Multivariable optimization
Multivariable optimization
(OP)
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 ?
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 ?





RE: Multivariable optimization
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.
RE: Multivariable optimization
RE: Multivariable optimization
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.
RE: Multivariable optimization
http://www.mathworks.com/
RE: Multivariable optimization
more info could be ordered from m.jaehnel@atlan-tec.com
RE: Multivariable optimization
m777182
RE: Multivariable optimization
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.