(226ao) Mathematical Model for Estimation of Heat Insulation Properties of Polymer Foams

Heat insulation of buildings is becoming increasingly popular in Europe and across the world. Rigid polymer foams are one of the most often used materials for this purpose. Due to their chemical composition and inner structure, they provide excellent resistance to convection, conduction and also heat radiation. However, a continuous effort is being made to further improve this resistance and to find cheaper ways to produce polymer foams. This contribution is focused on mapping of all important phenomena driving the heat transport in polymer foams and mathematical modeling of the morphology that would yield foam with the best possible insulation properties.

Currently, the most promising results are expected from foams with very small cell size, e.g., lower than 1 μm. In these foams, larger amount of phase interfaces should lead to more scattering and thus to the decrease of radiative heat flux. Moreover, when the cells are so small, their size becomes comparable to the mean free path of gas molecules. Therefore, the mechanism of heat conduction changes from diffusive to Knudsen mode, which results in the reduced gas conductivity. However, we show that the cell size must be chosen very carefully, because for polymers with low absorption coefficient the foam equivalent conductivity can significantly increase with the reduction of the cell size.

We have developed a computer program allowing to theoretically analyze and predict heat insulating properties of polymer foams. It simulates the coupled conduction-radiation heat transfer in computer reconstructed three-dimensional foams. The heat conduction is governed by Fourier law, in which the gas conductivity depends on the cell size because of the so-called Knudsen effect. We use the P₁-approximation to describe the heat radiation and derive the appropriate boundary conditions, which consider partial photon reflection on phase interfaces. Moreover, the reflectivity of phase interfaces accounts for wave interference phenomena in the thin polymer walls. Overall, our model doesn’t use any empirical correlations, the only input parameters are the foam microstructure and physical properties of pure phases.

By using this model, we can find optimal porosity, cell size, wall thickness and content of polymer in the struts. Additionally, we show that for certain types of foams, the equivalent conductivity depends on the total foam thickness and that the reduction of cell size could lead also to worse insulating properties. And finally, we propose ways, how to avoid these negative effects.