6.4 Nucleation Effects

As discussed in section 5.7, the rate of bubble nucleation per unit volume is assumed to take the form of the Arrhenius expression

(71)

The pre-exponential factor B_N determines the magnitude of the nucleation rate, while the activation energy E_N determines how rapidly the rate increases with temperature. To study what effect nucleation rate has on the behavior of a sample undergoing pyrolysis, pre-exponential factor B_N is varied while activation energy E_N is held at a fixed value. With a temperature of 730 K at the surface during the quasi-steady period of pyrolysis in the absence of bubbles (see Figure 16), the value of E_N is arbitrarily set such that the nucleation rate at 600 K drops to roughly 10 % of the rate at the surface. This study fixes the thin-film drainage time before bubbles are allowed to burst at 20 ms.

Surprisingly, the nucleation rate has little effect on the rate at which the polymeric sphere pyrolyzes, as shown in Figure 29. The 20 ms drainage case from the previous chapter, for which nucleation was limited to ensuring at least one bubble per element, is included for comparison, plotted in green as previously. The increase in sample radius, or swelling, is slightly larger for the highest nucleation rate, and the time of peak swelling corresponds to that of the maximum number of bubbles shown in Figure 30. Although early in the pyrolysis process the total number of bubbles present in the sample at a given time reflects the nucleation rate, by 40 s the number drops to values fluctuating between zero and the number of elements, the same range traversed in the absence of the Arrhenius nucleation model.

Figure 29: Sample radius vs. time for bubbling PMMA sphere with nucleation defined by B_N = 1 x 10¹¹ (s-cm³)⁻¹ (light blue), 2 x 10¹¹ (s-cm³)⁻¹ (navy), and 4 x 10¹¹ (s-cm3)−1 (purple) with E_N/R = 7500 K. Case without an Arrhenius nucleation model is shown in green. Drainage time for bursting bubbles is 20 ms.

Figure 30: Number of bubbles in sample vs. time corresponding to Figure 29.

Figure 31 shows a time sequence from 0 s to 76 s for bubble nucleation obeying an Arrhenius expression with B_N = 2 x 10¹¹ (s-cm³)⁻¹. At t = 8 s (the third frame), when the number of bubbles is at its peak of about 250, the bulk of the bubbles is well within the sample, with relatively few bubbles located at the surface and preparing to burst. Although the nucleation scheme requires the density of nucleation sites to increase with temperature and therefore with radial location within the sample, bubbles near the surface quickly travel through the melt of low viscosity to the surface and burst after 20 ms. According to the model, then, bubbles do not accumulate at the surface. This does not agree with experimental observations of combustion in microgravity in which a layer of growing, relatively monodisperse bubbles covers the surface of the spherical sample during the first few seconds of combustion. The fixed value of drainage time in the model clearly underestimates the time before bursting during this early period.

Figure 31: Two-dimensional view of bubbling PMMA sample with Arrhenius-based nucleation scheme with B_N = 2 x 10¹¹ (s-cm³)⁻¹ and E_N/R = 7500 K. Drainage time is 20 ms and time interval between frames is 4 s.