Model Reduction techniques.

1


Model Reduction techniques. Applications to
reactor scale-up.
Evgeniy Redekop, Palghat Ramachandran CREL
Washington University in St.Louis, MO
Project Activities
Introduction
Application to Stirred-Tank reactors
- Oversimplified and phenomenological models
often fail to predict chemical reactor
behavior accurately.
- Formulate and solve a detailed model of a
single phase Stirred Tank reactor accounting
for micromixing
As a starting point of the project a single phase
Stirred-Tank reactor was chosen because of the
following
- 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.
- Use preferably an open source software (e.g.
OpenFOAM, Scientific Python)? for all numerical
computations, so that modular extensions are
possible
- 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
- Derive the reduced model using such techniques
as Spacial Averaging, Proper Orthogonal
Decomposition, etc.
- Reliable reduced models can facilitate the
development and scale-up of new efficient
processes.
- Compare the results given by the reduced model
to the results given by the original detailed
model and the Compartmental Model
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
- Such models should meet the following criteria
- Apply the work to the multi phase reactors
(liquid-gas, liquid-solid, fluidized beds)?
Derived from the detailed model based on the
'first principles'
Predict the reactor behavior well enough
Require reasonable computational time
D. J. Lamberto et al. / Ch. Eng. Sc. 56 (2001)?
Proper Orthogonal Decomposition
Spacial Averaging
Model Hierarchy
POD reduction of the model involves (Shvartsman,
1998)
CDR equations are averaged over the cross section
in which the local diffusion prevails over the
reaction.
1. Formation of a database ensemble of
spatiotemporal data (obtained from integration
of the full model or experimentally)
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.
2. Extraction of an empirical eigenfunction
basis from the data
Fluidized Bed snapshots
- The model can be truncated to an arbitrary
accuracy
3. Projection of the original model onto the
low-dimensional space of the eigenmodes
- The reduced model retains all parameters of
the original model and is valid for a wide
range in a parametric space
Some of the advantages include
- POD modes form an optimal basis for a
decomposition, i.e. no other orthonormal set
converges faster
Balakotaiah, et al. / Ch. Eng. Sci 58 (2003)?
First three POD eigenmodes
References
- Analyticity of the reduced model is
advantageous for model analysis
- No a priory knowledge of the time/space scale
separation is necessary
D. J. Lamberto, et al. , Computational analysis
of regular and chaotic mixing in a
stirred tank reactor, Ch. Eng. Sci., 56, (2001)
- The model can be truncated to an arbitrary
accuracy
Application to a tubular reactor the method
yields hyperbolic equations more accurately
describing dispersion effects than traditional
parabolic models
D. Guha, M. Dudukovic, and P. Ramachandran,
CFD-Based Compartmental Modeling of
Single Phase Stirred-Tank Reactors, AIChE, 52
(5), (2006)?
- The technique can utilize experimental data
along with numerical simulation
S. Y. Shvartsman and I. G. Kevrekidis, Nonlinear
Model Reduction for Control of Distributed
Systems a Computer-Assisted Study, AIChE, 44
(7), (1998)?
- The technique is proven to be useful in a
variety of engineering disciplines
Application to a Stirred-Tank reactor LS averaged
model accounts for a micromixing and correctly
predicts multiplicity of steady states for non
isothermal regime
P. G. Cizmas, et al., Proper-orthogonal
decomposition of spatiotemporal patterns
in fluidized beds, Ch. Eng. Sci., 58, (2003)?
- The method is applicable to the complex
geometries of the flow
S. Chakraborty and V. Balakotaiah, Spatially
Averaged Multi-Scale Models for Chemical
Reactors, Adv. in Ch. Eng., 30, (2005)?
Energy spectrum of POD basis
P. G. Cizmasa, 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)?