# DOE to optimize a chemical reaction with impurity forming in series

This question contains chemistry, but the main focus is DOE design--hence posting on the engineering stack exchange.

The problem is a chemical reaction:

• A + B -> Product
• Product + B -> Impurity

The goals are

1. to minimize the impurity.
2. to minimize leftover A.

Reacting the mixture for a long time will get rid of A but increase the Impurity.

Reducing B will decrease the Impurity but increase leftover A.

To optimize the reaction, I'm planning a DOE with temperature, reagent amounts, and reaction time as variables.

I used software to generate a central composite design. However, the reaction time inputs for each experiment aren't physically reasonable. Look at some of the experiments:

+-------------+---------------+-------------------+
| Temperature | Reaction time | Amount of B       |
+-------------+---------------+-------------------+
| 20 C        | 4 hrs         | 1 mole equivalent |
+-------------+---------------+-------------------+
| 50 C        | 4 hrs         | 1 mole equivalent |
+-------------+---------------+-------------------+


At 20C, I think the reaction will proceed slowly enough that it won't be done after 4 hours. However, at 50C, I think the reaction will be done in less than an hour. The reaction output at 4 hours will be useless--all the B will have been long reacted (i.e. the maximum amount of impurity for that temperature/B input will have formed).

How can I handle the fact that reaction time inputs cannot be assigned randomly?

I know this is a while back, but my answer is generally applicable. From experience, these sort of experiments can make use of reaction time as a "free" variable. Before the DOE is even planned, take samples of the reaction at the three temperatures 20, 35 and 50 degrees at regular intervals. In this way you'll get a feel for the reaction rates. In fact, if you use an internal standard, you could generate actual rates. This reaction time information can be plugged into the DOE. My advice is to take more than three time samples because you can then expand your statistical analysis if necessary e.g. Design Expert allows you to add extra lines to your DOE or you can use a "historical data" approach.
So long as the timepoints are appropriately spaced, you should get a good model. When you have factors that are time bound or hard to change, then a split plot method can be used if that factor is not randomised to check there isn't a false analysis.

You could specify a bit more your knowledge on the system. Which is the limiting reagent? Do you expect a first-order or second order reaction?

It is my understanding that A is in small excess and that you have a certain knowledge of the reaction rates (as you can forecast that the reaction will be completed at 50 degC by not at 20 degC) You can define dimensionless time in a similar way to the Damköhler numbers such as:

$\tau = k\,t$

where $k$ is your expected rate constant and $t$ is time. This expression is valid for first-order reactions, but you can define similar one for arbitrary reaction rate expressions. By this approach, your are somehow combining temperature and time into a single variable and the variables for your DOE would be the dimensionless time $\tau$ and the reagent amounts.

Similarly, you can use "expected conversion of B" as an input variable instead of $\tau$.

• Thank you. I see what you mean for dimensionless time. I don't have exact knowledge of the rate constants--just some qualitative experience with the system. The system is a little messy to model. If I could know the rate constants, I'd model instead of a DOE. In any case, I think I will use conversion of B as an input variable instead of time. Jul 17 '17 at 23:12