At last! Parameter Estimation
With a model available now to simulate a reactor, we can estimate the model parameters. I used Mathcad's minerr routine with the Levenberg-Marquardt option to minimize the errors between the data curve and the estimated curve.
The Results for Model 1
The results are amazing! The parameter estimates for f and alpha were exact in every case that I examined. Those cases spanned 0.01 < f < 0.99 and 1 < alpha < 2. In all cases, the standard initial guesses were f = 0.5 and alpha = 1.
A perfect model
The main reason the results were so accurate is that the model for estimation was perfect. It was an exact duplicate of the model that simulated the data. No errors were included. Nevertheless, I expected some inaccuracy.
Errors can be of two types, measurement error and model error. Measurement error might be present due to a calibration error. Another possibility is that the measured signal is not linearly related to the tracer concentration.
I don't think the bias due to calibration error would adversely affect the estimation unless it is an error with just one of the detectors. It may be necessary to adjust one of the detector output signals (or the recorded data) until the area under the input and output curves are equal.
Most likely there will be model error. In that case, the estimates will only represent the data as best they can with the assumed model. Model error can be either macro or micro. An example of a macro error would be the existence of another branch or flow volume, or of cross flow. An example of micro error would be the chaotic flow of bubbles from the bed or of flowing eddies with non-uniform concentrations. The presence of micro error will appear as "noise" in the data, i.e. fluctuations about a mean curve. Such noise could obscure some subtle features of the response curve, making it difficult to obtain accurate parameter estimates.
As I proceed with other more complex models, I will try estimation with a more basic model to see how the method performs with macro model error. For micro error, I need data from an actual tracer study.