Archive for January, 2021

How Many Naturally-Occurring Elements are there? Corrigendum

January 28, 2021

And in non-COVID news…

… I received an e-mail from ‘Claire’ the other day pointing out that there was an error in one of my blog articles.

I try quite hard to be ‘right’ on this blog, so despite her politeness, I was distressed to hear this.

The error was in an article written on 15th February 2010 – yes, more than 10 years ago – entitled: Just How Many Naturally Occurring Elements are there?

Reading it again after all these years I was pleased with it. The gist of the article is that there is not a clear answer to the question.

It turns out that the nuclei of the atoms of some elements are so radioactively unstable that even though they do exist on Earth naturally, at any one time there are only handful of atoms of the substance in existence.

These elements seemed to be in a different category from, say, carbon (which has some stable isotopes) or uranium (which has no stable isotopes). But some of the isotopes of uranium have very long half-lives: 238U has a half-life 4.468 billion years – roughly the length of time that the Earth has existed.

So of all the 238U has which was donated to the Earth at its formation – very roughly half of it has decayed (warming the Earth in the process) and half of it is still left.

So I had no problem saying that 238U was ‘naturally-occurring’, but that it was a moot point whether Francium, of which there are just a few atoms in existence on Earth at one time, could really be said to be ‘naturally-occurring’.

So in the article I stated that I had stopped giving an exact number for the number of ‘naturally-occurring’ elements – I just say it is ‘about 100’ – and then discuss the details should anyone ask for them.

What was my error?

In the article I stated that Bismuth – atomic number 83 – is the heaviest element which has at least one stable isotopes. For elements with larger atomic numbers than Bismuth, every isotope is radioactively unstable.

What Claire told me was that in fact the one apparently stable isotope of bismuth (209Bi, the one which occurs naturally) had been found to be unstable against alpha decay, but with an exceedingly long half-life. The discovery had been announced in Nature in 2003: link

Click image for a larger version. Link to Nature here

What I want to comment on here is the length of the half-life: The authors estimated the half life of 209Bi was:

  • 1.9 (± 0.2) x 1019 years.
  • 19 billion billion years

This is an extraordinarily long time. For comparison the estimated age of the Universe – the time since the Big Bang – is estimated to be about:

  • 1.4 x 1010 years.
  • 14 billion years

Imagining that 1 kilogram of pure 209Bi was gifted to the Earth when it was formed roughly…

  • 0.4 x 1010 years.
  • 4 billion years

…ago. Then since that time less than 1 in a billion atoms (0.15 micrograms) of the 209Bi would have decayed.

We might expect a single nuclear decay in 1 kilogram of pure 209Bi every 5 minutes.

How could one measure such a decay? The authors used a transparent crystal of Bismuth Germanate (Bi4Ge3O12) which scintillates when a radioactive particle – such as an alpha particle passes through it. In this case, the crystal would ‘self-scintillate’.

But the background rate of scintillation due to other sources of radiation is much higher than the count due to the decay of the 209Bi.

To improve the discrimination against the background the authors cooled the crystal down to just 0.1 K. At this very low temperature its heat capacity becomes a tiny fraction of its heat capacity at room temperature, and the energy of even a single radioactive decay can be detected with a thermometer!

Combining light detection and heat detection (scintillating bolometry) helps to discriminate against spurious events.

And my point was…?

For all practical purposes 209Bi is stable. Anything with a half-life a billion times longer than the age of the Universe is at least stable-ish!

But Claire’s e-mail caused me to reflect that the apparently binary distinction between ‘stable’ and ‘unstable’ is not as obvious as I had assumed.

By this extraordinary measurement, the authors have reminded me that instead of saying that something is ‘stable’ we should really state that it may be stable, but that if it decays, its rate of decay is beyond our current limit of detectability.

So for example, we know that neutrons – outside a nucleus – decay with a radioactive half-life of just 10.2 minutes. But what about protons? Are they really unconditionally stable?

People have searched for the decay of the proton and established that protons may be stable, but if they do decay, their half-life is greater than 1.7 x 1034 years – or more than a million, billion, billion times the age of the Universe.

So now we know.

COVID-19: “We truly did everything we could.” Really?

January 26, 2021

Click for a larger version. See the text below for details of the four times at which the government failed to do “everything they could

Today (link) that Prime Minister Boris Johnson said he takes “full responsibility” for the government’s actions. Apparently he said:

“We truly did everything we could.”

Sadly, neither of these statements are true. If he took full responsibility, he would resign. And the government definitely did not do “everything they could“. In fact they did the opposite. They very consciously did “the least they could get away with“.

This article is about four occasions on which the government chose not to “truly do everything they could“.

The Big Picture

The graph at the top shows the course of the epidemic over the last year charting:

  • The number of people testing positive each day
  • The number of people admitted to hospital each day
  • The number of people who died each day after having recently tested positive

Looked at in this overview, it is easy to see the big picture: Positive cases lead to Admissions which lead to Deaths. There is – and never was – any confusion about this, except amongst those who sought – for ideological reasons – to spread confusion in an effort to downplay the seriousness of the epidemic.

Very roughly,

  • 2% of people testing positive will die.
  • 25% of people admitted to hospital will die.

I will ignore the Government’s appalling start to the pandemic in which – for example – on 3rd March 2020 Boris Johnson light-heartedly said he ‘shook hands with everyone’ as he visited a hospital with COVID patients.

Despite world-leading expert advice, the government were late to plan and act and so caused thousands of extra people to die.

They should have done better, but the pandemic came as a shock to many governments. So I will ignore this bad start because I remember that back in March 2020 the pandemic took me by surprise.

Government Failure#1: Early July 2020

Click for a larger version. See the text below for details of the four times at which the government failed to do “everything they could

From April through to June 2020, cases, admissions and deaths were falling with a halving time of 21 days. The weather was warm and sunny! The epidemic was shrinking and if the government had continued with its existing policies, we were on course for the incidence to be 10 cases per million at the start of September.

With an incidence of 10 cases per million, schools could have started safely at the end of the summer, and the track, trace and isolate programme could have had a chance to control the virus.

At the start of July, with the viral prevalence at around 0.1%, it became clear this trend was not continuing: link. However the government did not “do everything they could” to drive down the viral prevalence.

On 8th July, as cases were starting to rise, they announced an “Eat out to Help out” scheme.

In fact – as the government must have been advised – the virus was spreading amongst younger people.

Government Failure#2: Late August 2020

Click for a larger version. See the text below for details of the four times at which the government failed to do “everything they could

In late August 2020, cases exceeded exceeded 1000 per day, hospital admissions began to rise, and the rate of people dying had stopped falling.

At this point it was obvious that the virus was entering a second wave. The government had the best experts in the world but chose not to act because they felt – in my view – that there was an acceptable level of deaths which they could ‘get away with‘ – perhaps a few 10’s per day.

In fact the prevalence was growing at this point. Not acting at this point was a clear failure, especially with the re-opening of schools just a week or two away.

Government Failure#3: Late September 2020

Click for a larger version. See the text below for details of the four times at which the government failed to do “everything they could

In late September 2020, cases exceeded exceeded 5000 per day, hospital admissions were a factor 4 higher than in August, and more than 30 people a day were dying.

The epidemic was doubling every 15 days and it was completely obvious we need a shutdown. Even I noticed it. On 26th September I wrote “Here we go again…

But again, the government failed to “do everything it could” because it was in sway to a group of loud, right-wing back benchers who did not understand the nature of exponential growth.

Instead, they waited for a month to have an ineffective ‘Lockdown#2’ in November. I pointed out previously that this failure – this deliberate choice – led to the avoidable deaths of between 13,000 and 19,000 people.

Government Failure#4: Early December 2020

Click for a larger version. See the text below for details of the four times at which the government failed to do “everything they could

In Early December 2020, it was clear that although cases eventually began to fall, hospital admissions and deaths barely changed.

And then cases began to rise.

But again, the government failed to “do everything it could” because it sought popularity by suggesting people would be able to travel at Christmas.

With the expert advice they had to hand, the government must have known that would not be possible. But they encouraged societal mixing for economic reasons.

This list of failures is not exhaustive…

…because to create an exhaustive list of government failure would be exhausting. But off the top of my head…

  • The decision to send children back to school for one day at the start of January comes to mind as a catastrophic and obvious failure.
  • The complete absence of border quarantine for the last 9 months is … a curious omission.
  • The continuing utter failure of Test, Trace and Isolate is both a financial scandal and a moral disgrace.

In short, the Government have not “done everything they could“. Far from it.

We all make mistakes. And I am sympathetic to people who make them and learn. But this government has not learned.

I cannot understand how the ministers involved – most specifically the Prime Minister – can live with the shame of their repeated failures. Failures that have cost thousands of lives.

COVID-19: In the bleak midwinter

January 21, 2021

Apologies for not continuing with my usual coverage of the pandemic. But I am finding the scale of death distressing to contemplate.

Today (20th January 2021) there were 1820 deaths associated with COVID-19, a figure which should be compared with normal all-cause mortality of around 1700 deaths per day.

I cannot imagine what it is like to work in hospitals knowing that roughly 30% of those who are admitted with COVID-19 will be dead in about 7 days.

And the vast majority of these deaths were entirely preventable if the government had acted with public health in mind at all times. What kind of a government would knowingly not do that!

Cases >>>Admissions>>>Deaths. 

As best I can determine it looks like recent data is best described if:

  • After 5 days around 7 % of positive cases become hospital admissions.
  • After a further 7 days around 30% of hospital admissions become deaths.
  • So overall, around 2 % of positive cases become deaths after roughly 12 days.

The graph below shows the following quantities averaged over 7 days

  • () The actual number of recorded deaths (continuous black line)
  • (– – –)The number of positive cases multiplied by 0.022 (2.2%) and delayed by 12 days
  • (– – –)The number of hospital admissions multiplied by 0.31 (31%) and delayed by 7 days

The numbers are uncertain but it will looks like deaths will peak in the next few days.

Click for a larger version. The graph shows 7 day averages of actual deaths along with deaths predicted from positive cases (– – –) based on 2.2% mortality delayed by 12 days, and deaths predicted from hospital admissions (– – –)based on 30% mortality delayed by 7 days.

Also shown is a guideline (black dotted line – – –) indicating the death rate halving every 21 days as happened after the previous lockdown.

Administering the vaccine to the vulnerable should be effective at preventing deaths. But it will have little effect on viral spread because we are still so far from herd immunity – only 10% of the population have been exposed to the virus.

So what we should expect to see is that deaths predicted by cases should follow the black dotted line – because the vaccine will not stop widespread transmission until roughly 70% of the population are immunised.

However actual deaths should fall much faster.

But social restrictions will still be required even when the death rate becomes low because otherwise the virus will rage out of control.

If this happened, it would offer the virus many opportunities to find new mutations that could potentially evade the vaccine: this is not scaremongering: this is just what viruses do.

Herd Immunity. 

ONS data indicate that roughly 10% of the UK population have been exposed to COVID, and it has killed roughly 100,000 of my fellow citizens.

This indicates that early estimates of a potential death toll in excess of 0.5 million if the virus was allowed to run wild were – if anything – an underestimate.

Appalling as this Government’s response has been, we can at least be grateful that hundreds of thousands of lives have – for now – been spared.

In the bleak midwinter

January 19, 2021

So here we are in the bleak mid-winter – the place that everyone with External Wall Insulation loves to be.

As I remain-at-home-to-protect-the-NHS-and-save-lives I have spent a great deal of time staring at the following graph which shows the impact of the triple-glazing and External Wall Insulation.

Click for a larger vsrsion. Plotted in blue against the left-hand axis, the average daily consumption of gas (kWh per day) This is shown against the left-hand axis. Plotted in green against the right -hand axis is the average difference of the outside temperature from 19 °C (°C).

The graph shows two quantities plotted versus the number of days since the start of 2019.

  • In blue, I have plotted the average daily consumption of gas (kWh per day)
    • This is shown against the left-hand axis
  • In green, I have plotted the average difference of the outside temperature from 19 °C (°C)
    • This is shown against the right-hand axis

The dotted red line shows the weather now (circled in green) is colder than it was at this time two years ago.

However the amount of gas (circled in blue) that I am using to maintain the temperature of the house is now about half what was then: just over 50 kWh per day now versus just over 100 kWh per day then.

The carbon dioxide emissions associated with heating the house look set to be about 1.25 tonnes this winter – still a terrible figure – but much lower than 3 tonnes emitted in the winter of 2018/2019.

To go further we need to ditch the gas boiler and switch to a heat pump. Hopefully we will achieve this in the summer and then we can reasonably hope that next winter we will lower the carbon dioxide emissions associated with heating the house to about 0.4 tonnes – just 13% of what it was in 2018/2019.

 

 

Rocket Science

January 14, 2021

One of my lockdown pleasures has been watching SpaceX launches.

I find the fact that they are broadcast live inspiring. And the fact they will (and do) stop launches even at T-1 second shows that they do not operate on a ‘let’s hope it works’ basis. It speaks to me of confidence built on the application of measurement science and real engineering prowess.

Aside from the thrill of the launch  and the beautiful views, one of the brilliant features of these launches is that the screen view gives lots of details about the rocket: specifically it gives time, altitude and speed.

When coupled with a little (public) knowledge about the rocket one can get to really understand the launch. One can ask and answer questions such as:

  • What is the acceleration during launch?
  • What is the rate of fuel use?
  • What is Max Q?

Let me explain.

Rocket Science#1: Looking at the data

To do my study I watched the video above starting at launch, about 19 minutes 56 seconds into the video. I then repeatedly paused it – at first every second or so – and wrote down the time, altitude (km) and speed (km/h) in my notebook. Later I wrote down data for every kilometre or so in altitude, then later every 10 seconds or so.

In all I captured around 112 readings, and then entered them into a spreadsheet (Link). This made it easy to convert the  speeds to metres per second.

Then I plotted graphs of the data to see how they looked: overall I was quite pleased.

Click for a larger image. Speed (m/s) of Falcon 9 versus time after launch (s) during the Turksat 5A launch.

The velocity graph clearly showed the stage separation. In fact looking in detail, one can see the Main Engine Cut Off (MECO), after which the rocket slows down for stage separation, and then the Second Engine Start (SES) after which the rocket’s second stage accelerates again.

Click for a larger image. Detail from graph above showing the speed (m/s) of Falcon 9 versus time (s) after launch. After MECO the rocket is flying upwards without power and so slows down. After stage separation, the second stage then accelerates again.

It is also interesting that acceleration – the slope of the speed-versus-time graph – increases up to stage separation, then falls and then rises again.

The first stage acceleration increases because the thrust of the rocket is almost constant – but its mass is decreasing at an astonishing 2.5 tonnes per second as it burns its fuel!

After stage separation, the second stage mass is much lower, but there is only one rocket engine!

Then I plotted a graph of altitude versus time.

Click for a larger image. Altitude (km) of Falcon 9 versus time after launch (s) during the Turksat 5A launch.

The interesting thing about this graph is that much of the second stage is devoted to increasing the speed of the second stage at almost constant altitude – roughly 164 km above the Earth. It’s not pushing the spacecraft higher and higher – but faster and faster.

About 30 minutes into the flight the second stage engine re-started, speeding up again and raising the altitude further to put the spacecraft on a trajectory towards a geostationary orbit at 35,786 km.

Rocket Science#2: Analysing the data for acceleration

To estimate the acceleration I subtracted each measurement of speed from the previous measurement of speed and then divided by the time between the two readings. This gives acceleration in units of metres per second, but I thought it would be more meaningful to plot the acceleration as a multiple of the strength of Earth’s gravitational field g (9.81 m/s/s).

The data as I calculated them had spikes in because the small time differences between speed measurements (of the order of a second) were not very accurately recorded. So I smoothed the data by averaging 5 data points together.

Click for a larger image. Smoothed Acceleration (measured in multiples of Earth gravity g) of Falcon 9 versus time after launch (s) during the Turksat 5A launch. Also shown as blue dotted line is a ‘theoretical’ estimate for the acceleration assuming it used up fuel as a uniform rate.

The acceleration increased as the rocket’s mass reduced reaching approximately 3.5g just before stage separation.

I then wondered if I could explain that behaviour.

  • To do that I looked up the launch mass of a Falcon 9 (Data sources at the end of the article and saw that it was 549 tonnes (549,000 kg).
  • I then looked up the mass of the second stage 150 tonnes (150,000 kg).
  • I then assumed that the mass of the first stage was almost entirely fuel and oxidiser and guessed that the mass would decrease uniformly from T = 0 to MECO at T = 156 seconds. This gave a burn rate of 2558 kg/s – over 2.5 tonnes per second!
  • I then looked up the launch thrust from the 9 rocket engines and found it was 7,600,000 newtons (7.6 MN)
  • I then calculated the ‘theoretical’ acceleration using Newton’s Second Law (a = F/m) at each time step – remembering to decrease the mass by 2.558 kilograms per second. And also remembering that the thrust has to exceed 1 x g before the rocket would leave the ground!

The theoretical line (– – –) catches the trend of the data pretty well. But one interesting feature caught my eye – a period of constant acceleration around 50 seconds into the flight.

This is caused by the Falcon 9 throttling back its engines to reduce stresses on the rocket as it experiences maximum aerodynamic pressure – so-called Max Q – around 80 seconds into flight.

Click for a larger image. Detail from the previous graph showing smoothed Acceleration (measured in multiples of Earth gravity g) of Falcon 9 versus time after launch (s) during the Turksat 5A launch. Also shown as blue dotted line is a ‘theoretical’ estimate for the acceleration assuming it used up fuel as a uniform rate. Highlighted in red are the regions around 50 seconds into flight when the engines are throttled back to reduce the speed as the craft experience maximum aerodynamic pressure (Max Q) about 80 seconds into flight.

Rocket Science#3: Maximum aerodynamic pressure

Rocket’s look like they do – rocket shaped – because they have to get through Earth’s atmosphere rapidly, pushing the air in front of them as they go.

The amount of work needed to do that is generally proportional to the three factors:

  • The cross-sectional area A of the rocket. Narrower rockets require less force to push through the air.
  • The speed of the rocket squared (v2). One factor of v arises from the fact that travelling faster requires one to move the same amount of air out of the way faster. The second factor arises because moving air more quickly out of the way is harder due to the viscosity of the air.
  • The air pressure P. The density of the air in the atmosphere falls roughly exponentially with height, reducing by approximately 63% every 8.5 km.

The work done by the rocket on the air results in so-called aerodynamic stress on the rocket. These stresses – forces – are expected to vary as the product of the above three factors: A P v2. The cross-sectional area of the rocket A is constant so in what follows I will just look at the variation of the product P v2.

As the rocket rises, the pressure falls and the speed increases. So their product P v, and functions like P v2, will naturally have a maximum value.

The importance of the maximum of the product P v2 (known as Max Q) as a point in flight, is that if the aerodynamic forces are not uniformly distributed, then the rocket trajectory can easily become unstable – and Max Q marks the point at which the danger of this is greatest.

The graph below shows the variation of pressure P with time during flight. The pressure is calculated using:

Where the ‘1000’ is the approximate pressure at the ground (in mbar), h is the altitude at a particular time, and h0 is called the scale height of the atmosphere and is typically 8.5 km.

Click for a larger image. The atmospheric pressure calculated from the altitude h versus time after launch (s) during the Turksat 5A launch.

I then calculated the product P v2, and divided by 10 million to make it plot easily.

Click for a larger image. The aerodynamic stresses calculated from the altitude and speed versus time after launch during the Turksat 5A launch.

This calculation predicts that Max Q occurs about 80 seconds into flight, long after the engines throttled down, and in good agreement with SpaceX’s more sophisticated calculation.

Summary 

I love watching the Space X launches  and having analysed one of them just a little bit, I feel like understand better what is going on.

These calculations are well within the capability of advanced school students – and there are many more questions to be addressed.

  • What is the pressure at stage separation?
  • What is the altitude of Max Q?
  • The vertical velocity can be calculated by measuring the rate of change of altitude with time.
  • The horizontal velocity can be calculated from the speed and the vertical velocity.
  • How does the speed vary from one mission to another?
  • Why does the craft aim for a particular speed?

And then there’s the satellites themselves to study!

Good luck with your investigations!

Resources

And finally thanks to Jon for pointing me towards ‘Flight Club – One-Click Rocket Science‘. This site does what I have done but with a good deal more attention to detail! Highly Recommended.

 

COVID-19: Mid-January 2021

January 13, 2021

Friends, I have refrained from commenting on COVID so far this year because the situation is almost too terrible to contemplate.

That we should have sleep-walked into the pandemic last March seems to me forgivable: Even as I saw it coming I could not grasp what was really happening.

But today, deaths from the second wave have exceeded the 41366 deaths from the first wave.

That we should have walked with our eyes wide open into a worse situation now, after living with the pandemic for a year, seems to me to be utterly unforgivable, the product of mendacious incompetence.

Where we are now.

We have just reached a weekly average of more than 1000 pandemical deaths a day, a figure that should be compared with a figure of roughly 1700 deaths per day that we expect normally.

The virus itself is widespread amongst the population and economy has been devastated.

The situation is difficult in many countries: I invite readers to check out statistics on one or more of these countries.

Additionally, there are many free countries  which have coped well.

But the UK’s position is dismal.

What next?

As I have said many times previously, the three sort-of reliable streams of data give us three views of the spread of the infection.

  • The number of daily positive cases gives us our first view of viral spread.
  • Then the number of daily admissions to hospital gives us a second view. The criteria for hospital admission have probably remained similar throughout the pandemic.
    • Typically daily admissions comprise about 13% of positive cases but delayed by about 5 days from a positive test.
  • Then the number of daily deaths gives us a final view of the number of infections. This criterion has changed throughout the pandemic – but only to more clearly identify pandemical deaths.
    • Typically daily deaths comprise about 2% of positive cases but delayed by about 15 days, or 27% of daily admissions but delayed by about 10 days.

Understanding the simplicity of the progression: positive cases lead to admissions lead to deaths, is essential for grasping what is happening.

The factors and delays relating admissions and cases to deaths have changed through the pandemic – but not by much.

The graphs below show the 7-day retrospective average of daily deaths since July 2020. Also shown are the curves of

  • Positive cases multiplied by 0.02 (i.e. 2%) and delayed by 15 days
  • Daily admissions multiplied by 0.27 (i.e. 27%) and delayed by 10 days

The graphs are shown with both linear and logarithmic axes.

Click for a larger Image. The graph shows  positive cases multiplied by 0.02 (i.e. 2%) and delayed by 15 days and daily admissions multiplied by 0.27 (i.e. 27%) and delayed by 10 days alongside actual deaths per day.

Click for a larger Image. The graph shows  positive cases multiplied by 0.02 (i.e. 2%) and delayed by 15 days and daily admissions multiplied by 0.27 (i.e. 27%) and delayed by 10 days alongside actual deaths per day.

One can see that during the first part of the second wave, deaths are reasonably well “predicted” by the scaled and shifted curves of cases and admissions.

In summer, the actual deaths are below the admissions curve indicating – only about 10% of admissions died. This is likely because the disease was spreading amongst a younger fraction of the population.

But in December and January, the agreement has not been so good.

It could be that the delays between cases, admission and deaths are now shorter. Or that the mortality has changed.

For example, the graph below shows a calculation with delays of 4 days between cases and admissions and 4 days between admissions and deaths. This doesn’t describe the first part of the second wave so well, but seems to reasonably match what is happening now.

Click for a larger Image. The graph shows  positive cases multiplied by 0.02 (i.e. 2%) and delayed by 8days and daily admissions multiplied by 0.27 (i.e. 27%) and delayed by 4 days alongside actual deaths per day.

Alternatively, these changes in delay could be accompanied by changes in mortality. Until deaths have peaked, I won’t be able to clearly separate these two effects.

It looks like positive cases might now have peaked (Day 375), which suggests that deaths will peak (depending on the actual delays), sometime in the range Day 375 + (8 to 15) = Days 383 to 390 (i.e. 18th to 25th January).

After the peak

Also shown on all the graphs is a ‘guideline’ indicating how deaths might fall over the coming months.

The curve is an exponential curve starting at 1000 deaths per day and halving every 21 days – the same rate which applied after the first wave.

When we know the peak value of the death rate I will adjust the curve to match the actual deaths.

Since we now have vaccines, it is to be hoped that the death rate will decline much faster than after the first wave.

Let’s hope things go well. It does happen sometimes.

 

=================

Edited on 13/01 to add some countries which have done well at controlling coronavirus

 

Thinking more about batteries

January 9, 2021

Thanks to everyone who got in touch about the previous article on batteries.

Two communications in particular – from correspondents in Canada and Japan – gave me pause for thought, reminding me of my UK perspective.

And on reflection I realised again that I was really hoping to do two similar, but quite district, things with a battery, Both involved storing electricity and using it later: that’s what batteries do! But the first involved storing solar electricity and the second involved storing grid electricity.

Which of these tasks was relevant would change with the seasons, and the optimal storage of solar electricity depends quite a bit on the day-to-day variability of the weather.

Why use a domestic scale battery?

My aim is to reduce carbon emissions from my house. And just to be clear – my primary motive is moral, not financial. But nonetheless, money matters. I don’t have infinite resources and methods for reducing carbon emissions that don’t make financial sense are unlikely to catch on with less zealous carbon-noughts than I.

Defn: Carbonought/Carbonaut: a person seeking to live without net emission of carbon dioxide.

There are two distinct ways in which a battery can help.

Reason#1: Winter heating 

My gas boiler emits around about 0.2 kg for every kilowatt hour (kWh) of heat it provides. Over the winter of 2019/2020 it released 2.5 tonnes of CO2 just keeping the house at a modest 19 °C.

My recent improvements to the insulation will have reduced that by about half for the current winter – but that is still more than a tonne of CO2! A tonne! The gas boiler is more than 90% efficient at converting chemical energy to heat, so in order to reduce emissions. further I need to find another way to heat the house.

The answer is to “go electric”. As the UK electricity network has increased the contributions from renewable sources of electricity, the so-called carbon intensity (kg CO2 per kWh) has fallen steadily year-on-year. In 2020 the average was about 200 gCO2/kWh so using electric heating will not immediately result in lower CO2 emissions than burning gas directly. But the carbon intensity of UK electricity has fallen from about 450 gCO2/kWh in 2012 and is expected to be close to 100 gCO2/kWh in 2030 – and so switching to electric heating should result in a gradual year-on-year decline in emissions.

But a kWh of electricity costs around £0.24 whereas a kWh of gas costs only about £0.05 – so just going electric without doing anything else would cost more than 4 times as much!

My solution is to insulate the house reducing the overall heating load by about half, and then to meet the remaining requirement with a heat pump.

A heat pump is an electrical device that moves heat from outside to inside. Critically, one kWh of electricity can move two kWh of heat from outside resulting a net heating power of three kWh. This should reduce the annual bill for heating electrically to something similar to the previous gas bill.

But by using a battery, I can pay even less! I can buy the electricity to operate the heat pump at night – when it costs much less – and then use the battery to power the heat pump during the winter days. This arrangement should reduce the cost of heating the house to well below what I currently pay.

  • So in this case I would be using the battery to shift the time at which I purchased the electricity from the grid.

Reason#2: Summer Solar Electricity 

In the summer my house requires very little heating – but it still uses electricity. During the summer months I expect the solar panels on my roof to produce enough electricity to power the house – but they do not produce it at the correct time.

During the summer, the role of battery is to store solar electricity generated during the day which I can then use to power the house at night.

  • So in this case I would use the battery to shift the time at which I used locally-generated solar electricity.

If I can get this right I should not need to rely on grid electricity at all for many days at a time i.e. there would be practically zero carbon emissions during the summer – and additionally some electricity could be exported.

Examples

Let’s see the difference between summer and winter operation by using a few examples. 

The graphs below were generated using this spreadsheet – that simulates the interaction of solar panels and batteries.

Screen shot from a spreadsheet: Click for a larger view. The boxes in the top left with yellow backgrounds allow different parameters – such as battery size – to be changed. Various parameters are then calculated minute-by-minute through the day. Daily performance is summarised in the boxes with red backgrounds.

The graphs show two types of quantities versus hour-of-the-day.

  • For conceptual and programming simplicity, the time-axis shows hours after 11:30 p.m. rather than the more normal midnight. This allows me to plot the ‘off-peak’ electricity period (11:30 p.m. to 6:30 a.m.) as a contiguous area on the graph – it is shown as dark grey.
  • The battery capacity (- – -) and state of charge are shown in kW) against the left-hand axis.
  • Electrical demand (in kW), exported power (in kW) and solar power (in kW) are shown against the right-hand axis.

Case#1. At the moment (January in the UK) the house uses about 12 kWh of electricity each day – equivalent to a 0.5 kW continuous demand – and we don’t have a battery. The graph shows 0.5 kW of ‘demand’ represented by a thick black line which should be read against the right-hand axis.

The nominal daily cost of this electricity is £2.30 and the CO2 emissions amount to 2.33 kgCO2/day.

Case#1: Click for a larger version

Case#2. During the winter the solar panels generate on average 2 kWh per day. This weak solar generation (2 kWh) is shown as a yellow line in the graph below. This reduces the cost of electricity to £1.81 (saving £0.49) and reduces the CO2 emissions to 1.94 kgCO2/day (saving 0.39 kgCO2).

Case#2: Click for a larger version

Case#3. But sometimes – on clear cold days – the solar panels can generate 5 kWh per day as shown in the graph below. Because the instantaneous generation (up to 2 kW) exceeds the household demand – the bulk of the solar generation is exported (blue line). The price paid for this is generally low – EDF offer 1.8 pence per unit. But somebody somewhere will benefit from ‘my’ low carbon electricity!

Comparing Case#3 to Case#2, even though the panels generated 2.5 times more electricity than Case#2, the daily cost of electricity would only be reduced by a further £0.04 to £1.77 and ‘my’ CO2 emissions by a further 0.04 kg to 1.90 kgCO2/day.

Case#3: Click for a larger version

Case#4. So what would a battery do? Using the same 5.0 kWh of solar generation as in Case#3, if I had a 13.5 kWh Tesla Powerwall 2 battery then the solar generation would be captured and re-used later in the day. Very roughly such a battery would cost £10,000. Yes. That much.

The state of charge of the battery is shown as a green line in the graph below and should be read against the left-hand axis. Looking closely, one can see that once the solar generation exceeds the local demand – the excess is stored. Then when solar generation falls below local demand, the battery begins to discharge.

Case#4: Click for a larger version

By displacing expensive ‘peak’ electricity the daily cost is slashed to just £1.06 and the CO2 emissions reduced to 1.34 kgCO2/day. But all the stored electricity is used up before the end of the day.

Case#5. Now let’s imagine a summer day with 15 kWh of solar generation. Now the daily cost is slashed to just £0.20 and the CO2 emissions reduced to 0.35 kgCO2/day. Remember without solar panels or batteries the cost and emissions were £2.30 and 2.33 kgCO2/day. So we see that there are potentially large savings of both money and CO2 to be made.

Case#5: Click for a larger version

Case#6. At the end of the day described in Case#5, the battery still held 6.6 kWh of charge . So the next day – if solar generation were similar – would look like the graph below. The battery would discharge further over night and then be re-charged during the day.

Case#6: Click for a larger version

The daily cost is now £0.00 and the CO2 emissions are 0.00 kg. This is the situation in which the house is off-grid and all the electricity is used locally. Halleluiah!

Case#7. But now, at the end of the day, the battery holds 9.6 kWh of charge. So the next day – if solar generation were similar – would look like this:

Case#7: Click for a larger version

Again the daily cost and CO2 emissions are zero. But now we have reached a stable situation where the battery holds the same charge at the beginning and end of the day.

During the day – the storage capability of the battery was exceeded – and 2.9 kWh of electricity was exported to the grid. Potentially this export might generate a few pennies of payment but in general it represents a loss to me personally – but a boon for the planet because someone somewhere gets to use it as it displaces CO2-producing electricity generation.

Case#8. But now suppose the weather on the next day was dull – with just 4.0 kWh of generated electricity.

Case#8: Click for a larger version

In this case the battery will discharge – again the cost and carbon emissions are both zero – but at the end of the day the battery charge is low (1.6 kWh). In this case the optimal strategy depends on knowledge of the next day’s weather.

Case#9. Let’s suppose that Case#8 was followed by another poor generation day but that one just left the battery to discharge overnight – this situation is illustrated below in Case#9. Now one needs to purchase grid electricity at both peak and off-peak rates – this would cost £1.22 and emit 1.25 kg/CO2.

Case#9: Click for a larger version

Case#10. However if after Case#8 one had charged the battery from the grid overnight (see below) then one could avoid purchasing grid electricity at peak rate – this option would cost only £0.35 (less than a third of Case#9 cost) but emissions would be similar 1.26 kg/CO2.

Case#10: Click for a larger version

Case#11. But if one had charged the battery in readiness for a poor generation day (Case#10) and then by chance the weather had turned out fine, then one might have the situation as shown below.

Case#11: Click for a larger version

In this situation the cost and CO2 emissions are the same as previously (£0.35 and 1.26 kg) but now some of the electricity (1.9 kWh) is exported.

Case#12. If one could have anticipated Case#11, then one might have chosen to charge the battery less overnight:

Case#12: Click for a larger version

By reducing the charging rate, the cost and CO2 emissions are both reduced (£0.25 and 0.88 kg) and no electricity is lost by export to the grid.

Summary so far

In the summer the optimal financial strategy is to:

  • Have sufficient charge in the battery – through overnight charging – that no peak electricity need be purchased towards the end of the next day.
  • Have sufficient spare capacity in the battery so that no solar electricity will be lost to the grid – for negligible recompense.

However a strategy should not be so complicated that one has to spend significant effort programming the battery. Obviously bigger batteries makes these choices less critical,  but I think I need some real-world data before I can recommend a strategy for a UK summer.

In terms of reducing carbon emissions globally, then what I want is a situation in which – in addition to meeting my own needs – I export as much electricity as possible. – like Case#7.

Winter with a heat pump 

After installation of the external wall insulation, the house can now be heated 20 °C above the external temperature using less than 3 kW of heating power. Using a heat pump, this 3 kW of heating power can be supplied using just 1 kW of electrical energy.

Case#13. So after a heat pump has been installed, I can expect an additional roughly constant demand of between 0.5 kW and 1 kW on top of normal household use. So total demand will be between 1 kW and 1.5 kW.

Case#13: Click for a larger version

This is illustrated above:

  • For an additional 0.5 kW of demand i.e. 1 kW in total, the daily cost and CO2 emissions would be would be (£4.60 and 4.66 kg)
  • For an additional 1.0 kW of demand i.e. 1.5 kW in total, the daily cost and CO2 emissions would be would be (£6.90 and 6.99 kg)

Case#14. On a typical winter generation day (2 kWh), the solar panels will reduce this slightly:

  • For an additional 0.5 kW of demand i.e. 1 kW in total, the daily cost and CO2 emissions would be would be (£4.10 and 4.26 kg)
  • For an additional 1.0 kW of demand i.e. 1.5 kW in total, the daily cost and CO2 emissions would be would be (£6.40 and 6.59 kg)

Case#14: Click for a larger version

Case#15. In Winter in general a battery would be used to purchase slightly greener but much cheaper electricity at night.

For example in the situation below with an additional 0.5 kW of demand due to the heat pump i.e. 1 kW in total, the daily cost and CO2 emissions would be would be £1.40  and 4.00 kg – saving £3.30 with respect to the situation with solar panels but no battery.

Notice though that battery runs out of charge before the end of the day requiring the purchase of roughly 1.5 kWh of peak electricity.

Case#15: Click for a larger version

Case#16. Below I consider what I think is a “Reasonable Worst Case” corresponding to very cold winter day with prolonged temperatures around 0 °C.  There would be an additional 1 kW of demand due to the heat pump i.e. 1.5 kW in total, and even with the solar panels and the heat pump, the daily cost and CO2 emissions would be £3.70  and 6.33 kg.

Case#16: Click for a larger version

This comprises 24 kWh of demand from the heat pump which would deliver 72 kWh of heat. If that 72 kWh of heat had been provided by gas – as it is now – the daily cost and CO2 emissions would be £3.60  and 14.4 kg.

So in this reasonable worst case, the cost is similar to using gas – but there are big reductions in carbon emissions.

Can I ever reach zero emissions?

I think that ‘zero emission’ is possible in a certain sense, but it won’t be easy. And I may need to invest beyond what I have planned already. Let me explain:

  • In the summer it should be possible to operate ‘off grid’ for several days or weeks at a time, and indeed – with the battery fully-charged – to export a fraction of the solar-generated electricity. I think it would be fair to consider this exported electricity as “CO2 emissions avoided” because it is displacing the use of CO2-emitting generation for someone else. Let’s call the amount of CO2 production avoided X kg.
  • But in the winter it will be necessary to draw upon grid resources – and so CO2 will be emitted to generate that electricity. Let’s call the amount of CO2 production Y kg.

Click for a larger version: Illustration of how my come could become carbon neutral. Excess solar electricity exported in the summer avoids the emission of X kg carbon dioxide somewhere else on the UK grid. In winter when I need to draw electricity from the grid I will cause Y kg carbon dioxide to be emitted. If I can make X equal to Y then I think I claim that my home is carbon neutral.

To make the operation of this household ‘carbon neutral’ when averaged over one year, I need X to be equal to Y. Unfortunately at the moment I can only guess at X and Y in the most general terms – I hope to get the data I require this year. I am modestly confident that with a battery and a heat pump I can get the difference down to hundreds of kilograms rather than tonnes. But I am not sure I can get it down to zero.

But as the carbon intensity of UK electricity falls, year-upon-year, both X and Y will reduce and hopefully their difference will also get smaller.

Additionally, there is still space on my roof for more solar panels so perhaps in a year or two  I could increase X further. And of course we could just switch stuff off, turn down the thermostat and wear a pullover!

Summary

In writing this article – and the previous one – I have become convinced of the utility of domestic scale batteries and I hope to order one as soon as the last of the quotations comes in.

I am very excited by the prospect – External Wall Insulation AND a domestic battery. Truly I am living the dream!

 

Thinking about domestic batteries

January 3, 2021

My External Wall Insulation project is complete and the solar panels are installed, so I am left to simply gather data on how things are working: a retired metrologist’s work is never done!

So inevitably my mind is moving on to the ‘next thing’, which is possibly a battery, and I am left with nothing to do but write over-long articles about the possibilities.

  • [Note added on 9/1/2021: If you like this article, then try also the next article on the same subject – link – I think it is a little clearer and the spreadsheet has been improved.]

The idea of using a battery is very simple: store solar electricity and use it later! But as I tried to think about it, I found myself intermittently perplexed. This could be an age thing, or just due to my lack of familiarity with solar power installations, but it was not at all obvious to me how to operate the battery in harmony with the solar panels.

This is because energy can flow in several directions.

  • For example electricity from the solar panels could charge the battery, operate the domestic load, or be exported to the grid.
  • Similarly, the battery could charge itself from the grid, operate the domestic load or export energy to the grid.

Understanding these things matters because domestic scale batteries are not cheap.

  • A rechargeable AA battery with 5 Wh of capacity (3.3 Ah @ 1.5V) costs around £5.
  • If we scale that up to 13.5 kWh (the size of Tesla battery) then 2700 rechargeable AA batteries would cost about £13,500.
  • In fact there are some economies of scale, but the likely cost is still around £10,000.

After making several simulations I think I have a clearer idea how the scheme would work, so please allow me to explain.

Mode#1: Storing in the day.

At the moment the solar panels generate at the whim of the weather gods – and the iron diktats of celestial geometry.

In sunshine – even at mid-winter – the panels can generate at more than 2 kW and unless we are using that electricity in the house at the moment the Sun is shining, the power is exported to the grid.

Click for a larger version. Solar electricity (in kWh) generated daily since the solar panels were installed.

  • Over the last 50 winter days the panels have generated about 136 kWh
  • I have used about 60% of that, saving round 81.6 x 24.3 pence ~£19.83
  • But I have given away about 40% of the electricity I have generated.
  • I can arrange to sell that electricity to EDF, my electricity and gas supplier, for the grand price of 1.8 pence per unit i.e. the 54.4 units I have donated would be worth £0.98
  • However, if I could have stored those units and used them later I would have saved approximately £13.22.

So using a battery to store solar energy and then use it later to displace buying full-price electricity makes some financial sense. It also makes carbon sense, displacing grid electricity with low-carbon solar energy.

In winter, a battery would make the most of the meagre solar supply and in summer it would allow us to be effectively ‘off grid’ for many days at a time.

Mode#2: Storing at night.

But batteries can also be used to store electricity generated at night time – when it is cheap. EDF charges me 24.31 pence for each unit I use between 6:30 a.m. and 11:30 p.m. (‘peak’ rate) , but only 4.75 pence for each unit I use overnight (‘off peak’ rate).

On average, we use around 11 kWh/day of electricity, around 9 kWh of which is used during ‘peak’ time. So if I could buy that electricity at the ‘off peak’ rate (costing 9 x 4.75 = 42.75 p), store it in a battery, and then use it the next day, then I would avoid spending 9 x 24.31 pence = £2.19.

This strategy would save me around £1.76 per day, or around £640 per year – a truly staggering amount of money!

It would also be slightly greener. The exact amount of carbon dioxide emitted for each unit of electricity – a quantity known as the carbon intensity – depends on how the electricity is generated,

  • Electricity generated from coal has a carbon intensity of around 900 gCO2/kWh
  • Electricity generated from gas has a carbon intensity of around 500 gCO2/kWh
  • Electricity generated from nuclear, solar or wind has a carbon intensity of a few 10’s of gCO2/kWh

Depending on mix of generating sources, the carbon intensity of electricity varies from hour-to-hour, day-to-day and from month-to-month.

To estimate the difference in carbon intensity between ‘peak’ and ‘off peak’ electricity is quite a palava.

  • I went to the site CarbonIntensity.org.uk and downloaded the data for the carbon intensity of electricity assessed every 30 minutes for the last three years.
  • I then went through the data and found out the average carbon intensity for ‘Off Peak’ and ‘Peak’ electricity.
  • I averaged these figures monthly.

The data are graphed below.

Click for a larger version. Carbon intensity (grams of CO2 per kWh of electricity) for UK electricity evaluated each month since the start of 2018. The red curve uses data for ‘Peak Rate’ electricity and the blue curve shows data for ‘off peak’ electricity’. The black curve shows the difference between ‘peak’ and ‘off-peak’ and the dotted red line shows the average value of the difference.

The average ‘Peak Rate’ carbon intensity over the last two years is approximately 191 g CO2 per kWh, and the ‘Off-peak’ average is approximately 25 g (or 13%) lower.

I calculated that over the last year if I used 9 peak units and 2 off-peak units per day then the carbon emissions associated with my electricity use would have been 749 kg (~three quarters of a tonne) and the cost would have been £822.

If I had instead bought all those units at night, stored them in a battery, and used them the next day the carbon emissions would have been 661 kg – a saving of 88 kg and the cost would have been just £188 – a saving of £634.

Summary so far

So these two strategies involve using the battery to:

  • Store solar electricity in the day (which maximises my personal use of my personal solar electricity)
  • Store grid electricity at night (which appears to be amazingly cost effective and has about 13% lower carbon emissions)

Understanding how these two strategies can be combined had been hurting my head, but I think I have got there!

I think the operating principles I need are these:

  • Whenever solar electricity is available, use it.
  • If the solar power exceeds immediate demand,
    • If the battery is not full, store it.
    • If the battery is full, export it for whatever marginal gain may be made.
  • At night, charge the battery from the mains so that it is full before the start of the next day.

I have run a few simulations below assuming a Tesla Powerwall 2 battery with a capacity of 13.5 kWh. If you want, you can download the Excel™ spreadsheet here, or view typical outputs below.

  • Note: I hate sharing spreadsheets because as Jean Paul Satre might once have said “Hell is other people’s spreadsheets“. Please forgive me for any errors. Thanks

Battery only: No Solar

Click for a larger version. The dotted (—-) red line shows the battery capacity of a Tesla Powerwall 2 and the green curve shows the state of charge of the battery. Both should be read against the left-hand axis. The yellow curve shows the electrical power generated from the solar panels and the blue curve shows the power exported to the grid. Both are zero in this graph. Both should be read against the right-hand axis.

In the first simulation the battery charges from empty using 2 kW of ‘Off Peak’ electricity and fills up just before morning. It then discharges through the day (at 0.4 kW) and is about half empty – or half full depending on your disposition – the next evening.

So the next day the battery starts charging from about 50% full and then discharges through the day and is again about 50% full at the end of the day.

Click for a larger version. The dotted (—-) red line shows the battery capacity of a Tesla Powerwall 2 and the green curve shows the state of charge of the battery. Both should be read against the left-hand axis. The yellow curve shows the electrical power generated from the solar panels and the blue curve shows the power exported to the grid. Both are zero in this graph. Both should be read against the right-hand axis.

So based on this simulation, it looks like a stable daily charge and discharge rate could effectively eliminate the need to use ‘Peak-Rate’ electricity.

Each night the battery would store however much electricity had been used the day before.

Battery and solar in harmony 

Click for a larger version. The dotted (—-) red line shows the battery capacity of a Tesla Powerwall 2 and the green curve shows the state of charge of the battery. Both should be read against the left-hand axis. The yellow curve shows the electrical power generated from the solar panels and should be read against the right-hand axis.

The simulation above, shows what would happen if there were weak solar generation typical of this wintry time of year. As the solar electricity is being generated. the rate of discharge of the battery slows – is reversed briefly – and then resumes as the solar generation fades away.

A modest generation day – typical of a bright winter day or a normal spring day – is shown below.

Click for a larger version. The dotted (—-) red line shows the battery capacity of a Tesla Powerwall 2 and the green curve shows the state of charge of the battery. Both should be read against the left-hand axis. The yellow curve shows the electrical power generated from the solar panels and the blue curve shows the power exported to the grid. Both should be read against the right-hand axis.

At its peak the solar generation reaches 2 kW – and in the middle of the day re-charges the battery to capacity. When the battery reaches capacity – the solar generation covers the domestic load and the excess electricity is exported (blue curve).

On a long summer day solar generation might reach 3.6 kW but here I assume just a 2.5 kW peak. In this scenario, the battery barely discharges and solar generation covers the domestic load and exports to the grid during the day. Only in the evening does the battery discharge.

Click for a larger version. The dotted (—-) red line shows the battery capacity of a Tesla Powerwall 2 and the green curve shows the state of charge of the battery. Both should be read against the left-hand axis. The yellow curve shows the electrical power generated from the solar panels and the blue curve shows the power exported to the grid. Both should be read against the right-hand axis.

Battery and heat pump and solar 

The battery and the solar panels are just a part of the wider project to reduce carbon emissions which – if you have been paying attention – involves replacing my gas boiler with an air source heat pump. This uses electricity to move heat from outside into the house.

Back in the Winter of 2018/19 the gas boiler supplied up to 100 kWh/day of heating. In the slightly milder winter of 2019/20 the boiler used on average 70 kWh/day of gas for heating. This winter the External Wall Insulation and the Triple Glazing seem to have reduced this average to about 40 kWh/day – with a peak requirement around 72 kWh on the very coldest days.

Using a heat pump with a coefficient of performance of about 3, it will require 40/3 kWh= 13.3 kWh/day of electrical energy to supply these 40 kWh of heat energy. This amounts to an additional 0.55 kW running continuously.

I have simulated this situation below by increasing the load to 1.0 kW. In this case the battery will discharge a couple of hours early and we will have to buy a couple of units of full-price electricity.

Click for a larger version. The dotted (—-) red line shows the battery capacity of a Tesla Powerwall 2 and the green curve shows the state of charge of the battery. Both should be read against the left-hand axis. The yellow curve shows the electrical power generated from the solar panels and the blue curve shows the power exported to the grid. Both should be read against the right-hand axis.

And finally we come to the reasonable worst-case scenario. Here there would be effectively no solar power (dull winter days!) and the external temperature would be around 0 °C requiring around 72 kWh of heating i.e. 3 kW of heating power. This will require 1 kW of electrical power to operate the heat pump on top of the 0.4 kW of domestic load.

Click for a larger version. The dotted (—-) red line shows the battery capacity of a Tesla Powerwall 2 and the green curve shows the state of charge of the battery. Both should be read against the left-hand axis. The yellow curve shows the electrical power generated from the solar panels and the blue curve shows the power exported to the grid. Both should be read against the right-hand axis.

In this scenario we would require about 8 hours of full price electricity @1.4 kW i.e. 11.2 kWh which@ 24.3 p/kWh would cost around £2.70. So if there were 10 of these days a year it would cost roughly £27/year.

I could avoid purchasing this full price electricity by buying two Tesla Powerwall batteries to give a capacity of 27 kWh. But spending an additional £8000 to avoid paying £27 year does not look like a sound investment.

Click for a larger version. The dotted (—-) red line shows the battery capacity of two Tesla Powerwall 2 batteries and the green curve shows the state of charge of the battery. Both should be read against the left-hand axis. The yellow curve shows the electrical power generated from the solar panels and the blue curve shows the power exported to the grid. Both should be read against the right-hand axis.

Summary

Overall, I think I now understand how a battery would integrate with the way we use energy in this house, and I think it makes sense.

Regarding money:

  • Using a battery  I would appear to be able to save many hundreds of pounds each year by purchasing off-peak electricity instead of peak electricity.

Regarding carbon:

  • Without solar panels, the switch to ‘Off Peak’ electricity should reduce annual emissions from roughly 749 kg to about 661 kg – a saving of 88 kg.
  • With solar panels we should generate roughly 3700 kWh of low carbon electricity, all of which will be used either by me or by someone else, displacing carbon-producing generation. This would be true with or without a battery. But the battery allows me to personally benefit.
    • During the summer the battery should allow me to benefit from the full amount of solar energy generated, reducing grid use (and expenditure) to almost zero.
    • During the winter, where only about 2 kWh of solar generation is available each day, it should reduce carbon emissions by about 20% compared with using ‘Off Peak’ grid electricity.
    • In the worst case – when using a heat pump to heat the house on very cold days with negligible solar power – I will need to buy full price electricity for a few hours a day.

So when I replace the gas boiler with an air-source heat pump, we will inevitably rely on the grid for some full-price electricity on the few coldest days of the year. That is why I have been so keen to reduce the amount of heating required.

COVID-19: How many deaths will it take…

January 1, 2021

In the immortal words of his Bob-ness

How many deaths will it take till he knows, that too many people have died?

Personally I am in shock at the ongoing COVID death toll – currently over 70,000 but almost certain to exceed 100,000.

On the first day of this new year I have tried to gain some perspective on what has happened so far, what is happening now, and how things might evolve in the coming weeks and months

The story so far…

The story so far can be summed up in a single graph showing positive coronavirus (CV) tests, hospital admissions and deaths.

Click for a larger version. The graph shows 7-day retrospective averages of three quantities plotted on a logarithmic scale versus the number of days since 1st January 2020: The number of positive CV tests per day, the number of hospital admissions per day, and the number of deaths per day

We see the initial rapid rise in deaths in late March increasing to roughly 1000 people per day in mid-April.

Then we saw the effect of Lockdown#1 – with hospital admissions and deaths falling exponentially (a straight line on the logarithmic graph) for approximately 100 days.

From Mid-May onwards, the CV tests were widespread enough that the decline in positive tests followed the same exponential decay – halving time 21 days – as admissions and deaths.

Up until the start of July the virus was in decline.

There then followed a two-month period of confusion, where rising CV Positives were not matched by rising hospital admissions and deaths.

In retrospect it is clear that the virus was spreading amongst younger people who were less likely to be hospitalised or die.

Failure to act decisively at this point was – in retrospect – a mistake.

  • A further two months of restrictions would have substantially eliminated the virus from endemic circulation.
  • The incidence in September would have been low enough to Track, Trace, and Isolate, to function and for schools to re-open in relative normality.

Then in September, cases and admissions and death rates rose exponentially.

Failure to act decisively at this point was a mistake that is not just clear in retrospect: it was pointed out at the time. This choice by Government to ignore SAGE advice has resulted in the entirely predictable deaths of tens of thousands of people so far.

Instead the virus was allowed to spread and we now have the current situation in which the Government appear to be unwilling or powerless to act with sufficient strength to drive down viral prevalence.

They seem to be hoping that the roll-out of the vaccine will bring the COVID nightmare to an end. And this will probably happen.

But failure to act during December 2020 has allowed the viral prevalence to exceed 1% across wide swathes of the UK resulting in many more deaths than necessary.

…In the next few weeks…

I can hardly bear to look at the data at the moment. But as I write, despite Tier 4 measures or similar across most of the UK, there is no sign yet that the virus has been contained.

We can peer a short way into the future by assuming that 24% of people admitted to hospital die after 7 days.

We can peer a little further ahead by assuming that 2% of people testing positive for CV will die after 17 days.

The graph below shows these estimates of future deaths as dotted lines alongside the actual deaths so far.

Click for a larger version. Daily Deaths. The graph shows 7-day retrospective averages of three quantities plotted on a logarithmic scale versus the number of days since 1st January 2020: The number of deaths per day (continuous black line) and then two estimates for the number of deaths per day based on the number of positive CV tests 17 days earlier (dotted red line – – –)  and the number of hospital admissions 7 days earlier (dotted yellow line – – –) .

The basic structure of the second wave was well-predicted by both these estimates. But it is curious that the fall in the number of positive cases after Lockdown#2 ‘should’ have resulted in a significant fall in both admissions and deaths – but these falls did not materialise.

I cannot explain this, but one hypothesis would be that people were avoiding getting tests for fear that if they tested positive, they might then be obliged to alter their Christmas arrangements.

  • Looking at hospital admissions it looks as though death rates will rise to more than 600 per day in one week from now.
  • Looking at positive cases it looks as though death rates will rise above 800 per day in mid-January.

These deaths look to be ‘baked in’ and do not include the effect of any mixing – albeit pretty limited – at Christmas or New Year.

Looking further ahead

It is difficult to imagine one’s way to the end of January 2021. There will presumably be further restrictions, and possibly a full lockdown.

On the two graphs above I have penciled-in a basic structure of what we might expect based on the following assumptions for the state of play at the end of January 2021.

  • Hospital admissions  of 2500 per day: roughly the same as now.
  • Positive cases of 50,000 people per day: roughly the same as now.
  • Deaths of 1000 people per day – where we seem to be headed.
  • The rate of decline is the same as in Lockdown#1 – i.e. quantities halved every 21 days.

Of course there is real hope that the vaccines now available will make a significant difference over the next few months.

If administered at 1 million doses a week then the 12.9 million people over 60 should be vaccinated by Easter (4th April). A significant fraction of other vulnerable groups will also likely be immunised.

If all goes as planned, both the death rate and the rate of hospitilisations should fall much more rapidly than I have shown.

And so… 

And so it looks like sometime in the summer life will return to relative normality. Beer will be drunk in pub gardens with friends.

However I would not be surprised if there were further problems that impeded this expectation for any one of a thousand reasons.

Reasons might include difficulty producing the vaccine, the actions of vaccine refuse-niks, civil unrest due to widespread hardship and homelessness. Or possibly further viral mutations – or a new virus! I invite you write in your own ‘plot twist’.

And even when normality is resumed I would not expect life to ever be quite ‘normal’ again. As his Bob-ness said: “too many people have died.”

Dear reader: I trust the late stages of the COVID-19 pandemic will leave you and yours unharmed, wherever you are.

 

External Wall Insulation: Day-by-Day analysis

January 1, 2021

Happy New Year 2021!

Please note:

  • A list of related articles is given at the end
  • This article was amended around 12 hours after it was posted to take account of heating effect of the people living in the house!
  • Thanks to Ed, Simon and Geoff for noticing. And caring!

At around the time that the External Wall Insulation (EWI) was being applied to my house, I also had my electricity and gas meters upgraded to “Smart Meters”.

This gave me access to daily readings of gas and electricity consumption in kilowatt hours (kWh). I could get these readings in two ways.

  • The hand-held readout unit shows daily readings for the last 7 days.
  • After 3 days, the readings became available on the EDF web site, either to view directly as histograms, or to download as a spreadsheet.
  • Readings could be downloaded as daily data month-by-month, or half-hourly for any particular day.

From analysing this daily data I discovered something so obvious it was surprising!

What does the data look like?

The graph below shows the daily gas consumption (kWh gas consumed) plotted versus day-of-the-year

Click for a larger version. Graph showing daily gas consumption (in kWh) versus day of the year 2020

Initially I did not quite trust this new-fangled technology so I also plotted my weekly gas consumption readings expressed as a daily average. These are shown as solid blue lines on the above graph. Taking parallel overlapping readings showed me that in general I could trust the readings. Also shown are the 7 weeks of the EWI installation.

The daily readings do appear to generally make sense, but there were two days – days 298 and 329 – where gas consumption appears to have been zero: I think these are mis-readings.

To put the vertical scale into context, 24 kWh is the energy used by a 1 kW heater left on all day. So the vertical scale (100 kWh/day) is equivalent to just over 4 kW of continuous heating.

Each kWh of gas consumed results in the release of around 0.2 kg of carbon dioxide. So the full-scale 100 kWh/day would be equivalent to 20 kg per day of carbon dioxide. Total emissions over the period shown are just over 500 kg – more than half a tonne!

Weather

Whether the graph above represents good performance or not depends on how cold the weather was.

Our internal thermostat is set to 19 °C and so I assess the temperature ‘demand’ as being the difference between 19 °C and the average daily external temperature.

Below I have plotted the gas consumption data against the left-hand-axis, and additionally plotted temperature demand data against the right-hand axis.

Click for a larger version. Graph showing daily gas consumption (in kWh : left-hand axis) and temperature demand (in °C :right-hand axis) plotted versus day of the year 2020

The first thing to notice is how closely the curves correlate. Unsurprisingly, gas consumption directly follows temperature ‘demand’.

The second thing to notice is that after day 312, the gas consumption curve is much lower down: – there is a clear ‘gap’ between the temperature demand data and the gas consumption data.

Day 312 marked the point where the kitchen roof was sealed, marking the sealing of the building envelope.

Taking data only from Day 312 onward should allow me to assess the building performance by plotting gas consumption versus temperature demand. This graph is plotted below.

Click for a larger version. Graph showing daily gas consumption (in kWh) versus temperature demand (in °C). Notice that the best-fit line does not go through the origin.

The data make a pleasing straight line, something which is rarely a coincidence. But two things are puzzling.

  • The first puzzle is that the graph does not go through zero – or even near it! This implies that when the average daily temperature is 15 °C outside we require no heating!
  • The second puzzle is that the slope is 4.6 kWh per day per degree Celsius which is equivalent to 192 W/°C. This is considerably more than the 134 W/°C that appeared to describe the weekly data that I showed in my previous post.

The first puzzling thing 

I think the answer to the first point is that I have assumed that the heating inside the house is only due to gas consumption but in fact there are other sources of heating.

The electrical energy used in the house also warms the house. Indeed, in one perspective, one can view all electrical appliances as heaters, each with its own additional functionality as a computer, a light, or a radio etc.

So in the graph below I have plotted daily [gas + electricity] consumption versus demand. The data have the same slope – because electricity consumption is roughly constant day upon day – but the intercept is now closer to zero – indicating zero demand with an external temperature deficit of 2 °C. But this is still not zero.

Click for a larger version. Graph showing daily gas consumption + daily electricity consumption (in kWh) versus temperature demand (in °C). Notice that the best-fit line is closer to the origin.

However there are two more corrections. Over the period plotted, my newly-installed solar panels were generating on average 2.4 kWh per day, 60% of which (1.4 kWh) was used in the house rather than exported.

This 1.4 kWh/day does not show up on the electricity meter. Including this additional term we arrive at the graph below. This suggests zero demand with an external temperature deficit of 1.5 °C.

Click for a larger version. Graph showing daily gas consumption + daily electricity consumption + solar energy (in kWh) versus temperature demand (in °C). Notice that the best-fit line is even closer to the origin.

Finally, (and thanks to Ed, Simon and Geoff for pointing this out) we need to take account of the heating by the two human beings living in the house.

My wife and I each eat a nominal 2000 kilocalories per day (8.4 megajoules per day) and most of that energy ends up as heat. This 8.4 MJ per day corresponds to 2.3 kWh per day each i.e. we are each roughly  equivalent to a 100 W heater. So in a very real sense, our love will keep us warm. Allowing for this effect the best fit line now passes very close to the origin.

Click for a larger version. Graph showing daily gas consumption + daily electricity consumption + solar energy + body heat (in kWh) versus temperature demand (in °C). Notice that the best-fit line is even closer to the origin.

The best-fit line still describes the trend of the data well, but is now (within plausible experimental uncertainty) consistent with my belief that there should not be an offset.

Overall, I take this data as evidence for the validity of the obvious: that the heat from all the electrical appliances and the people in the house really does heat the house. I have known this intellectually, but this is the first time I have ever seen specific evidence that this is the case.

So I find this both completely obvious, but also somehow surprising – because I wasn’t looking for it!

The second puzzling thing 

  • So taking account of the first puzzling thing, I conclude – from analysing around 50 days of data – that the house takes 4.6 kWh/day/°C [192 W/°C] to maintain 19 °C.
  • But in the previous blog I concluded – from analysing two year’s worth of weekly data – that the house performed better, requiring less than 3.2 kWh/day/°C [134 W/°C] to maintain 19 °C.

Which of these do I believe? Sadly, I believe the worse of these two estimates.

The model in the previous blog article took no account of roughly 10 kWh/day of electrical heating or the 2 x 2.4 kWh/day of body heating by my wife and I. While this may not have been significant a couple of winters ago when we used almost 100 kWh/day in winter, it is significant now that daily gas consumption is only (roughly) 35 kWh/day.

Thermal Models #1 and #2

So I have now made a new thermal model Model#2 – which incorporates the electrical heating.

The graph below compares the two models #1 (- – -) and #2 (- – -). They both predict the gas consumption in terms of the weather, and a parameter that describes the house insulation. But Model #2 takes account of the fact that electrical power dissipation and human bodies also heat the house.

Average Daily gas consumption in KWh per day for the last two years. Also shown are two models. Model#1  (- – -) predicts gas consumption based on the average temperature ‘demand’. Model#2  (- – -) predicts the same thing but takes account of the heating of the house by (a) electrical appliances estimated at 400 W continuously or 9.6 4.8 kWh/day. The constants of proportionality for each model are changed to allow the model to match the gas consumption in the winters of 2018/19 and 2019/20. Click for a larger version.

Overall Model#2 is slightly better than Model #1 at matching the gas consumption data – but more importantly it is based on the basic reality that electrical appliances and body heat really do heat the house!

The models differ in the constants of proportionality that they require to describe the thermal insulation. In the graph above I have changed the constant of proportionality around Day 250 – when most windows were triple-glazed – and around Day 660 – when the EWI commenced.

  • In the winter of 2018/2019 model#1 suggests it took 280 W ( 6.7 kWh/day) of continuous power for each 1 °C above the external temperature.
    • Using model#2 the data is better described by a figure of 350 W ( 8.4 kWh/day) per °C
  • In the winter of 2019/2020 model#1 suggests it took 240 W of continuous power (5.7 kWh/day) for each 1 °C above the external temperature
    • Using model#2 the data is better described by a figure of 300 W ( 7.2 kWh/day) per °C
  • In this winter of 2020/2021 I had hoped the EWI would mean I needed only 134 W (3.2 kWh/day) of continuous power for each 1 °C above the external temperature.
    • Using model#2 the data is better described by a figure of 192 W ( 4.6 kWh/day) per °C

Summary

Looking at daily data, I found that a graph of gas consumption versus temperature demand did not go near the origin unless I took account of the heating due to the electrical appliances and the people living in the house.

  • Although this is obvious, this is the first time I have ever seen data which demonstrated this to be the case.
  • Essentially, I have turned my house into a calorimeter!

Taking account of this suggests that my house currently requires 192 W to heat it 1 °C above external temperature. This is 43% higher than 134 W/°C I had hoped for.

One obvious factor which I had not considered until it was pointed out in the comments on this article is that gas boilers are not 100% efficient. Typically, they are only around 90% efficient.

I will consider the impact of this effect, and other possible explanations in another article – watch this space!

Previous articles on this topic

2020

2019

 


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