Post date: Jul 26, 2013 5:53:28 PM
We switched projects and now Marty and I are teaming up to correct the accuracy of temperature measurement with thermocouples. Thermocouples are 2 dissimilar metals put together and joined at a junction. When the junction is heated, a Seebeck voltage is observed. The Seebeck voltage is very close to proportional to the temperature of the junction, and the thermocouple uses that fact to measure the temperature by first measuring the voltage. However, thermocouples are pretty inaccurate, and I have to devise a method of calibration for them.
That is where the arrhenius equation for diffusion coefficients come in:
D = (D_0) ( e^(E / (R * T)) )
D = diffusion coefficient
D_0 = diffusion coefficient at infinite temperature
e = natural constant
E = activation energy
R = universal gas constant
T = temperature
The basic idea is that we can calculate the diffusion constant for the process of Argon diffusing through silicate glasses at different temperatures and lengths of time. Using the data points, we can generate a graph and see what kind of calibrating algorithm would work best.
However, the calculation of diffusion constants is a little complex. It depends on the shape of the silicate glass - in other words its geometry. The 4 usual geometries include
Sphere
Infinite Cylinder
Infinite Plane
Cube
Each one has a different calculation for the diffusion constant. For example, the sphere's diffusion constant calculation goes like this:
f = 1 - (6/pi^2) * sum( e^(-n^2 * pi^2 * D * t / a^2) / n^2, n, 1, infinity)
Evaluating the infinite sum to solve for D is impossible, so we use an approximation which for spheres is this:
f = 1 - (6/pi^2) * e^(-pi^2 * D * t / a^2) for 0.85 < f < 1
f = (6/pi^(3/2)) * (pi^2 * D * t / a^2) ^ (1/2) - (3 * D * t / a^2) for 0 < f < 0.85
Using the approximation allows us to find D from the fractional loss of argon f over a period of time t through a sphere of radius a.
Looks and seems complicated, but this calculation is used quite often and is a very standard approach according to my mentor Marty.
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I've decided to write a calibration algorithm in Python just to test out the SciPy module and perhaps try my hand at some of its visual graph-making tools.
Python is not my language of choice due to slow speed, but it's started to grow on me with its clean syntax, simplicity, interpreter, and precision.