# ECO: why it matters where you start from

Over the summer Ofgem issued a ECO2 Cavity Wall Checklist which sets out the new rules of evidence that must be observed when wanting to overwrite the default RdSAP U-value for a wall with a higher U-value. In the consultation document they justify the new evidence requirements on the grounds that

we are concerned that wall U-values for cavity wall insulation measures (CWI) are being overwritten to values that we consider to be unreasonably high for the premises in question, and as a result the calculated savings for a measure are artificially inflated. This could lead to fewer households benefitting under ECO.

So at the heart of this is the claim that higher starting U-values will result in more savings (which may well be unjustified). But, surely if we are assessing the impact of insulation the starting U-value doesn’t matter? Does it?

Strangely, that is the one piece of the explanation which is missing from the consultation and from the final checklist; because while it is true that a given thickness of a given insulation will always have the same thermal resistance, it is not true that the benefit of that insulation will be the same.

Let’s unpack that a bit.

At its simplest, the thermal performance of a wall with regard to heat transfer, can be expressed as a thermal resistance. If, for a moment we ignore thermal bridging, we can calculate the thermal resistance of a wall by adding up the resistances of the layers of which is it composed. A cavity wall made up of an outer leaf of brick and an inner leaf of 100 mm dense aggregate block work with 20 mm of plaster to the inside might have a total resistance of 0.617 m²K/W. A similar wall with an inner leaf of 100 mm aircrete blockwork might have a resistance of 1.090 m²K/W. The wall with the aircrete inner leaf has a higher resistance than the wall with the dense blockwork inner leaf.

That’s fine as far as it goes, but it doesn’t allow us to quantify the important aspect of performance, the rate of heat transfer. For that we need the U-value, which is the reciprocal of the thermal resistance and expresses the rate of heat transfer in watts per metre squared kelvin (W/m²K). As an equation:

$U = \frac{1}{R}$

So our two walls have U-values of 1.620 W/m²K and 0.918 W/m²K respectively.

What happens if we add insulation to each of those walls, say 120 mm of insulation with a thermal conductivity of 0.030 W/mK? We can evaluate that by going back to the thermal resistance in each case. The insulation has a resistance of 4.0 m²K/W, so the wall resistances are now 4.617 m²K/W and 5.090 m²K/W, giving U-values of 0.217 W/m²K and 0.196 W/m²K. But it’s not just the final number we’re interested in, it’s the change. The wall with the dense block inner leaf has gone from 1.620 to 0.217, a drop of 1.403, the other wall went from 0.918 to 0.196, a drop of 0.722. So, although the final U-value is lower in the aircrete wall, the reduction in U-value is much greater in the dense block wall.

You can see that visually on the graph in this post, where the same thickness of insulation added to a high U-value gives a bigger drop than the same amount added to a low U-value. (There is a more detailed discussion of U-values in my book How Buildings Work.)

As ECO is all about savings, a higher starting U-value is better. I took a standard house type I use in SAP training and experimented with the effect on emissions of using the two starting U-values I calculated earlier. The annual carbon dioxide savings are 1,080 kg/CO₂ for insulating the dense block wall and 669 kg/CO₂ for insulating the aircrete block wall. The insulation is the same, but the difference of 411 kg/CO₂ is entirely down to that higher starting U-value.

# R-values or U-values?

NBS has recently published an article by Anthony Lymath explaining the terminology around U-values: it is a useful summary and worth a read. (There are some points which I would frame differently, but I’ll address those at the end of this article). As I was going through the piece I found myself reflecting once again on the relationship between U-values and R-values and the respective benefits of each method of representing the thermal performance of a building element.

At first sight, the R-value, which is the total thermal resistance of an element, is straightforward: all you need to do to calculate an R-value is add up the resistance of the layers in the construction. That also makes it easy to establish how much insulation you might need to reach the performance level required by national or local building regulations or codes, particularly when insulation is marketed by its own R-value. If you need to hit an R-value of 3.5 m²K/W and you know that you currently have a resistance of 1, then insulation batts at R2.5 will do nicely.

The R-value is also straightforward mathematically: the conductivity and thickness give you the resistance of a layer, so if you double the thickness of those R2.5 insulation batts they go from R2.5 to R5.

But achieving that simplicity has costs. First, the use of basic R-values means we lose sight of thermal bridging. Those R2.5 insulation batts are likely to be fitted between between timber studs, resulting in a 20% overestimate of performance (based on the 15% bridging fraction required by BR 443 Conventions for U-value calculations).

Secondly, in terms of analysis, the R-value is a dead end. Because it is an expression of resistance to heat transfer, rather than a measure of heat transfer itself, we can’t go any further with it. In contrast, we can use the U-value of an element in the thermal analysis of a building: if we know the U-values and areas of the building elements, and the temperature difference between inside and outside we can make a reasonable estimate of the rate of heat transfer. That, combined with data on thermal mass, solar and internal gains, enables us to model energy demand (something I cover in more detail in my forthcoming book How Buildings Work). The R-value can’t do that.

So the U-value lets us carry out the more detailed analysis we need to design energy efficient buildings. But the downside with the U-value is that it is not as straightforward to understand as the R-value. Mathematically, the U-value is the reciprocal of the R-value, which means that increasing the amount of insulation in an element will not give a nice straight line reduction in the U-value.

If we plot the R-values and U-values of a construction with increasing amounts of thermal insulation we see the R-value increaes in a straight line with a constant gradient. However, plotting the U-values of the same situation gives a curve, with rapid reductions of the U-value to begin with (say, increasing 50 mm of insulation to 100 mm), but less change once U-values are lower (say, increasing 250 mm of insulation to 300 mm). The graph goes a long way to explaining why we will probably never want to get U-values much below 0.10 W/m²K.

Personally, I think the R-value’s time is up: the benefits of simplicity are outweighed by the inability to go any further in analysing the performance of a whole building.

# Terminology

As I mentioned at the start, there are a few points which I would put differently:

1. Lambda value vs k-value. The article refers to a material’s thermal conductivity as its k-value. There was a time when that was the case, but now BR 443, calculation standards and Agrément certificates all refer to conductivity as lambda (λ). By sticking with lambda we also avoid the confusion with the kappa-value (κ-value), which is a measure of thermal mass.
2. Cold bridging vs thermal bridging. A repeating thermal bridge (to use the full term) occurs when a layer of one material is regularly interrupted by another material having a different thermal conductivity. In many cases the interrupting material has a higher conductivity, resulting in increased heat loss: hence the term cold bridge. However, there are cases where the interrupting material has a lower conductivity, resulting in reduced heat loss. The main occurrence is in stone walls, where the mortar will usually have a lower conductivity than the stone; the greater the proportion of mortar, the lower the U-value and the lower the rate of heat loss. Given that, thermal bridge is the preferred term.
3. Wall-ties as cold bridging. The effect of wall ties and other mechanical fasteners is not treated in the same way as thermal bridging. Instead we use separate correction factors which are added to the U-value at the end of the calculation process. (To be compeletely accurate: correction factors are ignored if they are less than 3% of the initial U-value.)