Ignition of a Thermally Thick Solid

The thermally thin case holds for d of about 1 mm. Let us examine when we might approximate the ignition of a solid by a semi-infinite medium. In other words, the backface boundary condition has a negligible effect on the solution. This case is termed thermally thick. To obtain an estimate of values of d that hold for this case we would want the ignition to occur before the thermal penetration depth, 5T reaches x d. Let us estimate this by 

From experiments on common fuels in fire, the times for ignition usually range well below 5 minutes (300 s). Taking a representative a = 2 x 10-5 m2/s, we calculate 

we might expect solids to behave as thermally thick during ignition and to be about 8 cm for t-lg = 300 s and about 2.5 cm for 30 s. Therefore, a semi-infinite solution might have practical utility, and reduce the need for more tedious finite-thickness solutions. However, where thickness and other geometric effects are important, such solutions must be addressed for more accuracy. 

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Let us now turn to the case of a thermally thick solid. Of course thickness effects can be important, but only after the thermal penetration depth due to the flame heating reaches the back face, i.e. t = ff, 6j(t ) = d. As in the ignition case, if tf is relatively small, say 10 to even 100 s, the thermally thick approximation could even apply to solids of d < 1 cm. Again, we represent all of the processes by a thermal approximation involving the effective properties of Tlg, k, p and c. Materials are considered homogeneous and any measurements of their properties should be done under consistent conditions of their use. Other assumptions for this derivation are listed below ... 

Because it is very tedious to measure Tig and kpc directly, it is much more common to determine ignition properties on the basis of an analysis of time-to-ignition data obtained over a range of heat fluxes. The analysis is usually based on a simple heat conduction model, which assumes that the solid is inert (negligible pyrolysis prior to ignition) and thermally thick (heat wave does not reach the back surface prior to ignition). An example of this type of analysis is discussed in Section 14.3.2.3.2. 

For common thermally thick combustible materials (greater than a few millimeters) the time to ignition is proportional to the product k p c (where k is the thermal conductivity, p is the density, and c is the heat capacity), which represents the thermal inertia of the sample. Thermal inertia characterizes the rate of surface temperature rise of the material when exposed to heat. Low values of thermal inertia lead to a rapid temperature rise for a given applied heat flux and hence, to a rapid ignition.4 Polymeric foams have much lower thermal conductivity and density than the corresponding solid materials, thus the surface temperature of the first heats up more rapidly than that of the latter. Foam surface may reach the ignition temperature 10 times faster than the solid polymer.5... 

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