# hpfem/esco2012-boa

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 \title{Towards New Polymer Foam Insulation Material by the Use of Mathematical Modeling} \tocauthor{R. Pokorny} \author{} \institute{} \maketitle \begin{center} {\large \underline{Richard Pokorny}}\\ ICT Prague, Dept. of Chemical Engineering, Technicka 5, 166 28, Prague 6, Czech Republic\\ {\tt Richard.Pokorny@vscht.cz} \\ \vspace{4mm}{\large Pavel Ferkl}\\ ICT Prague, Dept. of Chemical Engineering, Technicka 5, 166 28, Prague 6, Czech Republic\\ {\tt ferklp@vscht.cz} \\ \vspace{4mm}{\large Kosek Juraj}\\ ICT Prague, Dept. of Chemical Engineering, Technicka 5, 166 28, Prague 6, Czech Republic\\ {\tt Juraj.Kosek@vscht.cz} \end{center} \section*{Abstract} Some modern micro- and nano-structured materials (e.g., polymer nano-foams) represent an industrially important class of materials in today's technology, as they are widely used as heat insulation materials due to their low thermal conductivity and light weight. This contribution is thus concerned with two main topics: (i) the development of the algorithm which is able to solve the coupled conduction-radiation heat transfer problem on very fine discretization grid, and (ii) the study and modeling of various heat transfer phenomena which have to be considered at nano-scale. The algorithm of Voronoi tessellation was used for the digital reconstruction of the spatially two and three dimensional media with the morphology of the polymer foam. Finite difference (FDM), finite volume (FVM) and finite element method (FEM) were used for the processing of heat transfer equations. We also report on the experience with the new algebraic multigrid method, which is used for the solution of large systems of algebraic equations. This multigrid method, based on the aggregation of unknowns, allowed us to calculate the heat conduction on cubic domains containing up to $500 \times 500 \times 500$ voxels (finite volumes) in 3D and the coupled conduction-radiation problem on $2000 \times 2000$ pixels (in 2D). The concepts of both the gray medium and the spectrally-dependent medium are employed in the modeling of heat radiation. Intuitively, we expect that the effective thermal conductivity of foam decreases with increasing porosity, because polymers have about ten times higher thermal conductivity than gases. However, the higher porosity also means less polymer phase, which acts as a barrier for the thermal radiation. The combined effect of conduction-radiation thus results in the minimum effective thermal conductivity observed on the thermal conductivity versus porosity curve. In common heat insulations about 30\% of heat is transferred by radiation. \bibliographystyle{plain} \begin{thebibliography}{10} \bibitem{Notay} {\sc Notay Y}. {An aggregation-based algebraic multigrid method}. Tech. Rep. GANMN 08-02, ULB, Belgium, 2008. \bibitem{Wang} {\sc Wang M. and Pan N}. {Prediction of effective physical properties of complex multiphase materials}. Mat. Sci. Eng. R., 63:1 (2008). doi: 10.1016/j.mser.2008.07.001. \end{thebibliography}