Model Reduction techniques. Applications to reactor scale-up.

Evgeniy Redekop, Palghat Ramachandran

CREL Washington University in St.Louis, MO

Proper Orthogonal DecompositionSpacial Averaging

- Oversimplified and phenomenological models often fail to predict chemical reactor behavior accurately.- The detailed models of chemical reactors (CFD, CDR, etc.) require enormous computational time and are infeasible for optimization, scale-up, and control tasks. This is especially true in case of reactors characterized by complex reaction schemes and multiphase reactors.

- Reliable reduced models can facilitate the development and scale-up of new efficient processes.

- Formulate and solve a detailed model of a single phase Stirred Tank reactor accounting for micromixing

Project Activities

- Derive the reduced model using such techniques as Spacial Averaging, Proper Orthogonal Decomposition, etc. - Compare the results given by the reduced model to the results given by the original detailed model and the Compartmental Model- Apply the work to the multi phase reactors (liquid-gas, liquid-solid, fluidized beds)

- Such models should meet the following criteria:Derived from the detailed model based on the 'first principles'Predict the reactor behavior well enoughRequire reasonable computational time

- Use preferably an open source software (e.g. OpenFOAM, Scientific Python) for all numerical computations, so that modular extensions are possible

As a starting point of the project a single phase Stirred-Tankreactor was chosen because of the following:- While it is relatively simple device, the flow in it can exhibit a complex behavior effecting the reactor performance- Extensive literature on the subject is available containing both experimental and numerical data for comparisonA Compartmental model of a single phase Stirred-Tank reactor was recently proposedby Guha, et. al. (2006) which can be used asa reference point for an evaluation of theresults of this project

Fluidized Bed snapshots

First three POD eigenmodes

D. J. Lamberto et al. / Ch. Eng. Sc. 56 (2001)

POD reduction of the model involves: (Shvartsman, 1998)

2. Extraction of an empirical eigenfunction basis from the data 3. Projection of the original model onto the low-dimensional space of the eigenmodes

- No a priory knowledge of the time/space scale separation is necessary

- POD modes form an optimal basis for a decomposition, i.e. no other orthonormal set converges faster

- The model can be truncated to an arbitrary accuracy - The technique can utilize experimental data along with numerical simulation - The technique is proven to be useful in a variety of engineering disciplines

- The method is applicable to the complex geometries of the flow

Some of the advantages include:

P. G. Cizmasa, et al. / Ch. Eng. Sci 58

(2003)

CDR equations are averaged over the cross section in which the local diffusion prevails over the reaction.

- The model can be truncated to an arbitrary accuracy- The reduced model retains all parameters of the original model and is valid for a wide range in a parametric space

- Analyticity of the reduced model is advantageous for model analysisApplication to a tubular reactor the method yields hyperbolic equations more accurately describing dispersion effects than traditional parabolic models

Application to a Stirred-Tank reactor LS averaged model accounts for a micromixing and correctly predicts multiplicity of steady states for non isothermal regime

Introduction

Energy spectrum of POD basis

Lyapunov-Schmidt theory was suggested by Balakotaiah, et. al. (2005) as a unified framework for model reduction via spacial averaging. The method was applied to tubular, stirred-tank, monolith, and other reactor types.

Application to Stirred-Tank reactors

1. Formation of a database ensemble of spatiotemporal data (obtained from integration of the full model or experimentally)

ReferencesD. J. Lamberto, et al. , “Computational analysis of regular and chaotic mixing in a stirred tank reactor”, Ch. Eng. Sci., 56, (2001) D. Guha, M. Dudukovic, and P. Ramachandran, “CFD-Based Compartmental Modeling of Single Phase Stirred-Tank Reactors”, AIChE, 52 (5), (2006)S. Y. Shvartsman and I. G. Kevrekidis, “Nonlinear Model Reduction for Control of Distributed Systems: a Computer-Assisted Study”, AIChE, 44 (7), (1998)P. G. Cizmas, et al., “Proper-orthogonal decomposition of spatiotemporal patterns in fluidized beds”, Ch. Eng. Sci., 58, (2003)S. Chakraborty and V. Balakotaiah, “Spatially Averaged Multi-Scale Models for Chemical Reactors”, Adv. in Ch. Eng., 30, (2005)

Model Hierarchy

Balakotaiah, et al. / Ch. Eng. Sci 58

(2003)

V. Balakotaiah, et al., “Averaging theory and low-dimensional models for chemical reactors and reacting flows”, Ch. Eng. Sci., 58, (2003)