Mathematical Modeling of Nanomaterials - COMSOL.com

Excerpt from the Proceedings of the COMSOL Multiphysics User's Conference 2005 Frankfurt
Figure 1. The intumescent nanocomposite gel for the realization of retardant windows.
1.1.2 Model description
Our model represents the phase change from a virgin material (the gel) to a porous material
(the final structure with bubble inclusions and after the carbonization of the organic
compounds) by a moving mushy front, mainly driven by outgassing. Here we do not consider
a (hard) Heavyside phase front, usually used in problems of the Stefan type, which is more
natural from a physical point of view. For this, a generalized form of the transient heat
equation with a drain term representing the absorption of heat by endothermic reactions is
employed.
An additional mass conservation equation is coupled to this heat equation using a
temperature dependent empirical function for the mass loss due to outgassing.
This mass conservation equation is also used to derive the change of the porosity
(normalized inverse of the density) of the material and the effective dynamic thermal
conductivity (according to the Maxwell model).
The dynamic porosity is related to the averaged bubble growth and growth limits are
introduced by taking the stability of the bubbles into account. The stability of bubbles is
strongly related to the insulating properties of the porous phase and the final glass matrix
after the mushy transformation front has passed. The equations are solved in a domain Ω
using the finite element method.
To the derive the unknown empirical parameters as well as to validate the model, a number
of experimental measurements were performed in a specially designed high temperature
oven using (mechanically) stiff gel plates with a varying proportion of a hardening agent,
which determines the growth and stability of bubbles, and different thicknesses of the plates.
Genetic programming was applied to fit the model to the experimental data as well as the
boundary conditions, in addition the general model conduction +convection + radiation was
used to determine the free boundary.
1.1.3 Results
Our model provided a high correlation to the oven measurements using 1D and 2D numerical
simulations. In particular, our model allowed for a representation of the temperature
gradients in space over time and exhibited a good qualitative correlation in the phase change
behavior, which was analyzed in a different set of experiments such as thermo-gravimetric
analysis. The influence of the hardening agent was also reflected in the numerical
simulations. In Fig. 2 (left) the outgassing function depending on the temperature is shown
for particular sets of parameters. The temperature profile at the backside (none heating side)
of a stiff gel plate with a thickness of 20mm for different outgassing settings which are related
to the proportion of hardening agent in the plates is shown Fig. 2 (right). The plateaus in
curve reflect the active insulation properties due to outgassing of the aqueous nanocomposite.