If you have are making a design with a factor you are certain will have a quadratic relationship against a response but you have a wide range, can it be helpful to define a 3rd order model effect for that factor to better identify and model curvature.

I work in purification and we often deal with a recovery vs. purity balance. This relationship is usually quadratic around the optimal, where at a low condition, we see 100% recovery and 0% purity, at an optimal condition, we may see a 90% recovery and 90% purity, and at a high condition, we may see 90% purity, and 0% recovery (Sometimes 0% recovery also leads to inaccurate purity data).

I understand a 2nd order effect should be able to effectively model a quadratic relationship, however if I am uncertain how far to extend a design space, I worry that I could effectively overshoot by extending too much (Where the centerpoint is far beyond the optimal so that recovery is ~0% and so is the high condition), or undershoot by constricting the space too much (where the centerpoint is far before the optimal and leads to 100% recovery and 0% enrichment). I've had a few situations now where DoEs performing very well when characterizing a design space we are already aware of but tend to be inaccurate when I try to extend them to new design spaces where we are uncertain of the appropriate ranges. In the scheme of things, the expected total model curve would be a bell shape (With tails on each end) which I guess would be representative of a 4th order polynomial??? I am not really interested in characterizing that entire space as I just need a model that effectively predicts the range around the optimal, however with my current issue of not knowing where the optimal exists I am trying to determine a solution for future studies.

By increasing the order effect of this one factor, it creates more levels within the space so I assume it gives it a better opportunity to land on some intermediate conditions which flanks the optimal.

Is this thought process correct? I tested this within a DoE and it did not add any additional experiments but I know typically if that's the case, you are giving something else up.