Non-diffusive ignition of a gaseous reactive mixture following time-resolved, spatially distributed energy deposition

Abstract

Systematic asymptotic methods are applied to the compressible conservation and state equations for a reactive gas, including transport terms, to develop a rational thermomechanical formulation for the ignition of a chemical reaction following time-resolved, spatially distributed thermal energy addition from an external source into a finite volume of gas. A multi-parameter asymptotic analysis is developed for a wide range of energy deposition levels relative to the initial internal energy in the volume when the heating timescale is short compared to the characteristic acoustic timescale of the volume. Below a quantitatively defined threshold for energy addition, a nearly constant volume heating process occurs, with a small but finite internal gas expansion Mach number. Very little added thermal energy is converted to kinetic energy. The gas expelled from the boundary of the hot, high-pressure spot is the source of mechanical disturbances (acoustic and shock waves) that propagate away into the neighbouring unheated gas. When the energy addition reaches the threshold value, the heating process is fully compressible with a substantial internal gas expansion Mach number, the source of blast waves propagating into the unheated environmental gas. This case corresponds to an extremely large non-dimensional hot-spot temperature and pressure. If the former is sufficiently large, a high activation energy chemical reaction is initiated on the short heating timescale. This phenomenon is in contrast to that for more modest levels of energy addition, where a thermal explosion occurs only after the familiar extended ignition delay period for a classical high activation reaction. Transport effects, modulated by an asymptotically small Knudsen number, are shown to be negligible unless a local gradient in temperature, concentration or velocity is exceptionally large.

Keywords:

• thermomechanics
• hot-spot ignition
• asymptotic methods
• mechanical disturbances
• gasdynamics of explosions

Acknowledgements

The advice and experience of Professor William Sirignano are appreciated, as is the support of the AFOSR program manager, Mitat Birkan. Discussions with my late colleague and co-author, Professor J.F. Clarke, about thermomechanics in the context of reactive gas dynamics were inspirational.

Notes

1. A ‘non-instantaneous’ source is also considered.

2. The formulation lacks an explanation of how the hot spot is generated and what the pressure and velocity distributions might be.

3. The glossary in Appendix A provides additional information about the variables used in the analysis.

4. In contrast to the semi-infinite domain used in [6], the heated spot is finite, following the approach in [3].

5. A much larger reference power deposition is defined in Equation (28(28) ).

Additional information

Funding

This work was supported by a subcontract from the University of California, Irvine, with funds from an Air Force Office of Scientific Research (AFOSR) project ‘Reduced Basis and Stochastic Modeling of a Liquid Propellant Rocket Engine as a Complex System’.