Fluid-flow effects in the reactive decontamination of porous materials driven by chemical swelling or contraction

Abstract

Following the release of a chemical warfare agent, it is crucial for public health that the affected environment is entirely decontaminated. If the agent has seeped into a porous building material, the decontamination is achieved by applying a cleanser solution to the surface of the porous material, and allowing it to react in, neutralising the agent. Typically, the agent and cleanser solution are immiscible fluids and so the reaction occurs at the fluid–fluid interfaces within the pores. Previous studies have shown that the rate of decontamination of the porous material can depend on both the chemical reaction rate and the transport of cleanser to the reacting interface. These studies have all assumed that the two fluids have the same densities, so that diffusion is the only cleanser-transport mechanism. In this paper, we relax this assumption and investigate the effect of a fluid flow—generated by a change in density of the material (a swelling, or contraction) during the chemical reaction—on the decontamination process. This flow of fluid results in advection as well as diffusion of chemicals. Buoyancy effects are neglected. In particular, we show that when the agent is more dense than the reaction product, the decontamination process is slower, due to the adverse advection effect.

1 Introduction

We are interested in the decontamination of a porous material, such as brick, that has been exposed to a hazardous chemical agent. When this hazardous agent has soaked into a porous material, the usual method of decontamination is to apply a cleanser to the surface of the porous material, which reacts with the agent, creating a harmless reaction product. This method is classified as reactive decontamination.

It is of practical importance to public health that this decontamination is quick and effective. A cleanser chemical must be chosen depending on its suitability for neutralising the agent in question. Ideally, a cleanser should also be chosen to reduce the overall time required for the decontamination process [1]. The chemistry of decontamination processes can be complicated; cleansers are required to neutralise the agent as quickly as possible, while ideally not damaging surrounding material [1]. It is also beneficial in practice if cleansers are effective on a variety of warfare agents. Experimental research into the effect of different surfaces (including porous surfaces) on the decontamination include [2, 3]. The study of decontamination within porous media is experimentally challenging, since it is difficult to measure the transport and reaction processes within an opaque porous medium [2]. Moreover, the vast array of different combinations of agents, potential cleansers, and different types of porous media, notwithstanding the potential danger and security risk of working with hazardous agents, constrains the efficacy of experimental work.

Mathematical modelling of reactive decontamination has developed in recent years. While the result of this modelling cannot identify the best cleanser to use for a given agent directly, it provides useful insight on the characteristics of the cleanser, such as its strength, reaction rate with the agent, and solubility, that are beneficial for an efficient decontamination process.

A mathematical model for one-dimensional reactive decontamination was introduced by [4, 5], in which a contaminated porous medium, initially saturated with agent, is cleaned by applying a cleanser solution to the surface of the porous medium. Typically, an agent and cleanser pair are immiscible fluids. Often the agent is oily, while the cleanser is in aqueous solution [4]. Since the fluids are immiscible, the chemical decontamination reaction occurs at the fluid–fluid interface within the porous material. The product of the chemical reaction is distributed between the aqueous cleanser phase and the oily agent phase. The processes limiting the decontamination are therefore the transport of reactants to the reacting interface, which is assumed in [4] to be purely diffusive, and the rate of the chemical reaction itself. In the limit of an infinitely deep agent spill, [4] found that this partition coefficient strongly affects the decontamination rate. Other important parameters identified were the concentration of the applied cleanser and the rate of the chemical reaction. In particular, two parameter regimes were identified in which the decontamination was limited by the supply of cleanser and the removal of the reaction product, respectively.

The effect of the pore-scale processes on the decontamination was investigated in [6]. Two decontamination scenarios were considered, one of the same structure as that of [4] (although with the reaction product only soluble in the cleanser phase, and not the agent), and the other where a layer of agent only coated the walls of the pores—rather than saturating the porespace—and so both agent and cleanser fluid coexist within the pores. Homogenisation analysis was used in [6] (with a slight correction provided in [7]) to derive effective models for both scenarios. In particular for the case of an initially saturated porous medium, an effective reaction rate was found, depending on the average surface area of the reacting interface within the porespace.

In all these mathematical studies [4,5,6], it is implicitly assumed that the density of the agent and cleanser solution and product chemicals are the same, so that no fluid flow is generated by any changing fluid densities during the chemical reaction. This results in models with purely diffusive transport of the chemicals within the porous material. In practice, it is not likely that all the chemicals do have precisely the same densities. In this paper, we therefore investigate the effect of a change in fluid density during the chemical reaction, which can result in a fluid flow normal to the reaction front. We will therefore consider both diffusive and advective transport of chemicals.

The change in density of materials during a chemical reaction is a commonly observed phenomenon in other decontamination settings and is often referred to as a contraction or swelling of material. For instance, in the decontamination of food residues in the dairy manufacturing industry, swelling of the residue plays a role in its break-down [8, 9]. Similar swelling behaviour is observed in the removal of wool-grease from fresh wool [10]. Swelling is also an important phenomenon in the dissolution of glassy polymers [11], with particular applications in drug delivery [12]. We are unaware of any mathematical modelling of the decontamination of porous media in the literature that attempts to include the effect of flow driven by chemical swelling or contraction during the reaction. The present work is a first step towards remedying this issue. We present a decontamination model including these phenomena that is highly idealised but through which we may build understanding for the effect that the swelling/contraction flow has on the decontamination time. This work also establishes a framework by which swelling/contraction phenomena might be included in more detailed models.

We note that, like the model derived in [6], the decontamination model we derive in this paper is a type of Stefan problem [4]. Stefan problems classically occur in various physical systems [13], including phase change systems, notably freezing and solidification [14,15,16] and crystallisation [17]. Stefan problems can often exhibit instabilities [18], but we note that the form of Stefan problem studied in the present paper is stable. Solutions and limiting behaviours of Stefan problems with kinetic under-cooling (of which our model is one [4]), in the case of no fluid flow, are summarised in [19].

In Sect. 2, we extend the one-dimensional reactive decontamination model of [4] to include fluid-flow effects due to a density difference between the chemicals, additionally assuming, like [6], that the agent is neat and the reaction product is only soluble in the cleanser phase. We nondimensionalise the model, identifying a new dimensionless parameter, $\beta$, that characterises the fluid velocity. In Sect. 3, we then describe our numerical method for solving the decontamination model. Numerical and asymptotic solutions are presented in Sects. 4, 5, and 6 and interpreted in terms of the decontamination process. We summarise and discuss our main results in Sect. 7.

2 Mathematical model

2.1 Model derivation

Similarly to [4], we consider a one-dimensional porous medium of length ${\bar{L}}$. As illustrated in Fig. 1, the spatial variable ${\bar{x}}$ points into the porous medium, with ${\bar{x}}=0$ being the surface of the porous medium. Initially, the porous medium is saturated with neat agent with density $\rho _a$. At ${\bar{t}}=0$, a cleanser fluid is applied to the surface, ${\bar{x}}= 0$, of the porous medium, containing the cleanser chemical with molar concentration ${\bar{c}}_0$.

Fig. 1

Diagram of the decontamination problem with different densities of the agent and cleanser phases. Here, $\bar{L}$ is the depth of the agent spill, and $\bar{s}(\bar{t})$ is the position of the moving reaction front between the two fluids

With the assumption that the cleanser phase and the agent phase are completely immiscible, there is a free boundary at ${\bar{x}}=\bar{s}({\bar{t}})$, separating the two phases. Since the cleanser and agent chemicals only meet at the interface between the fluids, the decontamination reaction occurs only on this interface. As the reaction takes place, a product chemical is generated by the cleanser and the agent, which we assume is only soluble in the cleanser phase. Thus the agent remains neat at all times, but the cleanser phase is a mixture of cleanser chemical and product chemical. As agent is consumed, the reaction front ${\bar{s}}({\bar{t}})$ moves into the porous material. (One example of a decontamination scenario with the reaction product largely insoluble in the agent, is the decontamination of sulphur mustard with bleach or calcium hydroxide, detailed in Appendix A.)

The neat agent occupying ${\bar{x}}\in [{\bar{s}}, {\bar{L}}]$ has (constant) molar volume ${\bar{\chi }}$. The cleanser phase in ${\bar{x}}\in [0,{\bar{s}}]$ contains the cleanser and the product with molar concentrations ${\bar{c}}({\bar{x}},{\bar{t}})$ and ${\bar{p}}({\bar{x}},{\bar{t}})$, respectively. We assume the simplest possible stoichiometry of the chemical reaction, with

$$\begin{aligned} A+C\longrightarrow P, \end{aligned}$$

(1)

where A, C and P are the agent, cleanser, and product chemicals, respectively. The molar masses of the cleanser, product and agent are $M_c$, $M_p$, and $M_a$, respectively, given by

$$\begin{aligned} M_c{\bar{c}}=\rho _c,\quad M_p{\bar{p}}=\rho _p,\quad M_a=\rho _a{\bar{\chi }}, \end{aligned}$$

(2)

in terms of the mass concentrations $\rho _c$ and $\rho _p$ of the cleanser and product, respectively, and the agent density $\rho _a$. We note that for our chosen stoichiometry (1), we must have

$$\begin{aligned} M_a+M_c=M_p. \end{aligned}$$

(3)

The cleanser and product mixture density, $\rho$, is given by

$$\begin{aligned} \rho =M_c\bar{c}+M_p\bar{p}. \end{aligned}$$

(4)

We assume for simplicity that this mixture density $\rho$ is constant. This assumption is reasonable if the pure cleanser and pure product have similar densities.

If the agent density is different from the mixture density ($\rho \not =\rho _a$), a flow will be generated at the boundary. The agent, which is often an oily chemical, is likely to have a higher viscosity than the cleanser (often aqueous), so that it is reasonable to assume that the agent remains stationary, while the cleanser flows with velocity ${\bar{u}}$ in the ${\bar{x}}$ direction. Since we assume $\rho$ to be constant, the cleanser phase is incompressible. Therefore, in our one-dimensional geometry, the velocity ${\bar{u}}$ of the cleanser phase satisfies the mass conservation equation

$$\begin{aligned} {\bar{u}}_{{\bar{x}}}=0\qquad \text {for }0<{\bar{x}}<{\bar{s}}({\bar{t}})\text { and }{\bar{t}}>0, \end{aligned}$$

(5a)

where the subscript ${\bar{x}}$ denotes the partial derivative with respect to ${\bar{x}}$. The velocity ${\bar{u}} ({\bar{t}})$ is therefore spatially uniform, although it may vary in time. The value of $\bar{u}$ is determined by an appropriate boundary condition at $\bar{x}={\bar{s}}({\bar{t}})$ (discussed below). We emphasise that the only flow in the system is that generated by the change of density of the fluid at the reacting interface: there is no externally imposed flow.

Since the two fluids (cleanser and agent) may have different densities, we might expect additional flow effects due to buoyancy forces, which would invalidate our assumed one-dimensional geometry. These would not be important if the denser fluid were below the less dense fluid, but could have an effect if the less dense fluid was below the denser fluid. We estimate the maximum size of the Grashof number (the ratio of buoyancy to viscous forces) in the fluid to be $\text {Gr}=gd^3|\rho -\rho _a|/(\nu ^2\min (\rho ,\rho _a))\approx 10^{-5}\max (|1-\rho _a/\rho |,|1-\rho /\rho _a|)$ where $g\approx 10\,$m s$^{-2}$ is the acceleration due to gravity, $d\approx 10^{-6}\,$m is the approximate pore size for brick (concrete is likely smaller, plasterboard larger), and $\nu \approx 10^{-6}\,$m$^2\,$s$^{-1}$ is the kinematic viscosity of water (expected to be smaller than that of the oily agent phase). Since $\text {Gr}\ll 1$ so long as both $\rho _a/\rho ,\rho /\rho _a\ll 10^5$, we will reasonably assume throughout that buoyancy effects are negligible.

Within the cleanser phase, the cleanser and product chemicals diffuse against each other and are advected with the uniform net flow $\bar{u}$. The advection–diffusion equation for ${\bar{c}}$ is

$$\begin{aligned} {\bar{c}}_{{\bar{t}}}+{\bar{u}}{\bar{c}}_{{\bar{x}}}={\bar{D}}{\bar{c}}_{{\bar{x}}\bar{x}}\qquad \text {for }0<{\bar{x}}<{\bar{s}}({\bar{t}})\text { and }\bar{t}>0, \end{aligned}$$

(5b)

where the subscript of ${\bar{t}}$ denotes the partial derivative with respect to ${\bar{t}}$, and ${\bar{D}}$ is the effective diffusivity of the cleanser in the product, within the porous material (taking into account the effect of the porous material, which reduces the effective diffusivity compared with the diffusivity of the cleanser in its solution [20]). Since there is advection of the cleanser within the porous material, we might expect dispersion as well as diffusive transport of cleanser. However, we will see that the advective transport of cleanser cannot dominate over diffusion. As shown in [21], in this regime for which advection does not dominate over diffusion, dispersion effects are very small, and it is reasonable to neglect these in favour of the effective diffusion through the porous structure. The effective diffusivity ${\bar{D}}$ is therefore assumed to be constant, and in particular independent of the flow velocity. Given the solution ${\bar{c}}$ of (5b), we can then find ${\bar{p}}$ from (4), as

$$\begin{aligned} {\bar{p}}=\frac{\rho -M_c{\bar{c}}}{M_p}, \end{aligned}$$

(6)

but this will not be necessary for our investigation of the decontamination process.

To close the model (5) and to describe the motion of the free boundary ${\bar{x}}= {\bar{s}}({\bar{t}})$, we need four boundary conditions: one for ${\bar{u}}$, two for ${\bar{c}}$, and a final additional condition to describe the motion of the free boundary ${\bar{s}}({\bar{t}})$. We suppose that the cleanser is continuously being replenished at the surface of the porous medium, ${\bar{x}}=0$, at a constant molar concentration ${\bar{c}}_0$, and so, like [4], impose the Dirichlet boundary condition

$$\begin{aligned} {\bar{c}}={\bar{c}}_0\quad \text {at } {\bar{x}}=0 \text { for }{\bar{t}}>0. \end{aligned}$$

(7)

Although this is an idealisation, a similar effect may be achieved in practice by first applying a cleanser to the porous surface and then regularly wiping away fluid from the top and re-applying fresh cleanser at this given concentration. In particular, since we obtain a net flow of the cleanser fluid for $\rho \ne \rho _a$, this continual wiping away of excess fluid and re-application of cleanser at the surface is necessary to maintain the validity of our model.

At ${\bar{x}}={\bar{s}}({\bar{t}})$, we impose the chemical reaction between the agent and the cleanser. Like [4, 6], we assume that the rate, ${\bar{R}}$, of the reaction (1), is given by the law of mass action [22], namely

$$\begin{aligned} {\bar{R}}={\bar{k}}\frac{{\bar{c}}({\bar{s}}({\bar{t}}),{\bar{t}})}{{\bar{\chi }}}, \end{aligned}$$

(8)

where ${\bar{k}}\,\,[\text {m}^4\,\text {s}^{-1}\,\text {mol}^{-1}]$ is an effective rate constant. By conservation of mass of cleanser at the reacting interface, the flux of cleanser into the reacting interface must equal the amount consumed in the chemical reaction. We therefore impose the boundary condition

$$\begin{aligned} {\bar{D}}{\bar{c}}_{{\bar{x}}}-{\bar{c}}({\bar{u}}-{\bar{s}}_{{\bar{t}}})=-{\bar{R}} \qquad \qquad \qquad \text {at } {\bar{x}}={\bar{s}}({\bar{t}}) \text { for }\bar{t}>0. \end{aligned}$$

(9a)

By similarly considering the conservation of mass of the product chemical and the agent, respectively, at the reacting interface, we obtain

$$\begin{aligned}&{\bar{D}}{\bar{p}}_{{\bar{x}}}-{\bar{p}}({\bar{u}}-{\bar{s}}_{{\bar{t}}})={\bar{R}}&\quad \text {at } {\bar{x}}={\bar{s}}({\bar{t}}) \text { for }\bar{t}>0, \end{aligned}$$

(9b)

$$\begin{aligned}&- {\bar{s}}_{{\bar{t}}}=-{\bar{\chi }}{\bar{R}}&\quad \text {at } {\bar{x}}={\bar{s}}({\bar{t}}) \text { for }{\bar{t}}>0. \end{aligned}$$

(9c)

Since we may express ${\bar{p}}$ in terms of ${\bar{c}}$ according to (6), these three boundary conditions (9) close the model. However, it is helpful to rearrange these to a more convenient form. Specifically, we multiply each of these three expressions of conservation of mass of each chemical (9) by their corresponding molar mass, and sum the resulting equations, obtaining the expression of total conservation of mass at the reacting interface, namely

$$\begin{aligned} \begin{aligned} -\rho ({\bar{u}}-{\bar{s}}_{{\bar{t}}})-\rho _a{\bar{s}}_{{\bar{t}}}=0 \end{aligned} \end{aligned}$$

(10)

where we have used the identities (2)–(4). Rearranging (10), we obtain a boundary condition for $\bar{u}$, namely,

$$\begin{aligned} {\bar{u}}={\bar{s}}_{{\bar{t}}}\left( 1-\frac{\rho _a}{\rho }\right) \quad \text {at } {\bar{x}}={\bar{s}}({\bar{t}}) \text { for }{\bar{t}}>0. \end{aligned}$$

(11)

This condition (11) fixes the uniform velocity of the cleanser phase ${\bar{u}}(\bar{t})$. In particular, we note that there are three possible cases: (i) if $\rho >\rho _a$, the product created in the reaction takes up less space than the agent it is replacing, so the cleanser and product mixture flows into the porous material with ${\bar{u}}>0$; (ii) if $\rho <\rho _a$, the product created in the reaction takes up more space than the agent it is replacing, so the cleanser and product mixture flows out of the porous material $\bar{u}<0$; and (iii) if $\rho =\rho _a$, there is no flow, ${\bar{u}}=0$, and the system is completely governed by diffusion. We view (11) as a replacement for the boundary condition (9b), and so the remaining boundary conditions at ${\bar{x}}={\bar{s}}({\bar{t}})$ are (9a) and (9c). We note that, since the fluid velocity $\bar{u}$ is related to the speed $\bar{s}_{\bar{t}}$ of the reacting front via (10), the flux of cleanser chemical into the front, the left-hand side of (9a), is always non-positive, necessary since the right-hand side, $-\bar{R}$, of (9a) must be non-positive.

Using the expression (11) for ${\bar{u}}$ in (5), the equation for ${\bar{c}}$ becomes

$$\begin{aligned} {\bar{c}}_{{\bar{t}}}+{\bar{s}}_{\bar{t}}\left( 1-\frac{\rho _a}{\rho }\right) {\bar{c}}_{{\bar{x}}}={\bar{D}}\bar{c}_{{\bar{x}}{\bar{x}}}\qquad \text {for }0<{\bar{x}}<{\bar{s}}({\bar{t}})\text { and }{\bar{t}}>0. \end{aligned}$$

(12)

In summary, our decontamination model is (12), along with the boundary conditions (7), (9a), and (9c). Initially, at ${\bar{t}}=0$, we impose

$$\begin{aligned} {\bar{s}}=0, \end{aligned}$$

(13)

and so we do not require an initial condition for ${\bar{c}}$.

2.2 Nondimensionalisation

We now nondimensionalise our model and identify relevant dimensionless parameters. We make the rescalings

$$\begin{aligned} {\bar{x}}={\bar{L}}x,\quad {\bar{t}}=\frac{{\bar{L}}^2}{{\bar{D}}}t,\quad {\bar{s}}={\bar{L}}s,\quad {\bar{c}}={\bar{c}}_0c, \end{aligned}$$

(14)

noting that we have chosen the spill-lengthscale ${\bar{L}}$, and the diffusion timescale which balances all terms in the governing equation (12). Making these changes of variables in our model (7), (9a), (9c), (12), and (13), we obtain the dimensionless system

$$\begin{aligned}&c_{t}+ \left( 1-\beta \right) s_t c_{ x}= c_{xx}&\text {for }0<x<s( t)\text { and } t>0, \end{aligned}$$

(15a)

with the initial and boundary conditions

$$\begin{aligned} c&= 1&\text {at } x=0 \text { for } t>0, \end{aligned}$$

(15b)

$$\begin{aligned} c_x&=-(\lambda +\beta s_t)c&\text {at } x=s( t) \text { for } t>0, \end{aligned}$$

(15c)

$$\begin{aligned} s_{ t}&=\gamma \lambda c&\text {at } x= s( t) \text { for } t>0, \end{aligned}$$

(15d)

$$\begin{aligned} s&=0&\text {at }t=0, \end{aligned}$$

(15e)

where $\beta ,\gamma$ and $\lambda$ are dimensionless parameters defined by

$$\begin{aligned} \beta :=\frac{\rho _a}{\rho }, \quad \gamma :={\bar{c}}_0{\bar{\chi }}, \quad \lambda :=\frac{{\bar{k}}{\bar{L}}}{{\bar{\chi }}{\bar{D}}}. \end{aligned}$$

(16)

We note that $\beta$ is the ratio of the agent density to the mixture density, which describes the direction and magnitude of the flow (with positive flow for $\beta <1$, no flow for $\beta =1$, and negative flow for $\beta >1$). Meanwhile, $\gamma$ is the ratio of the molar concentration of the applied cleanser to the molar concentration of the agent, which describes the amount of cleanser added to the system compared with the amount of agent in the system, and $\lambda$, proportional to both $\bar{L}$ and $\bar{k}$, characterises the depth of the agent spill or the rate of the chemical reaction. We note that the group $\gamma \lambda$, which is the ratio of the reaction rate to the diffusive mass transfer rate, is the second Damköhler number.

In Appendix A, we list densities of various agents, and see that, if decontaminated with an aqueous cleanser, the parameter $\beta$ may be either smaller or greater than 1. We also estimate values of the dimensionless parameters in Appendix A, for two realistic decontamination scenarios involving the reaction of sulphur mustard with either a bleach solution or calcium hydroxide solution. For these cleansers, we estimate fairly small values of $\gamma =0.8$ and 0.03, respectively, although we also note that $\gamma$ may also take relatively large values ($\gamma =8$ for the cleanser DS2 in [4]). We also calculate $\beta \approx 1.3$ for sulphur mustard and argue that $\lambda$ may be roughly on the order of 1–$10^2$, although this may in fact be smaller.

2.3 Transformation to a fixed domain

Finally, we transform our dimensionless model (15) from the growing domain $x\in (0,s(t))$, to a fixed spatial domain $\eta \in (0,1)$, by making the change of variables

$$\begin{aligned} \eta =\frac{x}{s(t)}, \qquad c(x,t)=g(\eta ,t). \end{aligned}$$

(17)

With this change of variables the model (15) becomes

$$\begin{aligned} g_t=\frac{1}{s^2}\left( g_{\eta \eta }+ss_tg_\eta \left( \eta -\left( 1-\beta \right) \right) \right) \text { for } 0<\eta <1\text { and }t>0, \end{aligned}$$

(18a)

with the initial condition

$$\begin{aligned} s(0)=0, \end{aligned}$$

(18b)

and boundary conditions

$$\begin{aligned} g&= 1&\text {at } \eta =0 \text { for } t>0, \end{aligned}$$

(18c)

$$\begin{aligned} g_\eta&=-(\lambda +\beta s_t)sg&\text {at } \eta =1 \text { for } t>0, \end{aligned}$$

(18d)

$$\begin{aligned} s_{ t}&=\gamma \lambda g&\text {at } \eta =1 \text { for } t>0. \end{aligned}$$

(18e)

The benefit of this coordinate transformation is that this fixed-domain formulation (18) (on spatial domain $\eta \in (0,1)$) is more tractable numerically than the growing domain version (15) (on $x\in (0,s(t)$), since there is no need to re-mesh the spatial numerical domain at each time-step. We will analyse this system (18) both analytically in various limits, and numerically as in the following section. For ease of interpretation of our figures, however, throughout the remainder of the paper, we will plot c as a function of the physical variable x, rather than g as a function of $\eta$.

3 Numerical method and early-time asymptotics

In this section, we formulate a numerical method to solve the decontamination model (18). Since we have $s=0$ at $t=0$, we will need to make use of early-time asymptotic approximations to initialise our numerical simulations.

3.1 Early-time asymptotic analysis

We look for the early-time behaviour by setting $t=\epsilon T$ for small $\epsilon \ll 1$, with $T={\mathcal {O}}(1)$. Using the initial condition (18b) and the boundary condition (18c), we pose the first-order expansions

$$\begin{aligned} s=\epsilon s_1+{\mathcal {O}}(\epsilon ^2),\quad g=1+\epsilon g_1+{\mathcal {O}}(\epsilon ^2), \end{aligned}$$

(19)

where $s_1$ and $g_1$ are of ${\mathcal {O}}(1)$. At leading order in (18), we find that diffusion dominates (as the lengthscale s of the cleanser-occupied region is small at early times), and so obtain

$$\begin{aligned}&g_1(\eta ,T)=-(1+\beta \gamma )\gamma \lambda ^2 T\eta , \end{aligned}$$

(20a)

$$\begin{aligned}&s_1(T)=\gamma \lambda T, \end{aligned}$$

(20b)

or in terms of the original variable t,

$$\begin{aligned} \begin{aligned}&g(\eta ,t)= 1-(1+\beta \gamma )\gamma \lambda ^2 t\eta +{\mathcal {O}}(t^2),\\&s(t)= \gamma \lambda t+{\mathcal {O}}(t^2). \end{aligned} \end{aligned}$$

(21)

3.2 Numerical method

We solve (18) using the method of lines [23], with a finite-difference method for the spatial discretisation. Specifically, we use a second-order central difference scheme for the diffusion term in (18a), and a first-order upwind scheme for the advection term, so that our overall spatial discretisation is first order. We note that the upwind direction depends on the sign of $1-\beta -\eta$. Since $\eta \in [0,1]$, the upwinding direction is fixed for $\beta \ge 1$ (corresponding to a negative cleanser flow direction, away from the agent, in the physical coordinate system). However, for $\beta <1$ (positive cleanser flow), the effective advection term in (18a) changes sign within the domain, at $\eta =1-\beta$, and we take care to upwind correctly in these separate regions of the domain. Our spatial discretisation is summarised in Appendix B.

We note that our model (18) is stiff for small t for which s is small. For the time-stepping, we therefore use the inbuilt Matlab solver ode15s which is designed for efficient solution of stiff systems [24]. This is a multistep solver, using numerical differentiation formulas of order 1–5 [24]. We initialise our numerical solutions using the early-time approximations (21).

4 The effect of flow direction and magnitude

We recall that $\beta$, the ratio of the agent and cleanser-phase densities, determines the direction and velocity of the flow. In this section, we explore the dependence of the solution on $\beta$. We note that, as in Appendix A, the range of $\beta$ expected for liquid–liquid reactions is fairly limited, and likely to remain within an order of magnitude of 1. We explore this range and also a wider range of $\beta$ in order to understand the extent to which changes in chemical density can impact decontamination time. We also note that our model has relevance in situations where one of the fluids is gaseous or vaporised, in which case $\beta$ may vary by several orders of magnitude.

Fig. 2

Numerical solutions of (18) for various values of $\beta$. Throughout we hold $\gamma =2$ and $\lambda =4$ constant

Numerical solutions s(t) and $c(x,t)=g(\eta ,t)$ of (18) are presented in Fig. 2, exploring the effect of varying $\beta$ on the solutions. In Fig. 2a we show the reaction front s(t) for different values of $\beta$. For all values of $\beta$ we observe that the interface velocity decreases with time t, until the reaction front reaches the end of the contaminated region, $s(t)=1$. This is physically understandable, since as the cleanser-occupied domain grows, it takes longer for the newly added cleanser to reach the reaction front s(t). We note that, from our boundary conditions (15c) and (15d), the interfacial mass flux of cleanser into the reaction front s(t) is given by

$$\begin{aligned} \left( c_x-c((u-s_t)\right) |_{x=s(t)}=\frac{s_t}{\gamma }. \end{aligned}$$

(22)

From the shape of s(t) shown in Fig. 2a, we therefore see that throughout a decontamination process, the rate of cleanser mass flux into the reacting interface is reducing, as the interface velocity decreases. We also note that the shape of s(t) does not appear to change drastically with $\beta$, in particular there is no qualitative change in the shape of s(t) for $\beta <1$ (positive fluid flow) and $\beta >1$ (negative flow). Crucially, the reaction front is seen to move more slowly for larger values of $\beta$. This means that for decontamination scenarios in which the agent is denser than the cleanser–product mixture, we have slower decontamination than when the agent is less dense.

To understand why this is the case, we look in more detail at the concentration profiles for three values of $\beta$ in Fig. 2b–d, illustrating the cases of no flow ($\beta =1$), negative flow ($\beta =10$), and positive flow ($\beta =0.1$), respectively. We show c(x, t) for $x\in [0,s(t)]$ for a variety of times t up to the decontamination time at which $s(t)=1$. For $\beta =1$, in Fig. 2b, we observe similar behaviour to that found by [4]: at early times the concentration profile is approximately linear in x (as predicted by our early-time asymptotic analysis in Sect. 3.1) and for later time the concentration profile is slightly convex. When $\beta >1$ in Fig. 2c, we see that c has a much more strongly convex profile, compared to $\beta =1$. This is because we have a negative fluid flow in this case, so cleanser is advected away from the reacting interface. This also explains why we observe a slower-velocity reaction front in this case: cleanser must diffuse against the flow to reach the reacting interface. When $\beta <1$, as in Fig. 2d we find that c has a concave profile, since the positive fluid flow transports cleanser towards the reacting interface. The combined advective and diffusive transport of cleanser towards the reacting interface contribute to an increased speed of the reaction front.

Fig. 3

Variation of the decontamination time with $\beta$, for various $\gamma =0.05,$ 0.1,  0.2,  1,  and 2, with the arrow denoting increasing $\gamma$, and holding $\lambda =5$

We define the decontamination time of the system to be the time $t_\text {decon}$ at which all the agent has been reacted away, or equivalently that the reaction front has reached the end of the contaminated region, so that $s(t_\text {decon})=1$. In Fig. 3, we show the variation of the decontamination time with $\beta$, for a variety of values of $\gamma$ (and holding $\lambda$ fixed). For larger $\gamma$, corresponding to higher applied cleanser concentrations, we see that the decontamination time is smaller. This is intuitive, since we expect faster decontamination for stronger applied cleansers. The effect of varying $\gamma$ is investigated in detail in Sect. 5. We see that the increase of the decontamination time with $\beta$ appears to be slightly sublinear as $\beta$ increases. To investigate this behaviour, we now look at the case of large $\beta \gg 1$ in detail.

4.1 The limit of large $\beta$

Motivated by the above numerical solutions, we now investigate the limit $\beta \gg 1$, holding both $\gamma$ and $\lambda$ order 1. The decontamination time in Fig. 3 appeared to be close to linear in $\beta$, and so we make the change of variables $t=\beta \tilde{t}$, where $\tilde{t}={\mathcal {O}}(1)$. The system (18) then becomes

$$\begin{aligned} \frac{1}{\beta }s^2 g_{\tilde{t}}=g_{\eta \eta }+\left( \frac{\eta -1}{\beta }+1\right) ss_{\tilde{t}}g_\eta , \text { for } 0<\eta <1\text { and }\tilde{t}>0, \end{aligned}$$

(23a)

with boundary conditions

$$\begin{aligned} g&= 1&\quad \text {at } \eta =0 \text { for } \tilde{t}>0, \end{aligned}$$

(23b)

$$\begin{aligned} g_\eta&=-(\lambda + s_{\tilde{t}})sg&\quad \text {at } \eta =1 \text { for } \tilde{t}>0, \end{aligned}$$

(23c)

$$\begin{aligned} \frac{1}{\beta }s_{ \tilde{t}}&=\gamma \lambda g&\quad \text {at } \eta =1 \text { for } \tilde{t}>0. \end{aligned}$$

(23d)

For large $\beta$,we see from (23a) that the advection–diffusion process is quasi-steady over the chosen timescale. Specifically, expanding in powers of $\beta ^{-1}\ll 1$, we find that to leading order in $\beta ^{-1}$, (23a) has general solution

$$\begin{aligned} g=A\exp (-ss_{\tilde{t}}\eta )+B \end{aligned}$$

(24)

for A and B functions of $\tilde{t}$ to be determined. However, we notice that to leading order in $\beta ^{-1}$ the boundary conditions (23c)–(23d) require that both g and $g_{\eta }$ are zero at $\eta =1$, and thus both A and B must be zero. The issue here is that for the asymptotic analysis,we assume that g is of order 1 at $\eta =1$, whereas in fact the cleanser concentration is decaying exponentially in space, and becomes small here, as observed in Fig. 2c. Instead of simply taking the leading-order terms in the boundary conditions (23c)–(23d), we impose the full boundary conditions

$$\begin{aligned} g_{\eta }=-(\lambda + s_{\tilde{t}})sg, \quad \frac{1}{\beta }s_{\tilde{t}}=\gamma \lambda g \qquad \text {at } \eta =1 \text { for } \tilde{t}>0. \end{aligned}$$

(25)

The solution may then be expressed as

$$\begin{aligned} g=\left( 1+\frac{1}{\beta \gamma }\right) e^{-ss_{\tilde{t}}\eta }-\frac{1}{\beta \gamma }, \end{aligned}$$

(26)

where s satisfies the ODE

$$\begin{aligned} (\lambda +s_{\tilde{t}})e^{ss_{\tilde{t}}}=\gamma \lambda \beta . \end{aligned}$$

(27)

Rearranging, we find that the solution s is given implicitly by

$$\begin{aligned} \tilde{t}=\int _{0}^{s}\frac{\xi }{W(\lambda (1+\beta \gamma )\xi e^{\lambda \xi })-\lambda \xi }\,\textrm{d}\xi , \end{aligned}$$

(28)

where W is the Lambert W function [25]. In particular, we note that the decontamination time for which $s=1$ is approximated by

$$\begin{aligned} t_{\text {decon}}=\beta \tilde{t}_\text {decon}=\beta \int _{0}^{1}\frac{\xi }{W(\lambda (1+\beta \gamma )\xi e^{\lambda \xi })-\lambda \xi }\,\textrm{d}\xi . \end{aligned}$$

(29)

Furthermore, since $W(x)\sim \log (x)$ as the argument x becomes large, in the limit of large $\beta$, we obtain the expression

$$\begin{aligned} t_{\text {decon}}\sim \frac{\beta }{2\log (\lambda \beta \gamma )} \qquad \text { as }\beta \rightarrow \infty . \end{aligned}$$

(30)

Thus the growth in the decontamination time as $\beta$ becomes large is sublinear in $\beta$.

We note that a more formal asymptotic analysis of this large-$\beta$ limit is possible, including a boundary layer (of width ${\mathcal {O}}(\log (\beta ))$ about $x=0$, and with an outer region in which $c={\mathcal {O}}(\log (\beta )/\beta )$ is small. We have chosen above to present the analysis in a simpler and more intuitive way.

In Fig. 4,we compare numerically computed decontamination times with (29) (dashed curves) with excellent agreement even for relatively small $\beta$. The approximation (30) appears to have the correct growth behaviour for large $\beta$, although it is not a close fit. For small $\gamma$ especially, the approximation (29) is surprisingly accurate for small $\beta$. This may be because, for small $\gamma$, the rate of reaction is slow and thus the large-$\beta$ reduction (in which the chemical transport is quasi-steady) remains fairly accurate. The small-$\gamma$ limit is explored in more detail in Sect. 5.1 below.

Fig. 4

The large-$\beta$ behaviour of the decontamination time. Solid lines show numerical solutions of the full model (18), dashed lines are the approximations (29), and the dotted curves are (30). Throughout $\lambda =5$ is held constant

4.2 The limit of small $\beta$

Conversely, we note that in the limit $\beta \ll 1$ for which there is a strong flow of material into the reaction front, the model (18) reduces, at leading order in $\beta$, to

$$\begin{aligned} g_t=\frac{1}{s^2}\left( g_{\eta \eta }+ss_tg_\eta \left( \eta -1\right) \right) \text { for } 0<\eta <1\text { and }t>0, \end{aligned}$$

(31a)

with boundary conditions

$$\begin{aligned} g= 1 \quad \text {at } \eta =0 \text { for } t>0, \end{aligned}$$

(31b)

$$\begin{aligned} g_\eta =-\lambda sg \quad \text {at } \eta =1 \text { for } t>0, \end{aligned}$$

(31c)

$$\begin{aligned} s_{ t}=\gamma \lambda g \quad \text {at } \eta =1 \text { for } t>0. \end{aligned}$$

(31d)

Thus the system becomes independent of $\beta$ in this limit. Physically, we see that (since the product of the chemical reaction occupies a negligible volume compared with the amount of agent removed) the cleanser–product fluid flows down at the velocity, $s_t$, of the reaction front, simplifying the advection term in (31a). This explains the behaviour observed in Fig. 2d above.

5 The effect of applied cleanser strength, $\gamma$

We recall that $\gamma$ characterises the concentration of cleanser compared with agent. Since the agent is assumed to be neat, and so the molar volume of agent is fixed, $\gamma$ is a proxy for the strength of the applied cleanser solution. In this section, we explore the dependence of model solutions on $\gamma$.

5.1 The limit $\gamma \ll 1$

In particular, since the cleanser is in solution while the agent is assumed to be neat, a physically relevant limit is that of dilute cleanser, $\gamma \ll 1$, which we now study asymptotically.

We suppose that $\gamma \ll 1$ and $\beta ,\lambda ={\mathcal {O}}(1)$. We make the change of variable $\tau = \gamma t$, anticipating that for dilute cleansers, the time for the decontamination process will be slower. In terms of this new time variable, $\tau$, the model (18) becomes

$$\begin{aligned} \begin{aligned}&\gamma s^2g_\tau +\gamma (1-\beta )ss_\tau g_\eta =\gamma ss_\tau \eta g_\eta +g_{\eta \eta } \quad&0<\eta <1,t>0, \end{aligned} \end{aligned}$$

(32)

with boundary conditions

$$\begin{aligned} g=1\quad \eta =0, \end{aligned}$$

(33a)

$$\begin{aligned} g_\eta =-(\lambda +\beta \gamma \lambda g)sg \quad \eta =1, \end{aligned}$$

(33b)

$$\begin{aligned} s_\tau =\lambda g \quad \eta =1. \end{aligned}$$

(33c)

We pose the expansions

$$\begin{aligned} s=s_0+ {\mathcal {O}}(\gamma ),\quad g=g_0+{\mathcal {O}}(\gamma ), \end{aligned}$$

(34)

where $s_0$ and $g_0$ are of ${\mathcal {O}}(1)$. Substituting (34) into (32), we find that at leading order

$$\begin{aligned} {g_0}_{\eta \eta }=0, \quad 0<\eta <1,t>0, \end{aligned}$$

(35a)

with

$$\begin{aligned} g_0=1\quad \eta =0, \end{aligned}$$

(35b)

$$\begin{aligned} {g_0}_{\eta }=-\lambda s_0g_0 \quad \eta =1, \end{aligned}$$

(35c)

$$\begin{aligned} {s_0}_\tau =\lambda g_0 \quad \eta =1. \end{aligned}$$

(35d)

Along with the initial condition $s_0(0)=0$, we find the analytic solution of (35), namely

$$\begin{aligned} s_0(\tau )&=\frac{-1+\sqrt{1+2\lambda ^2\tau }}{\lambda }, \end{aligned}$$

(36a)

$$\begin{aligned} g_0(\eta ,\tau )&=1+\eta \left( \frac{1}{\sqrt{1+2\lambda ^2\tau }}-1\right) . \end{aligned}$$

(36b)

We note that $g_0$ is linear in $\eta$, since, physically, the transport of cleanser is dominated by diffusion and is quasi-steady over this long timescale. In particular, we note that there is no dependence on $\beta$ at leading order in this small-$\gamma$ limit, so that the flow of fluid has negligible effect on the decontamination.

Using the change of variable $\tau =\gamma t$, we may also obtain a leading-order expression for the decontamination time for small $\gamma$, namely

$$\begin{aligned} t_{\text {decon}}=\frac{2+\lambda }{2\lambda }\cdot \frac{1}{\gamma }+{\mathcal {O}}(1). \end{aligned}$$

(37)

5.2 Numerical results

Fig. 5

Cleanser concentration profiles at various times through the decontamination, for various values of $\beta$ and $\gamma$. In each figure, solid red lines show numerical solutions of the full model (18), while the dashed black lines in a and c are the small-$\gamma$ approximations (36b) (transformed back to the physical domain). In each figure, the arrow denotes increasing time. Throughout,we hold $\lambda =5$ constant

In Fig. 5,we show numerically computed cleanser concentration profiles, solutions of (18), for various $\gamma$. In Fig. 5a and c,we take small values of $\gamma$, and observe good agreement, with the cleanser concentration close to linear. For small $\beta$ in Fig. 5a, we see that the cleanser concentration profiles are slightly concave, lying above the linear asymptotic approximation, while for large $\beta$ in Fig. 5c,the solutions are slightly convex, below the linear approximations. For small $\beta$, changing $\gamma$ does not have a significant effect on the cleanser profiles, with the solutions in Fig. 5b quite similar to those in Fig. 5a. For large $\beta$, we do see a significant variation in the cleanser profile at larger $\gamma$: the profiles are much more steeply convex in Fig. 5d than in 5c, and the cleanser concentration is lower at the reaction front $x=s(t)$.

Fig. 6

Variation of the reaction front s(t) and decontamination time $t_\text {decon}$ with $\gamma$. Solid lines show numerical solutions of the full model (18) and dashed lines are the approximations (36a) or (37) (transformed back to the original time variable t rather than $\tau$). Throughout, $\lambda =5$ is held constant

The motion of the reaction front s(t) is shown in Fig. 6a, for various $\gamma$. The small-$\gamma$ approximation (36a) closely matches the numerically computed solution for small values of $\gamma$, capturing both the $\sqrt{t}$ shape of the curve and the decontamination time. The variation of the decontamination time with $\gamma$ is shown in Fig. 6b, for three different values of $\beta$, and the normalised error (normalising with respect to the size of $t_\text {decon}$) between the asymptotic solution (37) and the numerical solutions is also shown in Fig. 6c. We see convergence to the small-$\gamma$ approximation for all values of $\beta$ as $\gamma$ becomes small, with the convergence approximately linear in $\gamma$ as expected from our analysis in Sect. 5.1. We notice that the approximation appears to be best for smaller values of $\beta$, i.e.: in cases with positive flow of the cleanser phase into the reaction front. This is as expected from our analysis in Sect. 4.1, since we expect the decontamination time to grow as $\beta$ becomes large.

6 The effect of reaction rate or spill depth, via $\lambda$

The final dimensionless parameter in our model (18), $\lambda$, is proportional to both the reaction rate $\bar{k}$, and the depth, $\bar{L}$ of the contaminated porous medium, and we show numerical solutions of (18) for a variety of values of $\lambda$ in Figs. 7 and 8. Specifically, $\lambda$ may be understood as the ratio of the rate of the chemical reaction to the rate at which material is transported through the domain of lengthscale $\bar{L}$ (the diffusion rate on this lengthscale). For small $\lambda$, corresponding to either fast reaction rates or thin contaminated regions, diffusion of cleanser is fast relative to the reaction and so diffusion dominates the model (18). We therefore expect high cleanser concentrations, close to one, through the entire domain when $\lambda \ll 1$. Even for the relatively moderate value of $\lambda =0.2$, we observe this behaviour in Fig. 7a and c, for both large and small $\beta$. This effect is opposed in the large-$\beta$ case by the adverse flow of cleanser, and thus we see that c is lower at s(t) for the large-$\beta$ case (in Fig. 7c) than in the small-$\beta$ case (Fig. 7a). For large $\lambda$, corresponding to slow reaction rates or deep regions of contaminated porous media, the transport of cleanser is slow relative to the reaction rate, and so cleanser transport limits the decontamination process: the consumption of cleanser at the reacting interface becomes fast relative to the transport rate, and so the value of c at s(t) becomes small. This effect may be seen in Fig. 7b and d, and in particular it is enhanced in the case of large $\beta$, as we would expect due to the adverse fluid flow.

Fig. 7

Numerical solutions of (18) for various values of $\beta$ and $\lambda$. Throughout we hold $\gamma =1$ constant

We note that both our choice of timescale and lengthscale in our nondimensionalisation, given by (14), depend on the length $\bar{L}$ of the contaminated region. In dimensional coordinates, the decontamination front $\bar{s}(\bar{t})$ therefore collapses onto the same curve for any value of $\lambda$ (with the decontamination completed at the point $\bar{s}=\bar{L}$ along this single curve). Thus in this section (unlike in Sects. 4 and 5) we do not show the variation of s(t) with $\lambda$. In order to understand the effect of the spill depth on the decontamination time, we consider the variation of the scaled decontamination time

$$\begin{aligned} \hat{t}_{\text {decon}}=\lambda ^2t_\text {decon}, \end{aligned}$$

(38)

with $\lambda$. This scaled decontamination time $\hat{t}_\text {decon}$ is dimensionless, but behaves in the same way as the dimensional decontamination time ${\bar{t}}$ as the spill depth is varied, since the time-scaling in (14) is quadratic in $\bar{L}$.

Fig. 8

Scaled decontamination time against $\lambda$. The arrow denotes increasing values of $\beta$ or $\gamma$

In Fig. 8, we show the variation of the scaled decontamination time $\hat{t}_\text {decon}$ as $\lambda$ becomes large, varying $\beta$ in Fig. 8a and $\gamma$ in Fig. 8b. For all values of $\beta$ and $\gamma$, we observe a quadratic growth of the scaled decontamination time with $\lambda$. This is because for deep spills the system is limited by transport of cleanser through the domain. We also observe that when $\beta$ increases, the decontamination time increases while for increasing $\gamma$, the decontamination time decreases. This behaviour agrees with that predicted in Sects. 4 and 5.

7 Discussion and conclusions

In realistic decontamination scenarios, the cleanser–product mixture and the agent are likely to have different densities, and hence, a flow will typically occur during the decontamination due to the chemical swelling or contraction. For example, in Appendix A, we estimated the density ratio to be around $\beta \approx 1.3>1$ for the decontamination of sulphur mustard with either an aqueous bleach or calcium hydroxide solution. It is likely that other contaminant–cleanser pairings similarly have $\beta \ne 1$.

In this paper, we have derived and analysed a reactive decontamination model in porous media based on the model used in [4] but additionally including the fluid flow generated due to a change in fluid density during the reaction. Since the fluid flow due to the swelling or contraction is determined by the chemical properties of the contaminant and cleanser, it cannot be controlled in practice except by changing to a different decontaminant. It is therefore very important to understand what effect such swelling/contraction driven flows have on the decontamination dynamics. We found that a negative flow of cleanser, away from the reacting interface, occurs when the agent density is greater than that of the cleanser–product mixture, and that this slows down the decontamination since the cleanser chemical must diffuse against the direction of flow to reach the reacting interface. Conversely, if the agent is less dense than the cleanser–product mixture, we found that the resulting flow of material into the reacting interface advects cleanser along with it and so increases the decontamination rate.

However, perhaps surprisingly, we found that the fluid flow cannot dominate the decontamination model. This is because the flow velocity is proportional to the rate of consumption of agent, and thus intrinsically linked to the diffusion process. Consequently, the effect of including the fluid flow is limited: for large $\beta$ (a much denser agent than cleanser–product mixture) we found that the decontamination time increases fairly slowly with $\beta$, with a sublinear growth on the order of $\beta /\log (\beta )$. We also saw that in the limit $\beta \ll 1$ corresponding to a very dense reaction product (that takes up essentially no volume compared to the agent it is replacing), the system becomes independent of $\beta$, since the cleanser–product mixture simply flows at the velocity of the reacting interface, $s'(t)$. Furthermore, we found that in the dilute-cleanser limit of $\gamma \ll 1$, the leading-order decontamination behaviour is independent of $\beta$, and so of the fluid flow, since in this limit the cleanser-transport process is quasi-steady and dominated by diffusion.

Since the fluid flow cannot dominate the transport of chemicals, this transport of cleanser is relatively slow (on the diffusive timescale), and limits the decontamination process for deep spills. Although this is the decontamination method often used in practice, it may not be the fastest method of decontamination. It is possible that by forcing cleanser material into the porous structure (perhaps by pumping it in at high pressure) the decontamination time could be reduced. Especially in the case that cleanser phase is less viscous than the agent, this pumping in of the cleanser may result in viscous fingering instabilities, increasing the surface area of the reacting interface. However, pumping in this way may not be practical, or may simply result in displacement of the agent further into the porous material, essentially increasing the distance the cleanser chemical must diffuse in order to neutralise it.

Although we include fluid-flow effects, our decontamination model is less general than that of [4] in that we assume the reaction product is soluble only in the cleanser phase, and that the agent remains neat at all times. An important next step will be to allow the reaction product to also be soluble in the agent, in particular since this may result in either a flow in the agent fluid as well as the cleanser, or perhaps a stationary or reversing reaction front, as the agent phase is diluted. Another important decontamination scenario to investigate is when the reaction product has low solubility in both the agent and cleanser phases, and so forms a barrier between the two chemicals, severely limiting the reaction at the interface. Additionally, we should relax the assumption made in this paper that the cleanser and product chemicals have the same density, and allow the chemical-product mixture to have a chemical composition-dependent density. In this paper,we assumed that the fluids saturate the pore space of the porous medium and occupy distinct regions. In practice,the agent does not always saturate the pore space, but instead occupies some fraction of the pore space. In such cases, the capillarity drives the flow of agent and modellers must consider multiphase flows of agent, air and cleanser within the porespace. Changes of phase of the agent and/or cleanser could also be incorporated, such as agent vaporisation within the porous material. Density-driven flows could also occur for other pore-scale configurations of the two fluids, such as those studied in [6, 26], and the effect of fluid flow in these cases should also be investigated. Furthermore, the assumption that the geometry is one-dimensional should also be further investigated. Especially in the case of additional buoyancy-driven flows, or effects due to differences in viscosity, a two or three-dimensional geometry would be necessary to capture the flow effects.

The modelling and analysis in this paper provide a new understanding of the role of swelling or contraction of material during the reaction on the decontamination of porous media, infiltrated with a hazardous chemical warfare agent. Our work extends existing models in this area, such as [4], to include the advective–diffusive transport of cleanser chemical. In particular, we have quantified how an expansion of the material during the reaction results in slower decontamination, while the contraction of the material leads to an increase the decontamination rate. Our analysis suggests that using a cleanser with a higher density than the agent or that results in higher-density reaction product will reduce the decontamination time. We note that it is the volume occupied by the reacting and product chemicals that is important here, not the density of any solvent carrier for the cleanser phase. Although, in practice, it may not always be feasible to use high-density cleanser chemicals, it is an effect worth considering when choosing the best cleanser for a given agent. Additionally, the different flows of material that result for different densities of cleanser chemicals may require different practical remediation: more cleanser fluid may need to be frequently reapplied at the substrate surface in order to replace that drawn into the material by chemical contraction during the reaction, or conversely the swelling fluid containing the (potentially still harmful) reaction product may be expelled from the substrate surface and must then be carefully removed.

Appendices

Appendix A: Estimation of physically relevant parameter values

The majority of cleanser solutions are in aqueous solution, with density approximately $10^3\,$kg m$^{-3}$. Meanwhile, the densities several agents are listed in Table 1. We see that agents may be both more and less dense than an aqueous cleanser solution, and thus, both cases $\beta <1$ and $\beta >1$ are likely to be of physical relevance.

Common decontamination cleansers include bleach [28] and calcium hydroxide solutions [31]. In the remainder of this appendix, we estimate the dimensionless parameters $\beta , \gamma$, and $\lambda$ given by (16), for the decontamination of a particular agent, sulphur mustard, with either a bleach or calcium hydroxide solution.

Sulphur mustard, or HD, is generally relatively pure [29] and so our modelling assumption of a neat agent is reasonable. Sulphur mustard is an oily liquid, with low solubility in water [29, 31] and so when decontaminated with an aqueous bleach (containing hypochlorites) or calcium hydroxide (portlandite) solution, the cleanser and agent fluids are indeed largely immiscible. Furthermore, as in [4], the reaction products formed in the decontamination of sulphur mustard with either bleach or calcium hydroxide, are fairly insoluble in the agent (with a small partition coefficient ${{\mathcal {K}}}$ in the notation of [4]), and thus, our assumption that the agent remains neat is reasonable for these reactions.

We list approximate dimensionless parameter values for these chemicals in Table 2. From these values,we may compute $\gamma =0.8$ for the bleach solution and $\gamma =0.03$ for the calcium hydroxide solution. The density of the aqueous cleanser phase in either case is expected to be around $10^3\,$kg m$^{-3}$ so that, as in Table 1, in either case $\beta \approx 1.3$. The reaction coefficient ${\bar{k}}$ for the decontamination reaction in either case is difficult to find in the literature. Discussion in [31] suggests that the rate limiting step may have a rate $\kappa \approx 10^{-3}\,$s$^{-1}$ or “somewhat” smaller. From this,we estimate ${\bar{k}}\approx \bar{L}\bar{\chi }\kappa$ to be around order of $10^{-9}$–$10^{-8}\,$m$^4\,$mol$^{-1}\,$s$^{-1}$ or smaller. The dimensionless parameter $\lambda$ may therefore be in the range 1–$10^2$ or smaller.

Table 1 Densities of various liquid chemical warfare agents

Table 2 Approximate dimensional parameter values, with a sulphur mustard (HD) agent, and two different aqueous cleansers, namely a 5% bleach solution (bleach) and calcium hydroxide (Ca(OH)$_2$)

Appendix B: Details of the numerical method

In this appendix, we give details of the spatial discretisation used in our method-of-lines approach to the numerical solution of (18). We construct a uniform mesh $\{\eta _1,\eta _2,\ldots ,\eta _n\}$ on the interval [0, 1], where $\Delta \eta =\eta _{i}-\eta _{i-1}=\frac{1}{n-1}$ is the mesh size, noting that $\eta _1=0$ and $\eta _n=1$ are the boundary points. We note that $\eta _i=(i-1)\Delta \eta$. We let $g_i$ denote the approximate of g at $\eta =\eta _i$.

The boundary condition (18e) at $\eta =1$ gives an ODE for s

$$\begin{aligned} \frac{\text {d}s}{\text {d}t}=\gamma \lambda g_n, \end{aligned}$$

(B1)

and the boundary condition (18c) at $\eta =0$ fixes

$$\begin{aligned} g_1 = 1. \end{aligned}$$

(B2)

We discretise the equation (18a) in space to find temporal ODEs for $g_2,\ldots ,g_n$. We use a second-order central difference scheme for the diffusive term, and a first-order upwind scheme [34] to approximate the advective term, the form of which depends on the sign of the (spatially uniform) velocity, or equivalently (since $s_t>0$) on the sign of $1-\beta -\eta$. Specifically, the first-order derivative in the advection term of (18a) is approximated by

$$\begin{aligned} \left. g_\eta \right| _{\eta =\eta _i} \approx \frac{g_{i}-g_{i-1}}{\Delta \eta } \quad \text {if }1-\beta -\eta >0 , \end{aligned}$$

(B3a)

$$\begin{aligned} \left. g_\eta \right| _{\eta =\eta _i} \approx \frac{g_{i+1}-g_{i}}{\Delta \eta } \quad \text {if }1-\beta -\eta <0. \end{aligned}$$

(B3b)

We note that if $\beta \ge 1$, the velocity $ss_t(1-\beta -\eta )$ is negative for all $\eta \in [0,1)$, and so we use (B3b) to approximate $g_\eta$. For $i=2, 3,\ldots , n-1$, $g_i$ satisfies

$$\begin{aligned} \frac{\text {d}g_i}{\text {d}t}=\frac{1}{s^2}\left( \frac{g_{i+1}-2g_i+g_{i-1}}{\Delta \eta ^2}+s\dot{s}\left( (i-1)\Delta \eta -\left( 1-\beta \right) \right) \frac{g_{i+1}-g_{i}}{\Delta \eta }\right) , \end{aligned}$$

(B4)

where $\dot{s}$ is given by (B1). Using a ghost point to discretise the boundary condition (18d), we find the final equation

$$\begin{aligned} \begin{aligned} \frac{\text {d}g_n}{\text {d}t}=\frac{1}{s^2}&\left( \frac{-(\lambda +\beta \gamma \lambda g_n)sg_n-g_n+g_{n-1}}{\Delta \eta ^2}\right. \\&\quad -s\dot{s}\left( (j-1)\Delta \eta -\left( 1-\beta \right) \right) (\lambda +\beta \gamma \lambda g_n)sg_n\biggr ), \end{aligned} \end{aligned}$$

(B5)

for $g_n$, completing the spatial discretisation in the case $\beta >1$. The ODEs (B1), (B4) and (B5) must be solved simultaneously, along with the constraint (B2).

For $\beta <1$, the sign of $1-\beta -\eta$ is different for the two regions $\eta <1-\beta$ and $\eta >1-\beta$ of the domain. For $i\in \{2,\ldots ,n-1\}$ such that $\eta _i<1-\beta$, we use (B3a), and so have the discretisation

$$\begin{aligned} \frac{\text {d}g_i}{\text {d}t}=\frac{1}{s^2}\left( \frac{g_{i+1}-2g_i+g_{i-1}}{\Delta \eta ^2}+s\dot{s}\left( (i-1)\Delta \eta -\left( 1-\beta \right) \right) \frac{g_{i}-g_{i-1}}{\Delta \eta }\right) . \end{aligned}$$

(B6)

If we happen to have an $i\in \{2,\ldots ,n-1\}$ such that $\eta _i=1-\beta$, there is no advective term and we use

$$\begin{aligned} \frac{\text {d}g_i}{\text {d}t}=\frac{1}{s^2}\left( \frac{g_{i+1}-2g_i+g_{i-1}}{\Delta \eta ^2}\right) . \end{aligned}$$

(B7)

Finally, for i such that $\eta _i>1-\beta$, we use the same form of discretisation (B4) as in the case of negative flow, and likewise find the same form (B5) for the ODE for $g_n$.