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.

graph

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.)