Heating Requirements Revisited

An Eccentric Anomaly: Ed Davies's Blog

Back at the beginning of 2019 I had a go at finding a formula to approximate the heating requirement of the rental house I'm in now. I wasn't too satisfied with the results but I've recently added some features to my software which help with looking into this a bit more and also have a lot more data available to base the approximations on.

The recent cold and very windy weather has also helped focus my attention on the matter. The small back bedroom I use as a study and the downstairs bathroom have kept comfortably warm and the kitchen, which is on the west side of the house, has actually been warmer than normal but the rest of the house, which is all exposed to the easterly winds, has been pretty chilly.

When I looked in 2019 I came up with formula:

heatreq = (21.3839 - t)·(1 + 0.016284·v1.693317)

where t is the outside temperature (°C) and v is the wind speed (m/s). Effectively, the heatreq is a temperature difference in kelvin (or °C, if you like) with an added “wind-chill” factor.

I wasn't terribly convinced by these parameters, though. I like to keep the house quite warm but still the base temperature seemed a bit high; normally it's expected to be a few degrees below the actual inside temperature because of incidental temperature gains.

More generally, the connection between these values didn't seem to be very strong; the sum of the error squares wasn't that sensitive to how the heating requirement was divided up between the base temperature and the wind parameters.

For a while my software has had the facility to output integrals, etc, taken over (roughly) uniform periods of time. E.g., the command httpreader -l --daily metar-obs/EGPC/temperature --dtime W06 gives me daily averages, etc, for each day last week of the temperatures at Wick airport as a JSON file. It can do half-hourly, hourly, 3-hourly, 6-hourly, 12-hourly, daily, weekly (ISO8601 weeks starting Monday 00:00), monthly and yearly.

More recently, I've added the ability to plot these averages with my graphing program. Here is the boiler run time (as a proportion of the total time) and the estimated heating requirement, using the formula above, on a weekly basis since I moved my study into the small bedroom upstairs. As always, click for more pixels.

houseplot -l --y1 --tan1 --weekly --scale-factor 1 --boiler --y2 --blue --weekly --heatreq --y1range [0:] --y2range [0:] -s 2018-W36 -e 2021-W06 --svg -O formula1.svg

Not a great fit! The relative scaling of the two y axes is a bit arbitrary depending on the outlying maximum values so they can't be expected to line up too well. However, the winter-time heating requirement values are only a bit more than the boiler use whereas in the summer they're proportionally a lot larger.

It's not overly surprising that summer isn't well represented given that the formula was derived from November and December data.

A few iterations of playing with the values manually and adding a conversion factor to scale the heating requirement directly to an estimated boiler-run proportion, so the values can be put on the same axis, gives a somewhat tidier formula:

heatreq = 0.0106 × (18 - t) × (1 + 0.015v²)

houseplot -l --y1 --tan1 --weekly --scale-factor 1 --boiler --y1 --blue --weekly -t $/mul/0.0106/$/heatreq/18/0.015/2 --y1range [0:] -s 2018-W36 -e 2021-W06 --svg -O formula2.svg

This is obviously still far from a perfect match but it's getting reasonably indicative. Some of the discrepancies can be explained:

Some other cases I've looked into have been less obvious. E.g., there's a week towards the end of February 2019 with anomalously high boiler use. It was a mild week when I spent more time than you might otherwise expect at home (I was setting up to put the east-gable window in for the second time) but it's difficult to see why it made that much difference. Still, it's a useful reminder that this sort of formula fits not just the behaviour of the house but also that of the occupants.

Anyway, the right-hand end of this graph shows how out of the ordinary the recent cold and wind have been. It's a lot milder now, though still quite breezy.