How well does the calorimetric method perform in the presence of significant model error? To answer that question, I applied the method using a power law model with the reaction system data still produced by the original Langmuir-Hinshelwood (LH) competitive adsorption rate expressions.
The methodology used was the same as before, i.e. the single reactions were studied first and then the dual reactions were studied using parameters from the single reaction results. The best results were obtained with the following form for the rate expressions of either reaction:
The hydrogen concentration is not included in the above expression because the concentration is constant during a run. The power for the hydrogen concentration will be covered in the next blog.
The form above was selected in favor of a more traditional power law model which would have individual powers for both TMP1 and TMP2. Such a model would not accommodate the absence of the non-reacting, rate hindering pentene. However, the 0.35 power was determined from traditional power law models of the individual reactions.
With both reactants present, the above model resulted in the hydrogen flow curve shown below. As a reminder, the catalyst is charged at 15 min, so the reaction parameters only affect the remaining time of the run.
That doesn't appear to be a very good fit. Let's look at the fit to the concentration curves.
For design, the power law model might be adequate. It basically captures the main time constant and the relative rates for TMP1 and TMP2 hydrogenation. This information would allow reactor sizing and design of the product separation and recycle equipment. As we should expect, the model imperfection resulted in a lack of fit. However, the analysis method was still able to converge to a reasonable model.
The starting guesses for the parameters, particularly when both reactants were present, had to be closer to the optimum values for this power law model than for the LH models. However, for the single reactions, it was easy to adjust the guesses by a factor of two and rerun the optimization routine until convergence with a decent hydrogen flow plot was obtained. Results for bad guesses often resulted in the flow curve spiking high and then going to zero for most of the reaction time. For the combined reactions, the results from the single reaction studies provided good parameter estimates.
I have used power law models for design because of the simplicity of the parameter estimation process. However, the proposed dynamic method makes it easy to vary concentrations and determine LH parameters in a single run. Also, the Mathcad minimization routine appears to be very efficient even with the nonlinear LH model. Thus, I would definitely recommend exploring LH type models with this analytical method.