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basics:building_physics_-_basics:thermal_transmittance

# Thermal transmittance or U-value

As we already know, heat (fast-moving molecules) flows by itself from a system with a higher temperature towards a system with a lower temperature. We will now take a look at this heat flow in more detail and shall therefore assume the following: two systems have temperatures that are assumed to be temporally and spatially constant in each case 1) and which are separated from one another by a plane component. For now we will also assume that this surface is homogeneous and isotropic 2). The component's surface on the left borders the temperature reservoir with a higher temperature $T_h$ ('hot'), and its surface on the right borders the temperature reservoir with the lower temperature $T_c$ ('cool'); both surfaces are of an equal size, we will quantify this with the area $A$. Apart from this, the thickness of this component is measured and indicated with $d$. What exactly is between the two surfaces does not play any role in our subsequent considerations, what does matter is that the construction is rigid and durable 3). We will now experiment with different “fillings” for this component, whereby we will measure the heat flow $\overset{.}{Q}$ between the left and the right sides; this is the output and will therefore be specified using the measuring unit “watt” 4); we have already measured the temperatures, which will be kept constant in the process.

## The heat flow is proportional to the area of thermal contact

First of all, we can very quickly see that in all of the experiments the measured heat flow increases proportionally to the area of thermal contact of the component 5). The heat flow per unit area ˙q is called the heat flux density. Based on our conditions, this is the same everywhere. Thus:

$\hspace{2cm} \dot{Q}=A \cdot \dot{q} \hspace{6cm} \llap{[U1]}$

A simple and clear consequence of this is that if the exterior surface of a building design is increased (e.g. by splitting it up into many protrusions and recesses), then the heat loss will increase in relation to the increased area 6).

### The heat flow is (usually quite) proportional to the difference in the temperatures

For most assemblies of our separating component it turns out that the heat flow and thus also the heat flux density is proportional to the temperature difference between the hot and the cold system. These experiments show that usually the remaining proportionality factor also depends very little on other influences, such as moisture or the temperature level. However, this applies only within 'reasonable' limits, usually quite well especially in the building sector for many constructions with various materials. The 'largely constant' proportionality factor is called the thermal transmittance or U-value of the component. This is a measure of the heat transmittance of a component for each unit of area and temperature, so the measuring unit W/(m²K) is used.

$\hspace{2cm} \dot{q}= U \cdot ( T_h - T_c ) \hspace{6.7cm} \llap{[U2]}$

or with [U1]

$\hspace{2cm} \dot{Q}=A \cdot U \cdot ( T_h - T_c ) \hspace{6cm} \llap{[U3]}$

A simple and clear consequence of this is that if it becomes colder on the outside, then the heat loss at the same internal temperature increases in relation to the temperature difference. Obviously, that is why we have heating in winter: more when it becomes colder, and more if the area $A$ becomes larger. With [U3] we also already have one of the crucial formulas for calculating the heat loss of a building. It is also clear that if there are various such components, then the heat flows can be individually calculated according to this method and then simply added together. Thus here we already have the first 7) part of the heating energy balance. Now one must also 'reliably' determine the U-values of the components and make them available. Indeed there are lots of predefined values, measured and calculated. For example, we can find these in the certificates of certified Passive House products: components. And for the purpose of illustration, here is a table with “typical” U-values.

ComponentTypical U-value in [W/(m²K)] Remarks
Very old single-glazing (4 mm glass pane) 5.7 Window frost can form on the inner surface on cold days! It is uncomfortably cold and drives up the heating costs.
Thin concrete wall (8 cm normal concrete) 4.8 Strictly speaking, this was never permitted, but can still be found in some places despite this. Of course it leads to extremely cold interior surfaces, condensation and unaffordable heating costs!
'Thick' concrete wall (24 cm normal concrete) 3.5 Making the concrete thicker doesn't bring much. This was also never permitted 8)).
Old exterior wall (24 cm vertically perforated bricks of the 1960s) 1.4 For a long time, this was how we built mostly!
Exterior wall of the 1980s (30 cm lightweight perforated bricks)0.8 This is a quite moderate level of thermal protection, so subsequent improvement is generally worthwhile.
Exterior walls from around 2002 onwards (“EnEV” 36 cm lightweight masonry or also insulated timber constructions)0.45This isn't really good thermal protection, but such buildings are not yet due for modernisation - and with their mediocre level of insulation, their users will still have to continue paying for a few more decades 9).
New build exterior walls still commonly used today 0.3 Also this is not a sufficiently good level of thermal protection. We recommend doing it much better than this!

Typical exterior wall after EnerPHit retrofit (year 2020) 0.15 This is the maximum U-value we recommend for a building in Central Europe. It is also the economically optimum level of thermal insulation. Any higher value will only help finance the energy profiteers particularly in countries with petroleum and natural gas. Ausgeflockte Foamed?? 24 cm timber construction exterior wall with compound insulation system (phenolic rigid foam 12 cm) 0.09 This is barely 38 cm thick in total and can be created at reasonable cost - but usually it isn't at all needed in this quality. In this way we could easily build new constructions as nearly zero energy buildings today. Insulated 17 cm timber construction board with 6 cm vacuum insulation (prefabricated) 0.076 With 23 cm in total, this is thinner than the older “thin walls”. It is worthwhile even if the building plot is very expensive 10).

1)
at least in the time period and spatial area considered by us
2)
i.e. the same component characteristics and the same thickness; all these initial assumptions later turn out to be easily generalisable; however, in this way the fundamental questions can initially be identified, described and resolved much more clearly.
3)
i.e. it could be a homogeneous block of metal, but can also be two rigid shells on the left and right with the space in between filled with a fluid.
4)
How? Here is the simplest basic case first: if our system on the left does not lose any heat (it's not easy, but can be approximately achieved with some care), then the heat supply to the system on the left that is required to maintain its temperature will be exactly equal to $\overset{.}{Q}$ due to the law of energy conservation (first law of thermodynamics!) If there are other losses as well, then we can (similarly) gauge these in advance and deduct them from the other measured values after thermal contact via the component. What is especially convenient: if the heat is created via a resistance $R$ through which current $I$ flows, then the generated heat can be very accurately and easily determined in accordance with $P_{el}=V \cdot I = I \cdot R \cdot I = I^2 \cdot R$ ($V$ is the voltage here). Here we have “built” a very simple calorimeter.
5)
This is also easy to see based on our model concept with the vibrational energy of molecules being passed on
6)
As a consequence: construction becomes more costly; heating costs rise; the useable floor area becomes smaller. Isn't everyone complaining about increased living costs? Much of this is obviously planned and intended - not that it's 'forbidden', but we should be aware of this.
7)
and also decisive
8)
In contrast, just 2 cm insulation material on one side would bring down the U-value to around 1.1 W/(m²K). This is the intention behind the idea of “permanent formwork” in concrete constructions consisting of thermally insulating materials. In order to ensure that such formwork elements are stable, they should already be 4.5 cm thick - and as formwork they should naturally be on both sides, which would give us at least 9 cm and a U-value of less than 0.35 W/(m²K
9)
Note: during the legislation process at the time, we repeatedly pointed out in our expert reports and in hearings that this mediocre standard fixes regulations at high energy consumption values in the long run. This is called the “lock-in effect” in English.
10)
This is the thinnest wall with paradoxically phenomenal thermal protection at the same time. “No space” is not an argument for poor insulation!