Study Group Report 2009: reaction-diffusion models of decontamination (DSTL)
This is the final report on the problem of reaction-diffusion models of decontamination, brought to ESGI68 by Dstl. Click on the link at the bottom to download the full report as a pdf document.

Report coordinator
David Allwright (Knowledge Transfer Network for Industrial Mathematics)

Executive summary
A contaminant, which also contains a polymer is in the form of droplets on a solid surface. It is to be removed by the action of a decontaminant, which is applied in aqueous solution. The contaminant is only sparingly soluble in water, so the reaction mechanism is that it slowly dissolves in the aqueous solution and then is oxidized by the decontaminant. The polymer is insoluble in water, and so builds up near the interface, where its presence can impede the transport of contaminant. In these circumstances, Dstl wish to have mathematical models that give an understanding of the process, and can be used to choose the parameters to give adequate removal of the contaminant. Mathematical models of this have been developed and analysed, and show results in broad agreement with the effects seen in experiments.

Introduction
Dstl is the main research organisation of the Ministry of Defence. The problems described here are part of the remit of the Hazard Management team. Their brief is to develop methods to minimize the hazard resulting from the use of chemical, biological or radiological weapons. The team’s activities support both civil and military hazard management. For instance, patented decontamination formulations have been evaluated for uses such as cleaning railway rolling stock, removal of traffic film from road vehicles, and graffiti removal.

The nature of the problem
Consider a polymer solution for which the solvent is a pollutant, which we call A throughout (the Agent of contamination). A drop on a horizontal surface will adopt an equilibrium shape determined by its size and surface tension and gravity, in a time determined by its viscosity. If a layer of decontaminant, a solution of B in water, is applied above, then A will diffuse into the aqueous layer and undergo a chemical reaction with B which renders it harmless. The polymer P does not diffuse into the aqueous layer. This situation is illustrated in Figure 1.

Figure 1: Schematic diagram.

In some cases, the aqueous layer may contain a microemulsion of toluene and butanol. This has three effects. First, it increases the solubility of A in the aqueous layer, which makes it easier for the required reaction to occur. Second, it decreases the effective concentration of B in the aqueous layer, which reduces the reaction rate (or increases the amount of B that is required). And third, the organic component can enter the A region, where it can cause the polymer to swell.

The diffusion coefficients in the polymer solution depend on the polymer concentration, and so do the surface tension and viscosity. In some cases there may be some absorption of A into the solid surface, for instance if it is porous like concrete. The challenge is to determine the residual pollution level left on the surface as a function of time, and of all the physical parameters such as drop size, initial concentrations, reaction constants etc.

A second problem is to consider the same droplet but deposited on a vertical surface. In this case the decontaminant is delivered as a spray. There is gravity-driven flow of the decontaminant solution down the surface, which brings fresh decontaminant solution into contact with each drop, but also limits the contact time. Again, the question is how to determine the total required volume, and the time to deliver that volume, given the relevant physical and chemical properties.

Numerical results from one-dimensional model
The model above can be solved numerically. In each layer, a scaling depending on the thickness is used to replace the layer by a fixed interval. This results in an advection-reaction-diffusion system, which can be solved by an up-winding method. Some results from this are shown in Figures 2-3. First, results with polymer present in layer 1 are shown in Figure 2. Here the polymer has a diffusivity such that it does build up near the interface, but the concentration does not rise to more than 15%. However, in Figure 3 the diffusivity of the polymer is set lower (and more realistic), and here the concentration does built up to nearly 100% near the interface, which slows down the diffusion of A into the reacting layer, and so slows the reaction rate very much.

Figure 1: Numerical results from one-dimensional model including an initial 5% polymer in the contaminant layer.

Figure 2: Numerical results from one-dimensional model including an initial 5% polymer in the contaminant layer with small diffusion coefficient. Note that the plotted profiles in A are decreasing over time since, after a quick initial phase in which A diffuses out into the upper layer, the saturated value of A decreases, as the build up of polymer reduces the value at the interface.

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