With the addition of bubbles to this model, the gases from polymer degradation are no longer expelled instantly upon generation. Instead, these gases add to the volume of "nearby" bubbles, which obey specified rules of migration, coalescence, and bursting.
In a first look at bubble effects, a set of bubble nucleation sites with zero initial volume is placed randomly within the sample, with an initial density of one per finite element. Gases generated within each element are distributed to bubbles contained within or overlapping the element according to the volume of the diffusion region for each as described in section 5.3. New bubbles are nucleated within an element only when it is empty of any other bubbles. When bubbles touch each other, they instantly coalesce, forming a single bubble located at the center of mass of the original bubbles. Decreasing bubble velocity upon approach to the outer surface of the sample is neglected. Bubbles burst as soon as they touch the outer surface, adding their gases to the mass lost during the current time step and disappearing from further calculations.
Figure 13: Sample radius vs. time for a spherical PMMA sample exposed to an external heat flux of 60 W/cm2 with bubbles (orange) and without bubbles (red).
Figure 14: Mass loss rate vs. time corresponding to Figure 13. Black line denotes mass loss rate averaged over 0.5 s.
Figure 15: Average mass loss rate vs. time comparing cases with (orange) and without (red) bubbles.
Figure 16: Temperature vs. time at the sample center (lower lines) and on the outer surface for cases with (orange) and without (red) bubbles.
Figure 13 shows the radius as a function of time for a spherical sample divided into 41 finite elements (thus initially containing 41 bubbles). Results for the base case without bubbles, as described in the previous section, are included for comparison. The sample swells with the growth of internal bubbles and contracts when bubbles burst, causing considerable fluctuation in the plot on a fine time scale. These fluctuations are amplified in the plot of instantaneous mass loss rate vs. time in Figure 14. To compare plots of mass loss rate under various conditions, a time average over a period of time long with respect to the rapid fluctuations but short with respect to long-term variations may be used, as shown in the same figure.
A comparison of the time-averaged plot of mass loss rate with mass loss rate when gases escape instantaneously, shown in Figure 15, illustrates that the presence of bubbles slows the pyrolysis process and decreases the average mass loss rate. This is due to the insulating effects of the retained gases, whose thermal conductivity is considerably less than that of the polymeric melt. The effect is demonstrated in Figure 16, which shows that the temperature at the center of the sample is lower throughout pyrolysis for the case including bubbles than for the case without.
A view of bubbles within the spherical sample at various times during pyrolysis is given in Figure 17. During preheating and early gasification all bubbles are small; in later stages of pyrolysis some bubbles may become quite large in comparison to the size of the sample before they burst. Over the much shorter time period covered by the sequence in Figure 18, individual bubbles may be observed as they grow to the bursting point. Migration may also be observed in this sequence, though the axis of migration for a particular bubble is not likely to be aligned with the selected view.
Figure 17: Two-dimensional view of PMMA sample containing bubbles at 3 s intervals beginning at time t = 0.
Figure 18: Two-dimensional view of PMMA sample containing bubbles at 0.02 s intervals beginning at time t = 18 s.
If an element contains a portion of more than one bubble, the gases generated within that element during the next time step must be distributed among these bubbles. The model allows for distribution linearly by bubble radius, by bubble surface area, or by bubble volume. Gases may alternatively be distributed evenly among bubbles or by diffusion zones of the same thickness for all bubble portions within the element. Figures 19 and 20 show that the results are insensitive to how gas is distributed among bubbles, at least for this set of model assumptions. This is likely to be due in part to the low number of bubbles populating the sample during most of the pyrolysis process.
Figure 19: Sample radius vs. time for PMMA showing a lack of sensitivity to gas distribution.
Figure 20: Same as Figure 19 magnified at the end of pyrolysis. In order of increasing final times, gases are distributed to bubbles evenly, by bubble radius, by diffusion zone volume, by surface area, and by bubble volume.