# Software for calculating thermal bridges

Thermal conduction through a three-dimensional building component is described by Fourier's theory (according to Jean Baptiste Joseph Fourier):

\begin{align} &\Large{\sum_{i=1}^{3} \dfrac{\partial}{\partial x_i} \left( \lambda_i \dfrac{\partial T}{\partial x_i}\right) + \dot q(x,t)=\rho c \dfrac{\partial T}{\partial t}}\\\\ With\qquad&\\ T\qquad&\text{temperature}\\ \lambda\qquad&\text{thermal conductivity of the materials}\\ c\qquad&\text{thermal capacity of the materials}\\ \rho\qquad&\text{thickness of the materials}\\ \dot{q}\qquad&\text{any heat sources and heat sinks} \end{align}

It is not possible to solve this partial differential equation analytically. However, for a time-independent case which is one-dimensional with constant boundary conditions and no heat sources/sinks, an already known analytical solution results with this:

$$ \varPhi= U \cdot A \cdot \Delta \theta $$

This equation is used in energy balancing for taking into account the heat losses from regular building components. For calculating linear thermal transmittances, that is, for two-dimensional heat flow issues, the Fourier equation must be calculated numerically using computer-assisted methods. Typical heat flow software programs which can be used for building engineering applications are based either on the **finite element method (FEM)** or on the **finite difference method (FDM)**. The advantages and disadvantages of these methods are discussed in detail in [AkkP 16]. However, for the user it is crucial that the used software meets the requirements in DIN EN ISO 10211, Appendix A and that it can calculate heat flows and surface temperatures with the necessary accuracy regardless of the method used. It should be noted that not every software program can perform calculations in a non-steady state, i.e. time-independent, manner. However, for most applications in the context of energy balancing, steady-state calculation is sufficient.The software programs available on the market are usually characterised by the following programme structure:

- preprocessor
- processing unit
- postprocessor

Der preprocessor is responsible for the graphic modelling and definition of the boundary conditions. The actual calculation using the FEM or FDM is undertaken by the processing unit, while the graphical analysis and depiction is performed by the postprocessor. For modelling of the relevant details, the structural design must be known. Furthermore, the characteristic values of the materials, particularly the thermal conductivities of the materials present must be known; the tabular values in the norms DIN EN ISO 10456 and DIN 4108-4 are referred to in actual practice. The normative provisions for modelling the geometry, boundary conditions for temperatures and heat transfer resistances can be found in the norm DIN EN ISO 10211. The norm DIN EN ISO 10077 is used for calculation of windows.

## References

**[AkkP 16]** Thermal bridge free design ; Protocol Volume No. 16 of the research Group for Cost-effective Passive Houses, 1st edition, Passive House Institute, Darmstadt 1999 Link to list of PHI publications