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 enough Require 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-Tank reactor 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 comparison A Compartmental model of a single phase Stirred-Tank reactor was recently proposed by Guha, et. al. (2006) which can be used as a reference point for an evaluation of the results 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 analysis Application 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) References D. 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)‏