaining windows.
  • External Wall Insulation (EWI).

Notice that in the summers, gas consumption falls to around 5 kWh/day due to water heating and cooking, but that increases in the winter to as high as 100 kWh/day. It is the temperature-dependent part of this consumption that I think is related to the effectiveness of the insulation in the house.

Next we have a graph of the difference between the internal temperature (nominally 19 °C) and the average weekly external temperature over the same period. This is the ‘demand’ to which the gas central heating responds. Again, I have averaged the data over 5 weeks.

This data is plotted (green circles) on the same graph as the gas data, but should be read against the right-hand axis. Plotting the data in this way shows that the gas consumption is obviously related to external temperature.

Average daily gas consumption (in kWh) over the last two years in blue plotted against the left-hand axis and the difference of the average external temperature from 19 °C in green plotted against the right-hand axis. Click for a larger version and see the text for details.

It’s clear from the similarities between the two curves that the variation in winter gas consumption is due to the external temperature. I mention this extremely obvious fact because I was personally surprised by how similar the two curves were.

Model

My model of my house tries to predict the gas consumption based on knowledge of the weather, and a single parameter that describes all the ways that heat flows out of the house. This parameter tells me how many watts of heating power I need to keep the house 1 °C above the outside temperature. In case you are interested, the equations which summarise this are given in the figure below.

The mathematical model – in case you care. In the green background the heating power is expressed in kWh/day and in the blue background the heating power is expressed in W. Click for a larger version.

Initially I matched the model to the data using what is known in statistics as “the null hypothesis“i.e. I assume that nothing I have done has made any difference. So:

  • I adjusted the heat loss parameter of the model to match the data in the middle of the winter of 2018-2019.
  • The best fit is made by assuming I use roughly 280 W of gas heating for each °C that the external temperature fell below 19 °C. (That’s 6.7 kWh per day per °C)

The red dotted line (- – – ) shows the modelled gas consumption assuming that nothing I have done has made any difference. Click for a larger view.

With this assumption, it is clear that the model overestimates the gas consumption in subsequent winters: so it looks like the actions I took did have some effect i.e. the same demand has led to lower gas consumption. Phew.

To estimate how big an effect I modified the model so that the effectiveness of the insulation could be changed at two points: day 244 and day 657 shown as vertical pink lines in the graph below.

The red dotted line (- – – ) shows the modelled gas consumption assuming that the thermal performance of the house has improved from 280 W/°C in winter 2018/2019 to 240 W/°C in winter 2019/2020, to 134 W/°C in the current winter 2020/21 . Click for a larger view.

  • For the first section I assumed that I needed to use 280 W of gas heating for each °C that the external temperature fell below 19 °C.
  • For the second section I assumed that I needed to use only 240 W of gas heating for each °C that the external temperature fell below 19 °C – about 15% less.
  • For the final section – i.e. currently – I anticipate that I will need to use only 134 W of gas heating for each °C that the external temperature falls below 19 °C.

You can see that in the current winter (day 650 onwards) there is not yet enough data to say which value of heat loss parameter will best match the data.

Looking at previous years, the model does not match the data well in spring and summer – so I don’t feel I can definitively estimate the heat loss parameter until winter is fully upon us.

But the estimate of 134 W/°C – less than half what it was two years ago – does not (at this stage) look unreasonable.

Daily Data

I can apply the same model to the smart-meter data that I download from EDF which gives me my daily gas consumption. I can then compare this directly with the daily average temperature.

I expect this daily data to show added variability when compared to the weekly-averaged data because:

  • The ‘constant term’ – in my case roughly 200 W or 4.8 kWh per day – due to use of gas for cooking and heating water may look constant when averaged over 5 weeks. But it likely fluctuates from day-to day depending how much cooking or hot water is used.
  • In the short term, heat can be stored and released from the fabric of the building.
  • During the EWI works, there have been many days during which doors were open.

Also I must confess to a metrological faux pas in this article by using two units to describe the effectiveness of the insulation: watts (W) and kilowatt hours per day (kWh/day). Both are valid choices, but I will make penance for having used mixed units by showing graphs below in both sets of units!

With these caveats in mind, let’s look at the data.

Daily gas consumption in kWh/day (left-hand axis) and temperature demand (right-hand axis). The straight lines show weekly averages. Click for a larger version.

Daily gas consumption in terms of average power (W) (left-hand axis) and temperature demand (right-hand axis). The straight lines show weekly averages. Click for a larger version.

Notice the strong day-to-day correlation between gas consumption and average external temperature deficit.

I guess it is this strong correlation that allows gas supply companies to order the correct amount of gas in advance.

But the day-to-day data have oddities. For example, on some days the gas consumption seems anomalously low (e.g. days 298, 304 and 307), and on others anomalously high (e.g. days 294, 310).

Bearing these anomalies in mind we can divide the gas consumption data by the temperature deficit data to give the gas power per °C of ‘demand’. These graphs are shown below.

Gas power (in kWh/day) per °C of demand. The straight lines show weekly averages and the double line (===) shows the running average (±3 days). The dotted red line (– – – ) shows the level I hope to achieve. Click for a larger version.

Gas power (in W) per °C of demand. The straight lines show weekly averages and the double line (===) shows the running average (±3 days). The dotted red line (– – – ) shows the level I hope to achieve. Click for a larger version.

The daily data show too much variability to allow easy interpretation. But looking at the running average and the week-to-week data it looks like there is a trend downwards towards improved thermal performance over the course of the EWI works

However there are considerable uncertainties.

  • These data take no account of the non-thermal use of gas.
    • For example, heating 50 litres of water a day from 10 °C to 50 °C would use 2.3 kWh/day, the energy of which would substantially go down the drain
  • Similarly, at these low power levels, I should probably be taking account of:
    • The roughly 11 kWh/day of electrical power that we consume.
    • The roughly 4 kWh of heating provided by the warm bodies of myself and my wife.

So…

So it will take time before I can definitively evaluate the effect of the EWI. But by spring 2021, I suspect that I will have stared at the data long enough that the best way to analyse the data will have become clear to me. I hope so!

The reason it matters is that next year I plan to stop using the gas boiler altogether and switch to using an air source heat pump. Before spending yet more money on that I am keen to try to anticipate the likely demand so I can pick the right model!

Meanwhile, I am positively enjoying the EWI. I don’t know if this is a psychological effect of spending large amounts of money on something – or a genuine sensation caused by a more stable temperature and – I think – reduced air leakage.

And although still clad in scaffolding, the house itself is beginning to look rather smart. I’ll be sure to post some pictures when the Be Constructive team have left.

 

COVID-19: Let’s hope Lockdown#2 is working

November 13, 2020

Summary

Friends: Lockdown#2 began eight days ago on 5th November.

As I read the data now, they seem to confirm that the ‘Tiered measures’ taken prior to lockdown#2 were stabilising the pandemic with “R close to 1″.

This ‘frozen state” is characterised by stable numbers of positive tests, hospital admissions and deaths.

If Lockdown#2 is being effective, then in the coming week i.e. 8 to 15 days after ‘Lockdown’, we should see falling numbers of positive tests. There is no sign of that in this week’s data, and indeed the last two days have shown worryingly high case rates. But it is probably too early to tell what is happening.

We should sincerely hope that Lockdown#2 is effective, because I don’t know what else the government can do if it doesn’t work.

If Lockdown#2 is not being effective, then test numbers will remain broadly unchanged or rise, and inevitably, hospital admissions and deaths will remain at their current high rate, or rise.

In this ‘frozen state’ we avoid the exponential growth of the virus. Instead,  the viral prevalence will be ‘frozen’ at its current high concentration and cause the current death rate – more than 400 per day – to continue indefinitely.

There will also be the permanent ongoing risk that the death rate could again increase rapidly if adherence to restrictive measures lapsed – for example – over Diwali or Christmas – or perhaps just due to pandemic fatigue.

Let’s look at the data.

Data#1. Prevalence

Since late April the ONS prevalence survey has been randomly testing people in England each week to look for the virus. They then collate their data into fortnightly periods to increase the sensitivity of their tests. Details of their full results are described methodically in this ‘bulletin‘.

Data from the ONS: The latest fortnightly data point is highlighted in red. Click for a larger version.

The number of people tested and the number of positive tests are given in their table above. ONS estimate that at the end of the measurement period on 6th November 2020 on average 1.19% of the UK population were actively infected – almost the same as the previous two weeks.

The raw count of positive tests was:

  • 1,898 from 154,159 people tested in the two weeks to 6th November,
  • 1,866 from 182,068 people tested in the preceding two weeks, and
  • 1,025 from 161,738 people tested in the two weeks preceding that.

Curiously, this is the first time in this series that the number of people tested has fallen from one fortnightly period to the next. The data in the table above are graphed below.

Estimated prevalence of COVID-19 in England. Click for a larger version.

Note these estimates come from random survey tests (so-called Pillar 4 tests), not clinical tests.

I have shown two curves on the graph above.

  • The black dotted line (– – –) is the same curve I have plotted for the previous nine weeks. (Link)
  • It is a fit to the 3 black data points and shows what we might expect if viral prevalence were doubling every 15 days.
    • It looks to me like the exponential growth phase is over for now. Since these data end on November 6th, this must have been the result of the previous ‘Tiered Measures’ rather than Lockdown#2.
  • The blue continuous curve is the ONS model for what is ‘really happening’.
  • I cannot explain how this estimate lies consistently below the data on which the model is based. Clearly there is something I have still not understood.

Data#2. Tests and Deaths

The graph below shows three quantities on the same logarithmic scale:

  • the number of positive tests per day
  • the number of people newly admitted to hospital each day
  • the number of deaths per day.

The data were downloaded from the government’s ‘dashboard’ site.

  • Positive tests refer to Pillar 1 (hospital) and Pillar 2 (community) tests combined – not the Pillar 4 tests from the ONS survey.
  • The deaths refer to deaths within 28 days of a test.
  • Hospital admissions for the UK nations combined.

All curves are 7-day retrospective rolling averages of the data since July.

Data for positive casesdaily hospital admissions and daily deaths. Click for a larger version. The data indicate that the number of positive tests per day and daily hospital admissions are probably flat, but the rate of Daily Deaths are still rising.

The graph shows the data alongside exponentially decreasing and then increasing trends shown as dotted lines.

  • The declining trends correspond to quantities halving every 21 days – the rate at which the epidemic declined during Lockdown#1.
  • The increasing trends correspond to quantities doubling every 15 days.

Back in July, the three data sets initially fell with similar time-dependencies and then rose through the autumn.

  • As shown in the graph below, it looked like the rate of daily positive tests was no longer rising, but the data for the 12th and 13th of November show a large unexpected increase.

Recent data for the number of daily positive tests and their 7-day retrospective average. Click for a larger version.

  • It looks like the rate of daily hospital admissions may also be flattening.
  • The rate of deaths is still rising – exceeding a 7-day average of 400 deaths per day this week. This should flatten around 10 days after the rate of daily hospital admissions flattens
  • I have also drawn lines – guidelines not predictions – of how we might expect testsadmissions and deaths to respond if we managed to make the lock-down as effective as it was in the spring and summer.
    • These are guidelines not predictions.
    • Look to see if quantities fall faster than this, or slower than this.
    • I have drawn a couple of example lines for each quantity depending on when that quantity starts to fall.

So…

Currently the 7-day retrospective average of COVID-related deaths is 409 people per day, up from 333 people per day last week. These figures should be compared with the roughly 1700 ‘normal deaths’ each day.

With another week of data it no longer seems possible that the death rate at the end of Lockdown#2 will fall below 200 people per day. Indeed reducing the death rate to 200 people per day by the end of the year now seems challenging.

We should all hope that Lockdown#2 is as effective as it possibly can be. Because if it isn’t, we will all be in a pickle!

Stay safe.

External Wall Insulation: How it’s done.

November 11, 2020

As many of you will know, I am having External Wall Insulation (EWI) applied to my house.

As closer confidantes will confirm: I am obsessed with the project. Why? Because based on my calculations, it is the single-most effective thing one can do to an old house to improve its thermal performance and reduce carbon dioxide emissions.

And yet very few people seem to be doing it. My hope is that by simply talking about it – and by measuring how effective it really is – more people will consider it as an option.

The idea of EWI is simple – “just stick insulating materials to the outside of a house“. But the reality of doing this reliably and leaving the house weatherproof and looking good is complex.

There are some nice videos out there, such as this one below showing the Be Constructive team working on a previous house. There are more videos here. And if you want details, then check out the extensive EWIPro Complete Guide (pdf) and all the materials are available at the EWI Store.

But partly for my own satisfaction I thought I would outline each step with pictures rather than video. Also, the video shows the application of expanded polystyrene boards and the procedure for the polyurethane foam boards that I have used is a little different.

So here is my description the process. There is a gallery of photographs at the end of the article.

Step 1: Preparation

The job began by protecting all the working surfaces – the patio and the front and rear gardens – with protective plastic, and then all the windows were covered with a transparent adhesive film.

For my house, the Be Constructive team demolished an old chimney which no longer had a reason for existing, and removed almost 2 tonnes of loose render from the side wall. So much render was removed that the wall had to be roughly re-rendered before they could begin applying the EWI.

They then moved the boiler exhaust, external electrical fittings and drain pipes to take account of the fact that the house was about to grow by about 120 mm in all directions. This stuff is rather tedious – but essential.

Next came the preparation of the outside walls and the painting of a ‘stabilizing primer’. This penetrates porous surfaces and binds them, creating a surface to which adhesive can stick. This is particularly important for some building blocks which can be quite powdery.

Step 2: Boarding. Kingspan K5

Next the team installed so-called ‘starter track’. This plastic support is screwed into the wall at the level of the first layer of insulating boards – usually just above the damp proof course – and makes sure the boards are horizontal, and supports them while the adhesive mortar dries.

Different stages in the application External Wall Insulation. Click for a larger version.

Normally EWI utilises either expanded polystyrene (sometimes abbreviated as XPS or EPS) or Rockwool™, and boards made from these materials are available in a wide range of thicknesses.

However I had asked to use a board made by Kingspan called K5. I chose this because I could only put about 100 mm thickness around the house – and for a given thickness, K5 will give the best insulation.

I limited the insulation to 100 mm because that amount would still keep the walls underneath the existing ‘soffit’ under the eaves. Also – if the insulation were much deeper – I felt the windows might seem to be too recessed.

Only 100 mm thickness of Insulating Boards would fit under the eaves of my house. Click for a larger image.

For some reason, 100 mm thick boards of Kingspan K5 were not available and so the Be Constructive team glued pairs of 50 mm thick boards together to achieve the required thickness.

The ‘double’ boards were stuck to the wall using several thick blobs of adhesive mortar. Using a big blob of mortar perhaps 10 mm deep allows the outer surfaces of the boards to be made parallel even when the underlying wall is not.

In my illustrations I have deliberately drawn the boards as being not parallel. In fact the Be Constructive team actually took a lot of care into making the final surfaces vertical and smooth. This is important because it is very difficult to compensate for this after the fact.

The boards are ‘overlapped’ at corners and cut to shape around windows and other architectural features. Any gaps are filled in with expanding foam.

Insulating Boards are overlapped at corners. Click for a larger image.

Step 3: Mechanical Fixing.

Once the boards are stuck to the wall and the mortar has set, the boards are mechanically fixed in place. To achieve this a hole is drilled through the boards and into the wall. Then a plastic fixing is pushed into the hole. Finally a metal nail is hammered into the plastic fixing which locks the plastic fixing in place – like a rawlplug – and holds the boards against the wall.

Using metal nails adds a heat leak directly through the boards: each fixture increases the thermal transmittance of the board by about 3%. However there is not much that can be done about that. It would be unwise to rely solely on the mortar or just plastic fixings.

Step 4: Base Coat Layers

Now the boards are attached to the wall and functionally insulating the house. But they are neither weatherproof nor attractive.

Preparatory stages in the application of weatherproof render. Click for a larger version.

So the next step is to coat the boards with an adhesive mortar (called a ‘base coat’) in which a glass-fibre mesh is embedded. This mesh is essential to prevent cracking due to building movement.

For polystyrene insulation this is a simple process: the boards are rasped to create a smooth surface; a layer of base coat is applied; the mesh is pressed into place; and then the mortar is smoothed. This forms a surface on which the the final render can be applied.

For K5 insulation, the process is more complicated because the surface of the boards should not be abraded. So:

  • First a thin layer of the base coat is applied to boards to create a smooth surface.
  • Then a second layer of base coat is applied into which the fibre-glass mesh is pressed.
  • Finally a third layer of base coat is applied to form a surface on which the final render can be applied.

The base coat also meshes with the corner and reveal ‘beads’, and with extra fibre-glass mesh placed around the corners of windows.

Step 5: And finally

And finally we come to the point where render is applied.

The render is a mixture of stone with a specifiable particle size: 1 mm, 1.5 mm or 2 mm , together with a mortar and a silicone polymer. It can be coloured in a very wide range of colours.

Additionally, my house will have ‘faux’ bricks called ‘brick slips’ applied to match architectural details on neighbouring buildings.

I’ll be sure to post pictures when we have finished.

 

Anticipated general look of the front of our house after rendering. Click for a larger view.

Photo Gallery – click for a larger version

Does it work?

But does it work? Well, of course it works! It would be physically impossible for it not to work!

The question isHow well does it work?“. And specifically, “Does it work as well I anticipated in my modelling?

These are complicated questions to answer definitively – and they are especially difficult to answer quickly.

I will not have a definitive answer until later in the winter, but I will explain how I will answer the question in a follow-up article. For now I will just tease you with the answer that the data look ‘promising’.

Keep warm 🙂

COVID-19: Day 311: Lockdown#2

November 7, 2020

Friends: so Lockdown#2 begins and I think it is important to understand why Lockdown#2 is important.

Lockdowns in all their flavours are disasters. But so are large numbers of unnecessary deaths.

Why are the Daily Mail agitating in favour of more death?

The truth is this:

  • The data seem to show that the steps taken prior to lockdown were having an effect. It seems that the ‘Tiered Measures’ were stabilising the pandemic with “R close to 1″.
  • But “R close to 1″ is not enough. I emphasised back in May (Link) we need R very much less than 1.
    • If only the tiered measures were kept in place indefinitely – which is not a realistic prospect – they would have led to a situation with roughly 300 COVID deaths per day (compared with roughly 1700 deaths per day in a normal year) continued until a vaccine was available. Optimistically, this amounts amounts to 30,000 people dying unnecessarily early.
    • Also, keeping prevalence high in the population offers the permanent ongoing risk that the death rate could increase rapidly if adherence to restrictive measures lapsed.

Anyway: let’s look at the data

Data#1. Prevalence

Since late April the ONS prevalence survey has been randomly testing people in England each week to look for the virus. They then collate their data into fortnightly periods to increase the sensitivity of their tests. Details of their full results are described methodically in this ‘bulletin‘.

Data from the ONS: The latest fortnightly data point is highlighted in red. Click for a larger version.

The number of people tested and the number of positive tests are given in their table above. ONS estimate that at the end of the measurement period on 31st October 2020 on average 1.14% of the UK population were actively infected – almost the same as the previous week

The raw count of positive tests was:

  • 1,900 from 160,428 people tested in the two weeks to 31st October,
  • 1,474 from 182.677 people tested in the preceding two weeks, and
  • 641 from 142,366 people tested in the two weeks preceding that.

Note these are random survey tests (so-called Pillar 4 tests), not clinical tests. Their data – graphed below – suggest that the prevalence is now doubling roughly every 13.5 days – slightly slower than last weeks data indicated.

Estimated prevalence of COVID-19 in England. Click for a larger version.

I have shown two curves on the graph above.

  • The black dotted line (– – –) is the same curve I have plotted for the previous eight weeks. (Link)
  • It is a fit to the 3 black data points and shows what we might expect if viral prevalence were doubling every 15 days.
    • This week’s update lies slightly below the extrapolation.
    • It shows that 8 weeks ago it was already clear where we would be today.
  • The blue continuous curve is the ONS model for what is ‘really happening’.
  • I cannot explain how this estimate lies consistently below the data on which the model is based. Clearly there is something I have not understood.

Data#2. Tests and Deaths

The graph below shows three quantities on the same logarithmic scale:

  • the number of positive tests per day
  • the number of people newly admitted to hospital each day
  • the number of deaths per day.

The data were downloaded from the government’s ‘dashboard’ site.

  • Positive tests refer to Pillar 1 (hospital) and Pillar 2 (community) tests combined – not the Pillar 4 tests from the ONS survey.
  • The deaths refer to deaths within 28 days of a test.
  • Hospital admissions for the UK nations combined.

All curves are 7-day retrospective rolling averages of the data since July.

Data for positive casesdaily hospital admissions and daily deaths. Click for a larger version. The data indicate that the number of positive tests per day is flat and hint that the daily hospital admissions may be flattening.

The graph shows the data alongside exponentially decreasing and then increasing trends shown as dotted lines.

  • The declining trends correspond to quantities halving every 21 days – the rate at which the epidemic declined during Lockdown#1.
  • The increasing trends correspond to quantities doubling every 15 days.

We see that the three data sets initially fell with similar time-dependencies and then rose through the autumn.

  • It looks like the rate of daily positive tests is no longer rising.
  • It looks like the rate of daily hospital admissions may also be flattening.
  • The rate of deaths is still rising but should flatten around 10 days after the rate of daily hospital admissions flattens
  • I have also drawn lines – guidelines not predictions – of how we might expect testsadmissions and deaths to respond if we managed to make the lock-down as effective as it was in the spring and summer.
    • These are guidelines not predictions. Look to see if quantities fall faster than this, or slower than this.

So…

Currently the 7-day retrospective average of COVID-related deaths is 333 people per day. In my opinion, this is too high.

Looking at the trends on the graph it looks like the death rate at the end of Lockdown#2 will be around 200 people per day. In my opinion, this is still too high.

Personally I think we should keep going until TT&I can cope – and this probably corresponds to death rates in the range of 10 to 100 per day.

If TT&I worked, we could keep death rates this low indefinitely while allowing sustainable economic and family activity.

Stay safe.

The Drinking Bird

November 4, 2020

While on a Zoom call with colleagues the other day, someone mentioned that during their time at Bell Labs, they had worked opposite Miles V. Sullivan.

I did not recognise the name. Then he told us what Miles V. Sullivan had done – he had invented ‘the drinking bird’! If you have not seen one of these, it is a small glass toy shaped with a bird-like form which first stands upright, and then leans of over to sip water from a glass, a process which repeats apparently endlessly. It is a wonder to behold.

I remembered having stared at these in shop windows as a child. I had imagined that such an ingenious device must stem from antiquity – and indeed it does have ancestry. But in this implementation, Miles V. Sullivan patented it in 1945. And according to my source, used the money to fund his PhD studies.

After the Zoom call I immediately moved our son’s  Drinking Bird onto the laboratory bench kitchen table for further experiments. As I set it ‘drinking’ I was re-fascinated by the subtle interplay of multiple physical principles. And I resolved to read more about it.

Explanations

There are some excellent explanations out there (e.g. this one or that one) and here is a video from ‘The Engineer Guy’.

I have added a couple of clarifying points below:

Illustration of the operation of the drinking bird in four stages. See the text for details. Click for a larger version

If one looks at the ‘bird’ one sees something like the situation in A above: a red liquid with ‘nothing’ above it. To understand the operation of the device it is important to realise that (a) the space is not ’empty’: it is filled with the vapour of the liquid and (b) the red liquid is composed of a transparent liquid with just a tiny amount of red dye.

The pressure of vapour above a liquid depends very strongly on the temperature of the liquid surface: the more energetic molecules in the liquid can sometimes leave the surface, escaping the attraction of the other molecules, and become part of a gas (called a vapour when it is in contact with its own liquid). In a closed container, the molecule will bounce around and eventually return to the liquid surface. In the steady state, as many molecules will leave the liquid, as return to it. When this state of balance is achieved the pressure is called the saturated vapour pressure.

Illustration of the molecular nature of a liquid and its vapour. Molecules are represented by blue blobs. Their direction of motion is shown for some molecules by a red arrow. Molecules in the liquid jiggle back-and-forth and if they are near the surface and by chance have sufficient energy they can escape into the vapour. The chance of escape is very temperature dependent. Molecules in the vapour occasionally rejoin the liquid. When the rates of leaving and rejoining are equal, the vapour pressure is said to be ‘saturated’.

  • If we cool the liquid surface, the average energy of molecules in the liquid falls, and the rate at which molecules escape into the vapour falls and the saturated vapour pressure falls.
  • If we cool the container, then vapour will start to condense as liquid on the cold surface.

In B I have coloured-in the space with two different shades (pink and green) to show that the two volumes of vapour are in touch with two different liquid surfaces.

In C, we now have a non-equilibrium state. The beak has been cooled, and a few small droplets of liquid will have condensed. The vapour in the neck and head is now in contact with two liquid surfaces: the liquid in the neck, and the colder liquid droplets in the beak. This situation is complicated and not stable.

  • The increased rate at which molecules are removed from the vapour as they condense in the beak lowers the pressure in the head allowing the pressure in the base to push the liquid up the tube.
  • On balance, molecules will evaporate from the liquid surface in the neck and condense in the liquid in the beak. As they condense, they release latent heat, warming the droplets in the beak and effectively transferring heat from neck to the beak. The process will continue as long as the beak is cooled by evaporation of the water on its outer surface.

The illustration in D is just to show that what I can measure is just the location of the top of the column, and not actually the height of the liquid column supported by the pressure difference.

So it’s complicated. But I found even the explanation by The Engineer Guy disappointingly qualitative.

So I thought there might be room for an explanation which involved actually measuring something! And there were two questions I wanted answered.

  • How cold does the ‘beak’ get in operation?
  • What’s the liquid inside? How can one possibly tell?

The first of these questions turned out to be easy to answer, but the second one has proved really hard.

How cold does the ‘beak’ get in operation?

To answer this question I attached a thermocouple (about £18 from Amazon: a great Christmas gift) to the felt that was glued to the bird’s beak.

Experimenting on a ‘Drinking Bird’. The picture shows a thermocouple attached to its beak with an elastic band.

I then wetted the beak and measured the temperature versus time every 10 seconds until the answer stabilised – which took around three minutes. The temperature fell by 1.8 °C.

In this first experiment the bird’s beak was still, but in operation the birds beak initially moves through the air and I wondered if this motion was significant.

To answer this I used a small USB-powered fan to move air past the beak at about 1 m/s (measured with a small hand-held anemometer). And the moving air did have a significant effect. In moving air the temperature fell much more quickly and by a larger amount: approximately 4.3 °C. I was so surprised by this large number that I repeated the experiment and obtained similar data with a change of 4.5 °C.

Remember that in order to repeat these experiments you need to completely dry the beak which generally takes about a day.

The graph below summarises the data. So still air or moving air, the answer is more than the “three tenths of a degree”  that The Engineer Guy claims about 6′ 30″ into the video. If he was making the video in very moist air or at very low temperatures, it might be possible to get such a small temperature drop. But I think its probably just a (rare) slip.

Temperature change of the wetted beak of the drinking bird versus time for still and moving air. Click for larger version.

In fact I could have looked up these answers in a table for the operation of a wet-and-dry-bulb hygrometer (psychrometer) but where would have been the fun in that! And it is an interesting feature that you may care to verify, that above a threshold air speed, the temperature of the beak doesn’t depend on air speed.

Anyway, thrilled by having ascertained the temperature difference driving the bird’s drinking habit, I turned to the second question.

What’s the liquid?

I felt fairly confident that Methylene Chloride (Chloromethane) – the liquid which The Engineer Guy asserted was in the bird – was no longer used. It is a highly corrosive chemical commonly used in paint stripper and probably not approved for use in a delicate glass toy that could be played with by children.

The Wikipedia page suggests a number of possible liquids which could be used:

  • Ethanol
  • Methanol
  • Di Ethyl Ether
  • Carbon Tetrachloride
  • Chloroform
  • Chloromethane

But how can you tell which liquid? I suspected that for a given temperature difference, the height of the liquid column was characteristic of the liquid used. So…

  • Using standard tables in Kaye and Laby I looked up the vapour pressure of the different substances at different temperatures.
  • From this I worked out the latent heat of vaporisation and calculated the vapour pressures above each liquid at 19 °C and at 20 °C.
  • The difference between the vapour pressures at these two temperatures told me the pressure forcing the liquid column upwards for 1 °C temperature difference.
  • Finally, I looked up the density of the liquids. I then used the change in the height of the liquid column as an indicator of the pressure difference.

Some Equations. Just in case you care. Click for a larger version.

So now I knew that for a given temperature difference each liquid would rise up the tube by a different characteristic distance.

  • Chloromethane: 1917 mm/°C
  • Di Ethyl Ether: 363 mm/°C
  • Methanol: 87 mm/°C
  • Chloroform: 63 mm/°C
  • Ethanol: 43 mm/°C
  • Carbon Tetrachloride: 34 mm/°C

So we can see why (in the absence of safety concerns) chloromethane would be a good choice: for a given temperature difference it would rise up the tube more than five times further that the next best liquid, di-ethyl ether, and about 20 times further the next best candidate, methanol.

Now all I had to do was to measure the height of the column as I changed the temperature difference between the base and beak.

The Experiment

I followed the ‘Golden Rule of experimental physics: Do it quick! Then, do it right. And so it became apparent very quickly that this was going to be difficult.

I won’t trouble you with all the dead ends down which I traveled over the last few weeks! Instead I will show the apparatus I arrived at and explain some of its special features.

Overview of the final experimental arrangement. Click for larger version.

An overview of the experiment is shown above, and a close up of the bird held in its bath is shown below.

Close up of the drinking bird. Click for larger version

Below is a time-lapse movie compressing the four and a half hours of the experiment into 33 thrill-packed seconds. If you press pause on the movie, you can then scroll through the movie and read off the data just like I did!

Here are some of the special features which you might not appreciate at first glance.

  1. The most important point which explanations of the drinking bird miss is that there is no air in the container. The vapour pressure is determined by the temperature of the liquid with which it is contact. When the beak is cooled, vapour condenses inside the beak to form a tiny amount of liquid. The vapour pressures above a liquid varies exponentially with temperature and this is the origin of the special sensitivity of this device.
  2. I decided to induce the temperature difference not by cooling the beak but by warming the base in a bath of water. To keep the temperature uniform I stirred it with a cappuccino frother. But the experiment went on so long that I got ‘thumb fatigue’ and couldn’t hold down the button! Hence I bought a stirrer hot-plate for a very reasonable £65, but I only used the stirrer, not the heater.
  3. My thermocouple reader only has a sensitivity of 0.1 °C and this was not sufficient resolution for this experiment. To detect smaller temperature changes I used thermistors as thermometers. For the bath I used waterproof thermistors (£5.95 for 5) and for the beak I used miniature thermistors  (£5.81 for 10). I first calibrated one against the other by tying them together in waterproof tape and immersing them in a glass of water as I heated it by intermittently adding a few cubic centimetres ofhot water.
  4. To measure the column height I printed out a scale that I could attach to the neck. This does not give me the column height directly – just a marker of where the upper end of the column is. I was interested in the rate of change of the column height rather than its absolute value. I applied a crude correction for the fact that as the upper end of the column rose, the lower end (which I couldn’t see) got lower.
  5. I applied power very gently, heating the water up with just a watt or two of electrical power resulting in the temperature changing at roughly 1 °C over two hours – or 8.3 mK per minute. The power supply was a very reasonable £69. The idea was to balance the column exquisitely so that I would be able to make it go up or down by changing the heater power. But in fact the system has hysteresis – a given bath temperature does not give rise to a unique column height. More on this later.
  6. I really should have used a better ohm meter for the thermistor on the beak, but I didn’t have one!

Finally the results. First I plotted the bath and beak temperatures as a function of time through the experiment. I did this by scrolling through the video and taking data every 5 minutes. You can see how the bath warms deliciously slowly.

Beak and Bath temperatures as a function of time. Click for a larger version

The graph below shows the same information as the graph above but now I have added in green the observed height of the liquid column – this is referenced against the right-hand axis.

The shaded regions are where the liquid column was observed to climb or fall along the neck. Readings above 90 mm corresponded to partial filling of the bird’s head and could not be accurately measured.

Finally I plotted the observed column height versus the measured temperature difference. The results of the video experiment (Experiment 4) are plotted in green below alongside results from a previous experiment where I took data on both rising and falling liquid columns.

Observed column height in centimetres versus temperature difference between the beak and the bath. Click for a larger version

Discussion

The first feature of the results which surprised and frustrated me was the hysteresis – the height is not a unique function of the temperature difference.

I did not have the patience to investigate this fully, but I am pretty sure it is a real feature and not just a feature of poor experimental method. I still don’t understand precisely why it occurs, but I think it is connected with the fact that the vapour in the head is in contact with two liquid surfaces at different temperatures. The vapour pressure is therefore not well-defined and depends on the temperature of the liquid column which is different when rising or falling.

The vapour in the head is in contact with two liquid surfaces at different temperatures. Click for a larger version.

But from the measured slopes it seems the column rises roughly 285 ± 25 mm/°C. Looking at the list of likely liquids, although the uncertainty is large, I think the data rule out chloromethane (as expected).

My conclusion is that the liquid is most likely diethyl ether: flammable and dangerous, but not quite in the category of chloromethane. The difference between the expected rate of rise (363 mm/°C) and the experimentally measured rate of rise is quite large, but the disagreement is very much greater with the other candidate liquids. One could imagine that the reason why the measured sensitivity is lower than the theoretically expected one is because the vapour in the head is connected to two liquid surfaces – the cold one in the beak, and the warm one in the neck.

So…at this point you are probably thinking – will this article never end? That is certainly what I am thinking. So summarising abruptly, the drinking bird is a beautiful demonstration of subtle physics. And when I am not so busy insulating my house, I may well return to see if I can develop either a better theory, or make some better measurements.

COVID-19: What should we expect Lockdown#2 to achieve?

November 2, 2020

So we wearily turn to face a dank November in ‘Lockdown#2’.

But what has been unclear in the reporting that I have seen has been a reasonable estimation of what we might expect Lockdown#2 to achieve?

I looked at this question recently, (link) but that analysis was based only on data after July. My conclusion was that the best we could hope for was that rates of hospitalisations and deaths might fall as they had done during Lockdown#1, halving roughly every 21 days.

But I thought I would look back at the entire data set going back to March to see if there were any periods that might possibly justify more optimistic expectations.

Let’s see what the data tell us.

The story so far…

Click for larger version. See text for details.

The graph above shows 7-day retrospective averages for:

  • The number of positive PCR tests per day (Pillar 1 & Pillar 2)
  • The number of hospital admissions.
  • The number of deaths per day (in hospital within 28 days of a positive COVID-19 test).

The data are downloaded from the Government’s Data Dashboard Site.

The graph has a logarithmic vertical axis and shows the situation in the UK since from March. The vertical grid is marked every 7 days. On this type of graph, quantities which are increasing or reducing exponentially appear as straight lines.

However the rates of positive tests per day on the left and right of the graph should not be directly compared:

  • On the left tests were mainly Pillar 1 (in hospital) but
  • On the right there is now widespread Pillar 2 (Community) testing.

So let’s ignore positive tests and just look at hospital admissions and deaths – quantities which are probably still counted in similar ways.

Admissions and Deaths in Lockdown#1

Click for larger version. See text for details

Looking just at:

  • The number of hospital admissions,
  • The number of deaths per day,

…we see that after lockdown on March 23rd, admissions rose for a about a week and deaths continued to rise for about two weeks.

But starting 3 weeks after lockdown (~day 105), hospital admissions fell exponentially at a constant rate until late July (~day 200) – a period of of 95 days.

And starting about 5 weeks after lockdown (~day 119), the death rate fell exponentially at a constant rate until late July (~day 212) – a period of of 93 days.

Given the complexities involved in a  real-world epidemic, this ‘straight-line’ behaviour over this extended period is remarkable.

Over these periods:

  • The number of hospital admissions halved every 27 days
  • The number of deaths per day halved every 16 days

One should not pay any special attention to those particular halving times. Previously – looking just at data after the start of July – I estimated them both as being about 21 days.

Admissions and Deaths in Lockdown#2

Click for larger version. See text for details

What reduction – if any – in the rate of hospital admissions and deaths might we reasonably hope for in Lockdown#2?

The default guess should probably be that the rates will behave as they did in Lockdown#1. If this were the case then we would expect rates to rise for a week or two, and then fall, halving every 21 days (±6 days) or so. I have sketched dotted lines on the graph above to guide the eye as to some possible ‘trajectories’.

So optimistically we might hope that death rates (currently 260 per day) might fall below 100 per day by the end of Lockdown#2. But it is also possible that they might only just reach 200 deaths per day.

But do we really expect things to be the same as in Lockdown#1? Societal differences between now and then could affect these rates. Such factors might include:

  • General availability of PPE and hand-sanitiser is better now.
  • Masks are now commonly worn in enclosed public places.
  • Working from home has become more widespread.
  • Testing is much more readily available.
  • Children are back at school
  • Students are back at University
  • People are weary of restrictions
  • Misinformation is widespread.

Overall, I think expecting deaths and hospital admissions to reduce at the same rate as in Lockdown#1 is at the optimistic end of reasonable expectations. But that is just an impression, and I hope I am wrong.

How did we get here?

I don’t like to dwell on the past, but please allow me to dwell on the past.

How on earth did we end up here?

Click for larger version. See text for details

 

Looking back with the benefit of hindsight there have been three points when the Government might have acted. I have marked them on the graph above as A, B and C.

Opportunity A was in early July when the rate of positive tests began to rise. If I recall correctly, there was very little appetite (public or political) for further restrictions, and the significance of the rise in tests while admissions and deaths were still falling was not clear at the time.

Opportunity B was in late August when the rate of hospital admissions began to rise. The death rate at this time was below 10 per day. Ideally test, track and isolate would have acted as a damper to this growth, but that has been a complete failure. The return to school was an event of great importance and perhaps it was felt politically and educationally important to ‘get this done’. Notice that the return to school – slightly surprisingly – does not seem to have affected the rate of hospitalisations in subsequent weeks.

Opportunity C was in late September when positive tests, hospital admissions and deaths were all rising exponentially. Even I noticed that something was wrong. If the government had acted decisively then it would no doubt have been portrayed as over-reacting. But they would have been right, and as a consequence:

  • the furlough scheme could have ended now as anticipated,
  • thousands of lives would have been saved, and
  • infection rates might be low enough for test, trace and isolate to function.

Missing opportunity C was in my view a serious error.

 


%d bloggers like this: