Modeling Particle Cooling and Bubble Growth

Upon exposure to the atmosphere following the hydrogen explosion, CsMP experienced rapid cooling. It was assumed convective heat loss occurred only at the surface, with conductive cooling due to the heat gradient formed between the centre and surface^{36 (link)} dominating within the particle bulk. To further simplify the model, radiative heat loss at the particle surface was taken to be negligible. The radial and temporal heat profile of the particle due to just conductive heat loss was then modelled by solving the one-dimensional, spherically symmetric heat equation $$\begin{array}{c}\hfill {\rho }_p{c}_p\frac{\partial T}{\partial t}=\frac{k_p}{r^2}\frac{\partial }{\partial r}\left(r^2,\frac{\partial T}{\partial r}\right),\end{array}$$ where ${\rho }_p$ is density of the particle, $c_p$ the heat capacity of the particle, $k_p$ the thermal conductivity and r the radial coordinate. The physical particle properties (e.g. density, heat capacity, thermal conductivity) were assumed to be constant throughout the model. The convective heat loss at the particle surface was accounted for by imposing the boundary condition $$\begin{array}{c}\hfill k_p{\left(\frac{\partial T}{\partial r}\right)}_{r=R}=-q_c,\end{array}$$ where $$\begin{array}{c}\hfill q_c=h(T_{\infty }-T_s)\end{array}$$ is the convective heat flux. h is the heat transfer coefficient, $T_s$ is the surface temperature of the particle and $T_{\infty }$ the temperature of the surroundings^{23 (link)}. The heat transfer coefficient was calculated using the equation $$\begin{array}{c}\hfill h=\frac{\text{Nu}{k}_g}{2r_p},\end{array}$$ where Nu is the Nusselt number, $k_g$ the thermal conductivity of the air surrounding the particle and $r_p$ the particle radius. Calculation of the Nusselt number required the particle Reynolds number, Re, determined by $$\begin{array}{c}\hfill \text{Re}=\frac{2|(v_g-v_p)|r_p{\rho }_g}{{\eta }_g},\end{array}$$ where $v_g$ is the velocity of the surrounding air, $v_p$ the particle velocity, ${\rho }_g$ the air density and ${\eta }_g$ the air viscosity. In previous studies^{26 (link)}, the Nusselt number was calculated for a negligible internal temperature gradient by invoking the lumped capacitance approximation. This states that for Biot numbers close to zero (Bi $\to 0$ ) , the convective heat transfer to the surrounding gas limits the surface heat flux, and internal conduction is large enough to equilibrate the internal temperature gradient of the pyroclast. This contradicts our key hypothesis that radial variations in viscosity caused the unique internal texture observed in ‘Type B’ CsMP. For this reason the Nusselt number used in this study was calculated using newer data from Moitra et al.^{37 (link)}, that is not dependent on the lumped capacitance approximation: $$\begin{array}{c}\hfill \text{Nu}=a+b{\text{Re}}^{1/2}{\text{Pr}}^{1/3}\end{array}$$ with fitting parameters $a=76$ and $b=1.9$ , and Pr is the Prandtl number, taken to be 0.71 for ambient air^{37 (link)}.Figure 5

Schematic outlining the physical processes at each scale of the model. Particle-scale: the cooling of the isotropic, spherical particle from convective heat transfer to the surrounding environment and conductive cooling within the particle. Bubble-scale: The bubble growth model in which growth is limited by viscosity and halted once the temperature is lower then the glass transition temperature.