POSSIBLE SOLUTION(S)

Idea - not using differential equation model, but direct curve fitting. We turn to this laters.


We could solve just the differential equation for a(t) - exponential decay - and then use this with antidifferentiation (e.g., integrating factor) to get b(t) and then easily get c(t).


We set up differential or rate equations for the chemical reactions assuming rate of reaction is proportional to amount of chemical present.

We assume the reactions are first order.

We solve the differential equations for a, b, and c as functions of time with the rate constants k1 and k2 in each.

We grab off each solution:

We input and plot all the data.

We use the data to determine the rate constants k1 and k2.

We formulate a least squares function in the constants (now variables) k1 and k2 using the data from the chemicals -- first separately and then together.

We formulate a least squares function in the constants (now variables) k1 and k2 using the data from the chemical a.

We formulate a least squares function in the constants (now variables) k1 and k2 using the data from the chemical b.

We formulate a least squares function in the constants (now variables) k1 and k2 using the data from the chemical c.

Now we formulate a least squares function in the constants (now variables) k1 and k2 using the data from ALL the chemicals.

The rate constants k1 and k2 seem to differ depending upon which of the data sets we wish to "satisfy." Thus we take the values which satisfy "all" of the data.

We plug the values for the rate constants into the solved a, b, and c functions.

And plot the solutions - over the data.

This indicates we have a good model for the fit is almost perfect.

We optimize the objective function - no energy costs.

We optimize the objective function - with energy costs.

We now determine the sensitivity of the operation to energy costs.

Another approach from start -

We could fit polynomials to the data