The Mathematical Modelling of Unconfined and Confined Combustion of Explosives

Lead Research Organisation: University College London
Department Name: Mathematics


This doctoral project is based on a sequence of interrelated mathematical model problems that are concerned with increasing the understanding and quantifying of sensitive complex combustion properties and their associated implications in regard to human safety. The mathematical modelling is the key element prior to a combination of asymptotic analysis and numerical computation being applied. The first such problem is in terms of a finite-sized container specifically a long rectangle within which there is a comparatively small amount of solid material and the remainder is filled with an ideal gas. A discrete account of the interaction here will be formulated and solved appropriately. This should lead on to a continuum differential account which has wider application and then on to much more in terms of increasingly realistic modelling. Further details are as follows.
High explosives provide a low mass source of massive energy release, but this stored energy can pose a major hazard and even cause disaster if released accidently. Thus safe handling and storage is a constant. Understanding the circumstances in which an explosive can ignite, burn and detonate is essential if we are to predict the severity of likely hazards and understand the associated risks. When a high explosive is subject to significant heating as a result of either mechanical dissipation caused by accidental severe deformation or direct heating from a heat source it begins to react. The solid material reacts, i.e. burns to form high pressure gaseous products. As the reaction proceeds more and more gas is formed. The porosity of the explosive increases and as more and more surface area becomes exposed the reaction can accelerate and propagate with the increasing porosity and permeability until all the explosive is consumed or until some mechanism releases the pressure and the reaction is quenched. Violent reaction or even disastrous detonation can be achieved in some cases. The interplay between the burning (and thereby disintegrating) solid matrix and gaseous products is still ill-understood. Plainly there are confined locations where flames are interacting with flames from nearby surfaces, probably in highly complex ways.
Previous works have explored the two-phase problem of reacting solid and gaseous products from a macroscopic continuum viewpoint, but detailed treatments of the internal burning process and of how the hot gas heats the explosive up are lacking. The problem is compounded by the complexity of modern heterogeneous explosives in which crystals of pure explosive are embedded in polymer binders, which themselves can be reactive. The creation and propagation of flames in this type of explosive have not been modelled in detail; burn models in current use are generally empirically based macroscopic models rather than being based upon first principles. Their calibration often depends on the experiment or geometry being modelled. It is the need for models of explosive response to underpin safety cases that has driven the development of such empirical and semi-empirical models, which make assumptions about the physics and chemistry ongoing in a chemical reaction. Their empirical nature limits the applicability of such models as the validated predictive tools needed to make assessments where undertaking full experiments is costly and/or impractical. The proposed research begins the journey to overcoming this limitation. The recent advances in additive manufacturing of explosives provide motivation for the investigation of idealised explosive geometries, which are becoming realisable in practice while amenable to mathematical modelling. This interdisciplinary work links with EPSRC Research Areas including continuum mechanics, chemical reactions, fluid dynamics, non-linear systems, with industrial, defence and security aspects.


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Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/R512138/1 01/10/2017 30/09/2022
1916657 Studentship EP/R512138/1 25/09/2017 31/01/2022 Sally Said
Description The key findings of this award have been modelling the reaction-diffusion process and tracking the heat flow within an explosive material. To begin, single step reactions using the Arrhenius model were explored. We then extended the model to include three reaction terms. An alternative asymptotic approach was also employed to handle extreme parameter values. There is close agreement between results from the full reaction-diffusion problem and the asymptotic problem.

Another key finding of this award has been the exploration of gas effects. A first gas model is a discrete scheme which accounts for gas created in the system. Here we track the temperature of the gas produced from the reaction. This model has been extended to a continuous framework where we solve the equations numerically. There is some agreement between the discrete and continuous models for relatively mild parameters.
Exploitation Route Through the industrial partner and also though the international community who are interested in this area of research.
Sectors Aerospace, Defence and Marine,Chemicals,Manufacturing, including Industrial Biotechology