Diffusion Model

Reasoning why we use the model.

For the initial experiments a model for the diffusion in the Petridish could be useful. The structure is drew in Figure 1.

Figure 1: Full structure of diffusion device.

Mixed boundary condition is used which is a combination of the Dirichlet and Neumann conditions.

With at x=0 a so called Dirichlet boundary condition meaning that:

With Psat being the saturation pressure of the specific compound in this case being Methyl Nicotinate. Values for compounds can most of the time be conveniently be found in the chemical engineering Toolkit called Aspen. Generally data for different concentrations in different compounds at different temperatures can be extracted from Aspen. For Methyl Nicotinate this values is 0.1 at room temperature. For x=0.01 one can expect a so called von Neumann boundary which in this case means that:

Effectively meaning a gas wall for the system. The molecules bounce to the wall back into the system. A van Neumann boundary is a way to give a mathematical workable structure to that. For the problem definition to be complete and start programming a code to compute the answer We only need one more value being the diffusion coefficient. For this an empirical relation from the (Chen, 1962)

Now we have a complete and full problem and we will solve it using a numerical approach. I found a nice derivation on the internet but it was for temperature so it is in structure the same. So just pretend like T=C and a=D.

And you probably guessed but my value for h is ok in the model and in my code this translates to the line: dcdt(i,1)= a*x(i-1,1)+a*x(i+1,1) -2*a*x(i,1);

and YES a=h here and YES that IS confusing considering the other above statements. What I basically did is grab a bunch of Ordinary differential equations and couples them So that it becomes a PDE ( partial differential equation ). H*time step should also not be bigger than 0.5 and that does not happen, so the solution is stable.

For the boundaries I did the following:

dcdt(n+1,1)= a*x(n,1)+a*x(n-1,1) -2*a*x(n+1,1) ;

Looping the two last boxes to eachother in that way creating the von Neumann boundary. For the Dirichlet boundary I Just kept on setting the C value at x=0 to Psat.

Analytical solutions for this problem appeared a little more troublesome than expected. There are solutions ( in Sum row form ), but I was anticipating a nice function. I will get a book at the library to see if there are any better ones, but for now this works out fine. I can also use a Cranck-Nicolson approach to the problem to have another numerical solution for verification. And last but not least. The results:

In this graph we can see the result with at the 0 to 60 axis the 0 – 0.01 distance dimension. Thus the other 0 – 1500 is the time dimension ( 0 – 100 seconds ) so this graph shows that after Approximately 100 seconds we can expect steady state at the x=0.01 boundary.

References

Chen, N. H. (1962). New Generalized Equation for Gas Diffusion Coefficient. J. Chem. Eng. Data, 37–41.

Snifferometer

One of the other goals of the diffusion modeling was to model the device which we intended to build for the project. We made use of the finite element analysis simulator Comsol Multiphysics[3]

Figure 1: Snifferometer - Device with the modified yeast cells for sensing

Different models were analyzed for their suitability before coming to the final design in Figure. The device at the bottom has a membrance preventing the yeast coming in direct contact with external substances, above which is a layer of agar in which yeast is placed, the nutrients for it's growth is fed in through a channel along the sides of the tube using which the yeast cells can also be replaced as and when needed. A micro-optrode is then used to sense the photons emitted by the fluoroscent proteins.

Saliva Model

File:Test1.swf