Model 2: adding cross flow
Cross flow was added to Model 1, requiring a new parameter.
Parameter estimation results
Again, I was surprised with the excellent results: kgex was estimated exactly, along with f and alpha, for every case.
What was truly amazing was the range of kgex examined spanned four orders of magnitude . Expressed as a fraction of the total air feed rate, the range was 0.0001 to 0.4. So even very little cross flow can be detected by the method when the model is perfect. However, a very small exchange rate means that Model 1 would be sufficient.
Effect of Model Error
I used Model 2 to generate the data, but I used Model 1 to perform the parameter estimation. The results of one case are shown below. Note this case included a substantial amount of cross flow. The parameter estimates were severely affected by the model error.
Another case was conducted with a kgex that amounted to 1% of the air rate. The results are shown below.
With the lower kgex, the two models are more similar and thus the parameter estimates are better.
The inverse model "error"
The previous cases were concerned with estimation when a flow feature has been ignored. Now let's look at the opposite case: including a feature during estimation that doesn't exist in the reactor system. The cross flow was set to zero, thus making the model like Model 1. Model 2 was used for the parameter estimation. The results are below.
The estimation correctly found that cross flow wasn't present. In general, the best strategy is to estimate using a model that will capture the features that you suspect may be present. If not all features are present, the estimation should drive the associated parameters to zero or too a small number.