Understanding heat flow in my home: the key graph

A few weeks ago I realised – with a little help from readers – that in order to understand the thermal behaviour of my house, I needed to add up all the sources of heating within the house.

These heat sources included not just the gas I burned to power the central heating, but also the body heat of the humans in the house, and the electrical heating from all the electrical devices in the house.

This led me to plot an important graph and this article is about that graph and what I have learned from it.

The Graph

The idea behind the graph is simple, and please forgive me if you have read this before, but it is important.

When a house has a heating system controlled by a thermostat, the energy consumed in keeping the household temperature constant should be proportional to the temperature difference between the inside and the outside.

So one should find that if one plots a graph of:

• total energy dissipated daily inside the house (kWh/day)

against:

• the difference between the daily average external temperature and the thermostat-ed temperature (°C),

… the data should fall along a straight line through the origin i.e. zero temperature ‘demand’ should require zero energy consumption.

Initially, I found the data fell along a straight line, but it did not go through the origin (link). This was when readers reminded that I needed to take account of the heating from the humans in the house.

When I did this I found that best-fit straight line went pleasingly close to the origin.

Click for a larger version. Graph of total household energy use daily in kilowatt hours (kWh) against the difference between the daily average external temperature and 19 °C. Data runs from 14th November 2020 to 9th February 2021

Total Power in the Household.

Estimating the total heating power dissipated in the household is not entirely straightforward, and I have changed the way I have done it since the previous article.

• Gas:
  • In winter months, most of the heating arises from gas used by the boiler for domestic hot water and space heating.
  • Condensing-type boilers are typically about 90% efficient, i.e. only 90% of the gas consumed heats the house. To take account of this I multiplied the reading from the meter by 90%.
  • The boiler is a ‘combi’ boiler that heats water on demand to about 50 °C and then – in a shower or tap – it is mixed with cold water to yield water with a temperature of around 40 °C. When it flows down the drain of the shower its temperature has fallen to about 30 °C (Yes, I measured it). As it flows down the drain it cools further and I estimate that when it leaves the house it has cooled to roughly 20 °C. So about half the heat supplied to the water is retained within the fabric of the house, and about half is lost.
  • Some gas is used for cooking – probably only 1 or 2 kWh per day. This eventually heats the house with 100% efficiency.
  • I estimate water heating and cooking use only about 5 kWh/day and so in this analysis I have neglected the heat which flows out of the house with the waste water. I think this error might amount to my overestimating the household heating by perhaps 2 kWh/day. This error should be fairly constant in warm or cold weather and so I would not expect it to strongly affect the slope of the graph.
• Electricity
  • All the electrical items in the house – no matter what their nominal function (TV, laptop, radio, etc.) ultimately create heat. So I take the electricity reading from the smart meter and assume that all this energy is heating the house too.
• Humans
  • My wife and I have been in the house pretty much all the time since November, and so the heat from our metabolic processes also heats the house. Most of our 2000 kilocalories per day (8.4 megajoules per day = 2.3 kWh/day) ends up as heat i.e. we are each roughly equivalent to a 100 W heater. When our son was home at Christmas, I increased this term to take account of there being 3 people in the house.
• Solar#1
  • I have not taken account of direct solar heating of the house because (as I will explain below) I think at this time of year it is small.
• Solar#2
  • But I have taken account of solar electricity used in the house which does not show up on the smart meter. Typically, our solar panels are generating a couple of kWh per day that are used by appliances in the house, but which does not show up on the electricity meter. I have added 2 kWh/day to take account of this.

So estimating the total power in the household day-by-day is not entirely straightforward. But neither is it too hard with a little attention to detail.

The main parameter I want to extract from the graph is its slope: this tells me how much energy is required for each degree Celsius I raise the internal temperature above the external temperature. The slope can be expressed in two ways:

• 4.1 kWh/day per °C. So when the external temperature is 10 °C, and the internal temperature is 19 °C then I will need to use 4.1 kWh/°C x 9 °C = 36.9 kWh of energy each day to maintain the internal temperature.
• 171 W per °C. So when the external temperature is 10 °C, and the internal temperature is 19 °C then I will need to use 171 W/°C x 9 °C = 1539 W of power continuously to heat the house.

We will return to these numbers and their likely uncertainty later and in following articles.

Critically, the fact that ‘best-fit’ straight line goes through (or close to) the origin is very important, because this validates the basic assumption I am making in my thermal model of the house.

Variability and wind speed.

A correspondent privately expressed concerns that the wind might be affecting the rate at which his house cooled, and wondered whether it was significant for my house.

This seems plausible, because increased wind speeds certainly cool people more effectively. But does that hold for houses?

If such an effect were happening then it might account for some of the variability in the data – the scatter around the straight-line trend.

To look for this effect I subtracted the trend line from the data so I could look just at the scatter – the residuals as they are called in the statistics business.

Click for larger image. This shows the variability about the best fit straight line in the previous graph. The residuals have a standard deviation of 7.1 kWh/day. This is shown as a line and above and below the zero line.

The residuals seem to be pretty randomly scattered around the best-fit line with a standard deviation of 7.1 kWh/day.

I then looked at data from my weather station for the average daily wind speed.

I reasoned that if I plotted the residuals against the daily wind speed (see below) then (if wind cooled the house significantly) windier days should have residuals that tended to lie above the line of best fit – i.e. extra energy would be required to heat the house on windy days.

Click for larger image. This shows the residuals from the previous graph plotted against average daily wind speed.

We see that the scatter of the data compared to the original residual graph is hardly changed at all.

But the fact that the best-fit straight line is not quite horizontal indicates that the residuals show a weak correlation with wind speed. Days with an average wind speed of zero tend to have negative residuals i.e. low wind days tend to require slightly less heating.

• But the line of best fit can only ‘account for’ about 5 kWh/day of the variability at the very highest wind speed.

So overall, the idea that wind speed affects the heating required is not strongly supported. The idea cannot really explain ±7.1 kWh of daily variability.

On reflection this is not really surprising. Considering just the heat flowing through the walls, in order to leave the house, the heat must flow through four ‘thermal resistances’.

1. First, interior air must pass its heat onto the interior walls in a boundary layer of air.
2. Secondly the heat must flow through the brick walls.
3. Thirdly, heat must flow through the external wall insulation.
4. Finally, heat leaves the exterior surface of the house and warms the air in the exterior boundary layer.

Click for larger image.

Wind speed will only affect this fourth term, and this term is probably much smaller than the middle two terms in any case. So even if wind made a big difference to the last term, it wouldn’t affect the total ‘thermal resistance’ very much

Of course wind speed could also increase draughts in the house, and cool the house directly by removing warm air, but since the triple-glazing was installed, I haven’t noticed these wind-induced draughts.

Indeed the fact that there is not a strong correlation between extra heating required and wind speed sets a limit on the magnitude of this ‘air flow’ effect.

So what causes the variability?

The short answer is that I don’t know. I have considered a number of effects.

One possibility might be direct solar gain: heating of the house through sunlight shining through windows.

I checked for this effect by carrying out an analysis similar to the wind analysis but using the daily amount of electricity generated by the solar panels as a measure of sunniness.

Click for larger image. This shows the residuals from the previous graph plotted against daily solar power generation.

As with the wind speed analysis, we see that the scatter of the data compared to the original residual graph is hardly changed at all.

But the best-fit line has a small negative trend indicating that indeed sunnier days might require slightly less heating, but not enough to explain the scatter.

Other effects could include leaving doors open or the effects of energy lost by flowing hot water down the drain if we had lots of long showers. But none seem to be quite large enough to explain the observed variability.

Other more subtle effects might include the fact that cooking in the oven might possibly directly heat the thermostat – which is about 2 metres from the kitchen door. This might reduce the heat that the boiler supplies to the rest of the house.

But basically, I just don’t know.

The key result.

The key result from this analysis is that with its triple-glazing and snazzy External Wall insulation, the heating requirements for the house can be succinctly described by a single figure which can be written in two ways:

• 4.1 kWh/day per °C or 171 W per °C.

This is better than I estimated earlier in the winter (4.6 kWh/day per °C or 192 W per °C) but not as good as I had hoped for using a more naïve analysis.

In follow-on articles I will use this figure to look again at my estimates for the different ways in which heat can flow out of the house!

Previous articles about the house.

2021

• Domestic Batteries: Purchase Decisions and Realistic Models (1st February)
• In the Bleak Mid-Winter (19th January)
• Domestic Batteries: Thinking More about Batteries (9th January)
• Domestic Batteries: Thinking about Domestic Batteries (3rd January)
• External Wall Insulation: Day-by-Day Analysis (1st January)

2020

• External wall insulation: How well is it working? (23rd December) after a month or so
• External Wall Insulation: How well does it work? (16th November) just after installation.
• The Beginning of the External Wall Insulation Project (15th October)
• Passivhaus? Or Michaelhaus? (August 26th)
• Measuring the thermal conductivity of insulation (August 24th)
• I am becoming an insulation bore. (August 21st)
• My House: comparing models and measurements (July 28th)
• Estimating the expected thermal performance of my house (July 22nd)
• Be Constructive! (July 5th)
• House of Shame#1 (June 24th)

2019

• The OTHER front in the fight against climate change (September 22nd)
• Why does heating my house require 280 watts per degree Celsius above ambient? (August 18th)
• What it takes to heat my house: 280 watts per degree Celsius above ambient (August 16th)
• Should I still be using gas? (May 6th)