# U-value of a multi-layer build-up

With the basic principles that have already been presented, it can now also be shown how easily the U-value of a wall build-up consisting of several plane-parallel layers can be determined from the thicknesses and thermal conductivities.

At a constant (on the left side) higher temperature $T_h$ and a cold temperature $T_c$ on the far right, we will first wait for long enough until equilibrium has been achieved everywhere^{1)}. On account of the first law of thermodynamics, heat flow through each of the layers must then be the same everywhere. If it were not, then one or the other layer would absorb heat - then equilibrium would not be reached yet because the measured values in this layer would change. The constant heat flux density will again be indicated as $\dot{q}$. The temperatures at the layer boundaries on the cold (right) side of the layer are labelled $k$ with $T_k$. Let $T_0$:=$T_h$ and for the last layer let $T_n=T_c$ right at the end on the right. All temperatures are constant in dynamic equilibrium. For the k-te layer it appears as if it borders two temperature reservoirs with temperatures $T_{k-1}$ (warm side) and $T_k$ (cool side). The heat flux density flowing through it can therefore be determined according to [U2]:

$\dot{q} = U_k \cdot (T_{k-1}-T_k) \hspace{6cm} [U4]$

where $U_k$ is now the heat transfer coefficient of the k-te: layer. If this is divided by the value of $U_k$ (which is not zero), then we obtain

${\displaystyle T_{k-1}-T_k = \frac{\dot{q}}{U_k} = {R_k} \cdot \dot{q} \hspace{5cm} [U5]}$ ,

where we have introduced the heat transfer resistance $R_k$= $\frac{1}{U_K}$ in the last step. If we now add up all the temperature differences of the layers 1 to n, then all intermediate temperatures will stand out from this total and only the temperatures of the original reservoirs will be left

${\displaystyle T_{h}-T_c = R_1 \cdot \dot{q}+ R_2 \cdot \dot{q}+ ... + R_n \cdot \dot{q} \hspace{3cm} [U6]}$ ,

and we can now exclude/factor out the common factor ˙q from this total. In this way, what we end up with is that the total heat transfer resistance of a multi-layer building assembly becomes equal to the total of the individual overall heat transfer resistances of all layers:

${\displaystyle R = R_1 + R_2 + ... + R_n \hspace{5,8cm} [U7]}$ ,

Heat transfer resistances therefore are simply added up. From this we can also determine the resulting U-value $U$ of the overall build-up by calculating the reciprocal:

${\displaystyle U = \frac {1} {R_1 + R_2 + ... + R_n} \hspace{5,8cm} [U8]}$ ,

The individual R-values of the layers can be calculated based on [$λ1$] from the individual layer thicknesses $d_k$ and the material properties (especially thermal conductivities). This is also the calculation of the U-value of multi-layer building assemblies presented in the standards.

${\displaystyle \hspace{2cm} R_k= \frac{d_k}{\lambda_k} \hspace{6.5cm} [\lambda 1]}$

All this can be packed into e.g. a spreadsheet calculation - and the calculation process will then be fixed and new “variants” of a build-up can be quickly determined. In the PHPP there is such a worksheet: “U-values” which also has a couple of other features. However, “manual” calculation is also fast once the thicknesses have been measured and the specific values of materials have been looked up.

The consequence is that every U-value of an old building assembly, no matter how poor, can be improved in practice by adding another insulating layer. Even just a few centimetres will help a lot in case of poor values of old building assemblies (see e.g. our instructions relating to insulation of a radiator niche (German only). However, in most cases, because there is quite a lot of space on the outside of building assemblies in existing buildings, and insulation materials are among the most inexpensive and most easily processable building materials, it is worthwhile to consider the application of sufficiently thick insulation layers ^{2)} on the old building assembly. If executed correctly, this will reduce heat losses by several times^{3)}. In fact, the main costs for such measures are only the “set-up costs” - scaffolding and of course weather protection on the outside are usually necessary. However, because all this will be needed anyway at some time, even if only for painting the façade, what is important is to take advantage of such opportunities to get rid of cold and uncomfortable interior surfaces of walls and high heating costs.

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