gemini://ur.gs/blog/u-values/

A friend and I are currently going through a bit of a journey together, booted along by the expected price of energy for the coming winter. We both want the same thing - a well-insulated house with a heat pump to keep us warm, and solar panels to power the heat pump. No gas, directly or indirectly! Our starting points are a bit different, though.

He's recently bought an old Victorian end-terrace with solid brick walls and no insulation, while mine is a detached new-build with regular cavity walls and plenty of insulation. This naturally means his house loses a lot more heat than mine. That means he'd need a much bigger heat pump to stay warm - but how much bigger? And how much would adding insulation help?

Fortunately, we'd taken the excellent Eco Refurbishment course, run by People-Powered Retrofit and Carbon Co-op, in 2021, so we had a vague intuition about how to work these things out, even if the specifics were gone from memory. As you'd expect from the title, U values are the magic.

Conceptually, a house is a box, and the heat pump is a little daemon that takes heat from outside and puts it into the box. The heat naturally diffuses right back out; U values tell you how quickly the heat leaves the box. It's actually surprisingly obvious from the units: W/M²K. A U value might look like "0.13W/M²K", or "1.5W/M²K".

Breaking it down a bit, that's "Watts per Square Metre per Kelvin". Watts are a measure of heat; square metres are a measure of area; kelvin here refers to the difference in temperature between inside and outside.

The U-value itself, i.e., the number, is a measure of how much heat a given "sandwich" (all the layers that make up an exterior wall, floor, or roof - bricks, cavity, insulation, plaster, etc) is losing. Lower is better - you're losing fewer watts - and higher is worse. If you know all the layers of your house's wall/floor/roof construction, you can calculate this yourself, but it's much easier to just go look up numbers for typical construction. Let's assume a uniform 0.18 for the moment.

A box (house) has six sides (ignoring pitched roofs for simplicity), so we can just sum up the area of those sides to work out how many square metres we're talking about. If the house is a 5x10x5M box, that would be 250 square metres. Remember, this is the surface area of the outside of the box, not the area of the floor plan; assuming two floors, that would be 100M² for this house.

As for those kelvin, it's a delta. If it's 5°C outside and you want it to be 20°C inside, K is 15. If it's -10°C outside and you want it to be 20°C inside, well, then K is 30. Let's stick with 15 for the moment.

So putting it all together, we can say we're losing 0.18W of heat from each of 250 square meters, multiplied by the delta kelvin of 15 - that's 675W. Sadly, if your house is "solid brick construction", your U value is 2W/M²K, so you end up losing an order of magnitude more heat - 7,500W!

These numbers are pretty close to the difference between my house and my friend's house. There's lots of refinements you can do - windows and doors lose more heat, for instance, and it's not actually a flat roof - but it gives us a first-order approximation.

It's enough for us to know that insulation taking the U value from 2W to 0.18W would allow my friend to have a heat pump running at about 675W instead of at 7,500W, saving a massive 6,825W. This permits a smaller heat pump to be fitted, which saves a lot of money on installation. It's not a break-even proposition though - maybe you save £5K on the heat pump and solar panels, but the insulation job might cost £10K or more, depending on how you do it.

However, energy isn't free, and it's getting more expensive at the moment, so we should consider running costs. Powering the pump from solar panels is fine, but if we weren't doing that we could store the solar panel output in a big battery and buy less electricity from the grid.

Electricity is charged in kilowatt hours. Getting from W to kWh is easy - divide watts by 1,000 (to get to kW) then multiply by the number of hours you're interested in. 7500W is 180kWh per day. 675W is 16.2kWh per day.

This is heat output - how much heat the house is losing each day. Heat pumps don't use one kWh of electricity to produce 1kWh of heat - that's what immersion heaters do! Instead, they have a "coefficient of performance" (CoP), which tells you how much heat they can pump into the house for each unit of electricity they use up. For an air-source heat pump attached to a hot water system (an "air-water ASHP"), 3.5 is a good guess.

Divide the heat output by the CoP to get power input, then multiply by the energy provider's cost per kWh, and we learn that the uninsulated house loses about £20 of heat per day(!), while the insulated house loses just £1.85 of heat per day - or to put it another way, the insulation saves you around £6,625 per year.

Now, if energy bills double to 80p/kWh, then you're saving basically double that - £13,250/yr. More than the insulation!

This does assume you keep the whole house at 20°C at all times and that the mean outdoor temperature is 5°C and that the floor loses energy at the same rate as the walls and that there are no party walls and a host of other things, but none of them will change the result by an order of magnitude. The verdict is plain: time to get the insulation in.

There's one other consideration that had us talking at cross purposes for a while, and that's heat pump capacity. Say you start with a house that's 5°C indoors - the same as outdoors. If the heat pump can only output exactly as many watts as the house is losing, then the house will never get warm. In practice, you need to over-size the pump - and the larger it is relative to the house's heat loss, the faster it'll get to the desired temperature.

It's likely to be providing hot water too, and installers are paranoid about under-sizing, so they tend to choose much bigger units than you actually need. It's not a problem for running costs - once the house is up to temperature, it can just run at 25% or 50% of capacity - but it does mean that even if you calculate you only "need" a 4kW (heat output) pump, you're likely to be recommended a 12kW one anyway.

Questions? Comments? Criticisms? Contact the author by email: gemini@ur.gs