Multivariable optimization

[wildcat2]

I want to optimize a (polymer) process consisting of say 5 isothermal reactors in series (the actual process has a recycle stream and other complications but these can be dealt separately). The process is continuous.
How can I maximize the production of polymer by playing with the 5 reactor temperatures while keeping the product on spec ?
Reaction kinetics increase with temperature and decrease with polymer concentration. For a given configuration the reactors' volumes are known. The production in reactor i is Pi= Vi x ri where ri is the reaction rate in reactor i.
Quality is defined as Melt flow index (MFI) which for a given formulation depends on Molecular weight (MW) which decreases with temperature. (Assume that the relations with temperature are known). The overall MW is :
MW= Sigma (Pi x MWi)

I have heard of dynamic programming whereby you have to optimize from the end and progress backwards to the first stage (Bellman principle). But I don't know how to apply it.
I have also tried using Excel Solver but I don't think Solver is capable of tackling this kind of problem.

Can someone help ?

 

[jmw]

Somewhere there is a thread on Process Intensification which may have some of the answer for you. I think there is a technique based on Taguchi methods for optimising multivariable processes. Try this link:

http://www.rkroy.com/

but a web search under Taguchi and Burman may reveal more.
One thing is how you measure quality. Are you using an Polymer MFI sensor for feedback control? these are usually a short capillary device with a dP transmitter. This measures the viscosity which is a function of the molecular weight. Other methods involve disolving the polymer in solvent and measuring the resultant viscosity... Other applications are using rotational viscometers. MFI and rotational are useful because they can handle the process conditions right after the reactor which gives fast feedback.
I'd be interested in how you control your reactors.

 

[jmw]

PS there is a polymer forum where you might get a good response to your thread.

 

[wildcat2]

Thanks jmw for the tip on Taguchi. I will investigate.
For MFI we use the offline method. MFI test takes only 15 min. so it is not too difficult. For Polyethylene some people have used a benchtop NMR. Bruker claimed that they can determine MFI quite accurately. Such spectrometric method work on compositional change. I guess the basic premise is that there is not much variation in MW. In polymers the 2 key parameters affecting MFI are MW and oligomers/ mineral oil additive.
The solution viscosity you describe involves dilution of polymer in a solvent and measuring the resulting viscosity to estimate MW. This may take longer than the MFI test although the determination of MW is faster than by GPC.
The measurement of viscosity as a measure of MFI is at best tricky as the temperature/ shear correction is not easy.

I have operated a Rheometric n-line melt indexer whereby polymer melt is pumped through an instrument. We ended scrapping this expensive instrument because the reading is way off real MFI. It can give relative variation.

 

[CARF]

It might be a good idea to visit the Mathworks site, excellent software for dynamic (process) modelling.

http://www.mathworks.com/

 

[MRJDEIMOS]

If you are interested to get a DECHEMA PDF about this topic get in touch with me. The BASELL group presented such kind of application for their Spheripol Process Extruder. It is based data driven process models. For this they are using historical data and offline laboratory analysis to build this kind of nonlinear models. As described the methode starts from the end of the process. During validation Root square error of 98 and 96 could be archived.

more info could be ordered from m.jaehnel@atlan-tec.com

 

[m777182]

It will not be so easy as it looks like. You will need a lot of basic data about your material and equipment properties, which, I am sure, you do not have at hand now.Further you will need a good instrumentation to follow the output properties and maybe the MFI is not the only one.Next you should get good kinetic models of your process.Then you will have to play with heat transfer within your reactor- delta Ts', Re numbers, quality of mixing etc. At the end you will start optimizing the last stage of your reactors and then proceed towards the first one. Local feedbacks make things even more difficult from the viewpoint of convergence of the mathematical solution. The procedure is called Bellman's dynamic programming. If you are not really familiar with programming tools(matlab or similar), if you have not spent some time with kinetics and you are new in heat transfer and mixing technologies, it would be better to find some speciallists with reputation on this area. The best you can do then is to define and explain exactly what you would like to do. Your contribution to the team work will be then providing reliable process data and you will get them using Taguchi,factorial,central composite or whatever experimental design. If you succeed in getting the data your partners will appreciate your work. And do not forget, that experimenting for yourself is most difficult because if you cheat or trim results you will have to live with it.
m777182

 

[wildcat2]

Thanks m777182:
The optimization model I am working on is deterministic (based on first principles) and not of the data mining type (which relies on actual plant data). Of course I have to make certain simplifying assumptions (temperature uniformity within a CSTR, transverse uniformity in a PFR, good mixing etc...). The application of the Bellman's principle as you stated is far from simple, especially when the model is non-linear and non-analytic.
One way to apply dynamic programming in this context has been championed by Prof. Luus (U.Toronto). He was kind enough to provide the Fortran code for his method. However the task of linking to my Excel spreadsheet is too daunting.

Thanks again for your input.