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Using a cooling profile

Phil_Kay

Hi,

A customer asked me this. I think it might be quite simple but my brain is struggling on this for some reason.

They have done a very simple DoE. 2 factors: Temperature, T, and time, t. The response is yield.

The reason for doing this is to understand stability of the product (therapeutic cells) between processing stages. Basically, storing at high temperatures for a long time causes a loss in yield.

They did the experiment at small scale so they could heat/cool more or less instantaneously. They have a good model from the DoE that will tell them what yield they should expect if they hold the cells at any temperature, T, for any time, t. They have used MC simulation in JMP to look at variation in yield given typical variation in T and t.

At production scale cooling takes longer. They have a typical cooling profile (T vs t). They would like to use this cooling profile and their DoE model to predict what yield they should expect at the end of the cooling stage.

Is that clear?

Hope you can help,

Phil

Accepted answers

Mark_Bailey

In addition to Mike's suggestion, they could use the S = f(T,t) linear model and calculus to determine the loss in the slower processing step. They could use the numerical integration Integrate() function in JSL or an external tool like WolframAlpha for the general symbolic solution. The linear model is pretty simple, so I think I would symbolically integrate the linear model, then plug the fitted parameter values into the general solution as a column formula, then use Profiler to answer questions.

All comments

MikeD_Anderson

This sounds similar to a thermal budget situation I helped a customer out with a while back. The trick I used was setting up DOE with constraints that prevented the second stage of the thermal process from being cooler than the first (it could increase in temperature or stay the same). The time dimension came from how quickly the part was moved between stages of the heating process. In your example, I think you'd just have to switch the constraints.

An example DOE is attached.

Sequentially Constrained I-Optimal Design.jmp

Mark_Bailey

MikeD_Anderson

Riffing off @Mark_Bailey a little - my gut says you should be able to use FDE to consolidate the thermal profile information into something a standard DOE should handle... nothing to back that up, just something that popped into my head while reading Mark's response.

M

Phil_Kay

Thanks for your replies. I also wonder if FDE could be used somewhere. But treating it as an integration problem is the main thing, I think.
Thanks, Phil