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Diffusion
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Introduction

The main goal of AROMA is to find the origin of a certain target molecule, i.e. its source. Therefore it is crucial to understand what kind of sources can be found in practical applications and how the respective molecule of interest diffuses
from this source. The diffusion behaviour will clearly affect how close to the source the robot has to be in order to being able to sense the presence of the compound.

In order to evaluate the distance at which a source can be detected (and thus the feasibility of an application) we developed a model for the three-dimensional diffusion of specific compounds from their source.

In the most basic form, to compute the molecule concentration $c(\vec{x},t)$ at a point in space, one has to find a solution to Fick's second law:

\begin{equation}
\frac{\partial}{\partial t} c(\vec{x},t) = D * \Delta c(\vec{x},t)
\label{eq:Fick_gen}
\end{equation}

Furthermore the solution to the above equation also has to satisfy some source-dependent boundary conditions. We differentiated between three kinds of scenarios:

- The diffusion from an instantaneous point source, representing for instance the case when a balloon filled with a certain molecule explodes
- The diffusion from an infinite point source, representing the case of a leakage in a gas pipeline
- A source, diffusion and degradation (SDD) scenario, representing the case where a landmine buried for a long period of time in the ground

Solving for the finite source

In the scenario of a finite source, meaning there is an initial amount of the compound at the source which gradually spreads out.
In 1D, Fick's second law (\ref{eq:Fick_gen}) is:

\begin{equation}
c_t = D * c_{xx}
\label{eq:Fick_1D}
\end{equation}

The fundamental solution to this equation (\ref{eq:Fick_1D}) is well known (REF[1],REF[2]):

\begin{equation}
c(x,t) = \frac{M}{\sqrt{\pi4Dt}}*e^{-\frac{x^2}{4Dt}}
\label{eq:Fick_1D_sol_f}
\end{equation}

where x is the distance from the source, M is the initial concentration at the source and D is the diffusivity constant.

This solution satisfies the following boundary and initial conditions:

- $c(0,x) = M*\delta(x)$
- $\int_{-\infty}^\infty c(x,t) dx = M$
- $\lim_{t\rightarrow 0^+} c(x,t)=0 \quad \text{for} \quad x\neq 0$
- $\lim_{t\rightarrow 0^+} c(0,t)=\infty $

It is easy to verify that this fundamental solution prefectly corresponds to the case of having a finite source! As the 3-Dimensional diffusion equation is linear (\ref{eq:Fick_gen}), it is obvious that (\ref{eq:Fick_3D_sol_f}) satisfies the
three dimensional case, given a finite source:

\begin{equation}
c(\vec{x},t) = \frac{M}{(\sqrt{\pi4Dt})^3}*e^{-\frac{x^2+y^2+z^2}{4Dt}} \Leftrightarrow c(r,t) = \frac{M}{(\sqrt{\pi4Dt})^3}*e^{-\frac{r^2}{4Dt}}
\label{eq:Fick_3D_sol_f}
\end{equation}

where r is now the distance from the point source.

Solving for the infinite source

The scenario of having a leakage in a gas pipeline can exemplarily modelled by an infinite source because there will be a constant generation at the place of the source. Again, in this scenario the temporal evolution is of great importance.

The solution above (\ref{eq:Fick_3D_sol_f}) only satisfies the boundary conditions of a finite source but it is not trivial to manipulate the equation such that it satisfies the necessities when dealing with an infinite source.

Being able to solve for the case of an infinite source, we first express the Diffusion law in spherical coordinates. Under the assumption of isotropic diffusion one can drop the angular dependencies which yields to:

\begin{equation}
\frac{\partial c(r,t)}{\partial t} = D * (\frac{\partial^2 c(r,t)}{\partial r^2} + \frac{2}{r} \frac{\partial c(r,t)}{\partial r})
\label{eq:Fick_spher}
\end{equation}

Using the coordinate transformation $u=c*r$ [REF 3] which again yields to $u_t = D*u_{rr}$ and following the steps as shown in ([REF 1]) yields to:

\begin{equation}
c(r,t) = \frac{c_0*a}{r}*[1-erf(\frac{|r-a|}{\sqrt{4Dt}})]
\label{eq:Fick_genc_sol}
\end{equation}

where $erf$ represents the error function, a is the outer radius of the source and r is the distance from the center of the source.

Equation \ref{eq:Fick_genc_sol} yields the intensity distribution for the case of an infinite source, providing concentration $c_0$ at a nonzero radius a as this solution now satisfies:

- $c(a,t)=c_0$
- $\lim_{r\rightarrow \infty} c(r,t)=0$
- $c(r,0)=0 \quad \forall r \in (a,\infty) $

To sum it all up, we used equation \ref{eq:Fick_3D_sol_f} and \ref{eq:Fick_genc_sol} to model the temporal diffusion of the molecule of interest, depending on our assumptions on the source.

Solving for the source, diffusion and degradation (SDD) scenario

In its initial form, Fick's law describes the diffusion but without any additional effects. However, often also the degradation of the concentration has to be taken into account, especially when considering long-term scenarios such as
landmines which are buried into the ground for multiple years.

When adding the degradation term, the differential equation changes to:

\begin{equation}
\frac{\partial c}{\partial t} = D*(\frac{\partial^2 c(r,t)}{\partial r^2} + \frac{2}{r} \frac{\partial c(r,t)}{\partial r})-k*c(r,t)
\end{equation}

where k equals the degradation factor and D is the diffusivity constant. The boundary conditions are:

\begin{equation}
-D*\frac{\partial c}{\partial r}(R,t) = Q
\end{equation}

\begin{equation}
c(r,0) = 0 \qquad \forall r \geq R
\end{equation}

Those boundary conditions represent a constant flux Q (i.e. generation) at the boundary of the radially symmetric source with radius R and the zero initial concentration before the whole process starts. REF 4.

Since we consider long term scenarios, we will assume that the concentrations reached steady state, denoted by adding subscript ss, meaning that the differential equation as well as the boundary conditions simplify to:

\begin{equation}
D*(\frac{\partial^2 c_{ss}(r)}{\partial r^2} + \frac{2}{r} \frac{\partial c_{ss}(r)}{\partial r})-k*c_{ss}(r) = 0
\end{equation}

\begin{equation}
-D*\frac{\partial c_{ss}}{\partial r }(R) = Q
\end{equation}

\begin{equation}
c_{ss}(\infty) = 0
\end{equation}

As shown in REF 4, the solution to this problem is:

\begin{equation}
c_{ss}(r) = \frac{Q*R^2*exp(\sqrt{k}*(R-r)/\sqrt{D}}{r*D*(1+\sqrt{k}*R/\sqrt{D}}
\end{equation}

Furthermore, one can calculate whether the assumption of actually having reached steady state is valid through calculating the local accumulation time $\tau(r)$ which provides an estimate for the time required to reach steady state at a given
location [REF 4]. $\tau$ is given by:

\begin{equation}
\tau(r) = \frac{r-R}{2*\sqrt{Dk}}+\frac{R}{2k(R+\sqrt{D/k})}
\end{equation}

Conclusion

To summarize, the modelling of the diffusion behaviour of the source is the first necessary step in order to understand how the robot will behave. Our different source models allow us to model various different situations from gas leaks which
were just created a few hours before to landmines which are buried in the ground for years.

Throughout our project, we focussed on optimizing AROMA for sensing and locating landmines. Therefore AROMA can be used to clear minefields in order to lower the number of of civilians killed or injured by them. For instance in 2016, the
Monitor recorded 8,605 mine/ERW (explosive remnants of war) casualities from which 78 percent affected civilians.[REF 6].

Almost all of the landmines contain TNT and therefore they also releas by products like DNT (Dinitrotoluene) in the air. We found out that DNT is the best target molecule for reaching our goal and therefore all of the following modelling steps
will be based upon this molecule.

Furthermore the task to detect landmines/DNT perfectly corresponts to our SDD diffusion scenario. The plot below shows the steady state concentration generated by a single mine placed at the origin. It takes approximately 115 days till this
steady state concentration is reached at 30m from the source. Therefore the assumption of this scenario is valid as we expect that clearing the field rather happens years after the deployment.

In order to understand how the molecules are taken up by our robot we needed to create a model of the bubbling. Check it out on our bubbeling webpage.

Comment

Of course one could have also just computed a numerical solution by for instance using a forward Euler approach based on equation \ref{eq:Fick_gen} and approximating the second derivative via the central difference as: $\frac{\partial
f}{\partial x^2} \approx \frac{f(x+h)-2f(x)+f(x-h)}{h^2}$

Bibliography

- [1] "Chapter 3 of Lecture Partial Differential Equations and Mathematical Biology", by Professor Junping Shi, Spring 2004
- [2] "Diffusion from an Instantaneous Point Source with a concentration-dependent coefficient", by R. E. Pattle, 24.7.1958
- [3] "The mathematics of diffusion", by J. Crank, Second Edition, 1975
- [4] "Numerical differentiations Spreadsheet Calculator" by Kim Gaik Tay et al., 2013
- [5] "Local accumulation times for source, diffusion, and degradation models in two and three dimensions", by Peter V. Gordon, The Journal of Chemical Physics, 2013
- [6] https://reliefweb.int/report/world/landmine-monitor-2017