Estimating ignition time of solid exposed to increasing-steady thermal radiation

Abstract

This contribution presents an alternative methodology to predict ignition of solid under increasing-steady heat flux, which may be irradiated from the smoke layer with time-dependent temperature. An approximate analytical solution, using ignition temperature criterion, is proposed and two typical heating conditions, linear- and t²-steady heat fluxes, are emphasized. The critical increasing rate of heat flux and the critical transition time, guaranteeing non-ignition in the first stage, are quantitatively assessed. Two explicit ignition time correlations are derived for low and high heat fluxes in the second stage, and the thresholds separating the application regimes of the two correlations are also provided. PMMA (polymethyl methacrylate) and a previously developed numerical model are employed to validate the reliability of the analytical model and the accuracies of the approximations used. The results show that under the designed four sets of increasing-steady heat fluxes, the analytically predicted surface temperatures and ignition times match the simulation results well. Using the calculated ignition time, the linear dependency of ignition time on the squared critical energy is also found valid in current study. Meanwhile, the effect of critical temperature on ignition time predictions is quantitatively examined by parametric study.

Appendices

Appendix 1

Solution for $\theta_{1}$ in Eq. (12):

Define Laplace transformation:

$$ L\left[ {\theta_{1} \left( {x,\tau } \right)} \right] = \Theta_{1} \left( {x,p} \right) $$

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where $p$ is the frequency parameter in Laplace transformation. Then, the heat transfer problem related with $\theta_{1}$ can be expressed as:

$$ \left\{ \begin{gathered} p\Theta_{1} = \alpha \Theta_{{1{\text{xx}}}} \hfill \\ \left. { - k\Theta_{{1{\text{x}}}} } \right|_{{{\text{x}} = 0}} = \frac{{\dot{q}^{\prime\prime}_{{\text{c}}} }}{p} \hfill \\ \left. {\Theta_{{1{\text{x}}}} } \right|_{{{\text{x}} = \infty }} = 0 \hfill \\ \end{gathered} \right. \, $$

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Solving this ordinary differential equation, $\Theta_{1}$ can be derived as:

$$ \Theta_{1} = \frac{{\dot{q}^{\prime\prime}_{{\text{c}}} }}{k}\frac{{e^{ - {\text{qx}}} }}{pq} $$

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where $q = p/\alpha$. With inverse Laplace transformation, $\theta_{1}$ can be obtained as [51]:

$$ \theta_{1} \left( {x,\tau } \right) = \, L^{ - 1} \left[ {\Theta_{1} \left( {x,p} \right)} \right] = L^{ - 1} \left[ {\frac{{\dot{q}^{\prime\prime}_{{\text{c}}} }}{k}\frac{{e^{ - {\text{qx}}} }}{pq}} \right] = \frac{{2\sqrt {\alpha \tau } \dot{q}^{\prime\prime}_{{\text{c}}} }}{k}\left[ {\frac{1}{\sqrt \pi }e^{{ - \frac{{x^{2} }}{4\alpha \tau }}} - \frac{x}{{2\sqrt {\alpha \tau } }}{\text{erfc}}\left( {\frac{x}{{2\sqrt {\alpha \tau } }}} \right)} \right] = \frac{{2\dot{q}^{\prime\prime}_{{\text{c}}} \sqrt \tau }}{{\sqrt {k\rho C_{{\text{P}}} } }}{\text{ierfc}}\left( {\frac{x}{{2\sqrt {\alpha \tau } }}} \right) $$

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Appendix 2

Solution for $\theta_{2}$ in Eq. (12):

The heat transfer problem related with $\theta_{2}$ can be expressed as:

$$ \left\{ \begin{gathered} p\Theta_{2} - A_{1} e^{{ - B_{1} x}} = \alpha \Theta_{{2{\text{xx}}}} \hfill \\ \left. { - k\Theta_{{2{\text{x}}}} } \right|_{{{\text{x}} = 0}} = 0 \hfill \\ \left. {\Theta_{{2{\text{x}}}} } \right|_{{{\text{x}} = \infty }} = 0 \hfill \\ \end{gathered} \right. \, $$

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Solving this ordinary differential equation, $\Theta_{2}$ can be derived as:

$$ \Theta_{2} = \frac{{A_{1} }}{{p - \alpha B_{1}^{2} }}e^{{ - B_{1} {\text{x}}}} - \frac{{A_{1} B_{1} }}{{q\left( {p - \alpha B_{1}^{2} } \right)}}e^{ - {\text{qx}}} $$

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With inverse Laplace transformation, $\theta_{2}$ can be obtained as [51]:

$$ \begin{gathered} \theta_{2} \left( {x,\tau } \right) = L^{ - 1} \left[ {\Theta_{2} \left( {x,p} \right)} \right] = L^{ - 1} \left[ {\frac{{A_{1} }}{{p - \alpha B_{1}^{2} }}e^{{ - B_{1} x}} - \frac{{A_{1} B_{1} }}{{q\left( {p - \alpha B_{1}^{2} } \right)}}e^{ - {\text{qx}}} } \right] = L^{ - 1} \left[ {\frac{{A_{1} }}{{p - \alpha B_{1}^{2} }}e^{{ - B_{1} x}} } \right] - L^{ - 1} \left[ {\frac{{A_{1} B_{1} }}{{q\left( {p - \alpha B_{1}^{2} } \right)}}e^{ - {\text{qx}}} } \right] \hfill \\ = A_{1} e^{{ - B_{1} x + \alpha B_{1}^{2} \tau }} - \frac{{A_{1} }}{2}e^{{\alpha B_{1}^{2} \tau }} \left[ {e^{{ - B_{1} x}} {\text{erfc}}\left( {\frac{x}{{2\sqrt {\alpha \tau } }} - B_{1} \sqrt {\alpha \tau } } \right) - e^{{B_{1} x}} {\text{erfc}}\left( {\frac{x}{{2\sqrt {\alpha \tau } }} + B_{1} \sqrt {\alpha \tau } } \right)} \right] \hfill \\ \end{gathered} $$

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