One-, two- and three-dimensional transient heat conduction

The current legislation of several countries, e.g. Czechia and Slovakia, requires the above assessments in the project documentation for the building permit and also in the energy certification of buildings.

The issue of 1-, 2-, and 3-dimensional steady-state heat conduction is described in detail in the publication ‘Wärmebrücken’ [5].

One-, two-and three-dimensional transient heat conduction

In contrast to steady-state heat conduction, the transient modelling is still highly optional. It mainly serves to improve the quality of projects in terms of validating proposed solutions, optimising them and, eventually, finding new, innovative and qualitatively better solutions. In certain circumstances, however, it can also serve as a tool for demonstrating compliance with the requirements of the relevant standards and thus also enter into (forensic) expert practice or expert activities. Not everything can be measured and, if so, these are often timeconsuming and costly processes. In such cases, computer-aided transient heat conduction calculations can be a suitable alternative.

The modelling of heat conduction in the non-steady (transient) state is based on Fourier's second law, which is described by Eq. (see [1]): =��(��2�� ����2 + ��2�� ����2 + ��2�� ����2)

where a is the thermal diffusivity factor in m2/s.

The above relationship is also called the second Fourier differential equation of heat conduction. It is a partial differential equation from the coordinates of space and time as independent variables and temperature as the dependent variable. It gives the dependence between the temporal change in temperature (left side of the equation) and the local change in temperature (right side of the equation). The constant of proportionality in this case is the thermal diffusivity factor or thermal diffusivity (please not to be confused with thermal conductivity coefficient). According to this equation, the temperature changes most rapidly over time in those bodies that have a higher value of thermal conductivity. The thermal diffusivity factor expresses the change in temperature at a particular location in a substance relative to the change in temperature at the surface. The higher the thermal diffusivity factor, the faster the temperature inside the substance changes relative to the temperature change at the surface. To describe the temperature field for three-dimensional heat conduction, then:

=��(��2��(��,��,��) ����2 + ��2��(��,��,��) ����2 +

��2��(��,��,��) ����2 ) For a steady-state temperature field, the left-hand side of the equation equals zero and has the following form:

(��2��(��,��,��) ����2 + ��2��(��,��,��) ����2 + ��2��(��,��,��) ����2 )=0