The total concentration (Ct) of chlorine in the concrete exposed to water containing chloride varies over time according to the equation known as Fick's second law
∂Ci/∂t = Da∂2Ci/∂x2
The integration of the equation provides the following:
C (x,t )= Cs[1-erf (x/2 √Da·t ]
Where :
• C (x,t) is the concentration of chloride ( kg/m3) at a distance x (m) to the generic instant of time t (s);
• Cs is the concentration of chloride on the surface (kg/m3);
• Da is the apparent diffusion coefficient (m2/s)
For Cs and Da determination , experiment is carried out by which the chloride concentrations Ci inside the concrete are measured at different distances xi from the surface after a certain time t of immersion in a 3% NaCl solution . The desired parameters Cs and Da, are obtained by Curve-fitting to the error-function solution to Fick’s 2nd law.
erfc is available in excel, set it up with Cs and Dc in specific cells
then to get the curve fit you can produce a column of predicted values next to the column of actual vaules.
Next column is actual - predicted
Next column is this value ^2. Have a cell that sums this up
Then use the solver function to minimise the sum, by changing the Cs/Dc values. You may need to put a constraint in that they don't go negative.
As an addon to the previous answer, if you're looking at profiles on a real structure, they rarely follow a specific fixkean curve, one of the things I did was produce a cumulative frequency of the values at any set depth, then fit a fickean curve to key percentiles, eg 5, 25, 50, 75, 95 percentile, then fit the curve to percentile data and get a series of chloride diffusion parameters for each profile.
As mentioned could be set up in Excel, perhaps using an iterative approach to optimizing solution.
Non-linear fitting is provided in a variety of stat packages including DATAPLOT available from NIST:
https://en.wikipedia.org/wiki/Dataplot
The LMFIT library in python is an easy to use well-documented library for data fitting. Here is a link to the examples gallery: https://lmfit.github.io/lmfit-py/examples/index.html.
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