4.1.4 Effective Thermal Conductivity

The effective thermal conductivity of a material consisting of multiple components depends on the geometrical configuration of the components as well as on the thermal conductivity of each. In the literature on thermal decomposition of materials, many authors [26], [27], [29] choose to represent thermal conductivity by a volume-weighted average, or parallel arrangement, of the properties of the N_c components,

(11)

This expression describes an upper limit of k* in which the components connect the heat source directly through the material. This is a good description in cases in which the component with highest conductivity is well-connected throughout. A series description of effective thermal conductivity,

(12)

has been used to model intumescent coatings as a thermal resistance network [12]. In this limit, the components are treated as layers running parallel to the surface, providing an insulating effect that is dominated by the component with lowest conductivity.

In the case considered here, low thermal conductivity bubbles are completely surrounded by the polymer melt, which has much higher thermal conductivity. The parallel arrangement could therefore be argued to be a reasonable approximation. However, observations of the burning spheres show that a bubbly layer, with bubbles tightly packed, forms on the sample surface within the first second or two following ignition. The resulting thin polymer films will not conduct heat as well as wide regions of melt. For reasonable treatment of both conditions, the geometric mean approximation [30]

is chosen in this work to estimate an effective thermal conductivity for each element. This combination of gas and polymer thermal conductivities predicts a value between those given by the parallel and series limits.