Modeling of Granular Mixing using Markov Chains and the Discrete Element Method
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Transcript of Modeling of Granular Mixing using Markov Chains and the Discrete Element Method
AIChE Annual Meeting, San Francisco. Nov. 2006 1
J. Doucet, N. Hudon, F. Bertrand and J. Chaouki
Modeling of granular mixing using Markov chains and the discrete element method
J. Doucet §, N. Hudon*, F. Bertrand § and J. Chaouki §
§ Department of Chemical Engineering
Ecole Polytechnique de Montréal, P.O. Box 6079, Station Centre-Ville,
Montréal, QC, Canada H3C 3A7
*Department of Chemical Engineering
Queen’s University, Kingston, ON,
Canada K7L 3N6
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Organization
Motivation and previous work Theory and definitions Application to a cylindrical drum Discussion
Effect of the time step of the chain Number of states Learning time Connection between Markov chain properties and mixing
Conclusion
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J. Doucet, N. Hudon, F. Bertrand and J. Chaouki
Motivation
Simulation of granular mixing are CPU intensive Deterministic vs probabilistic
Mixing may be viewed as the successive application of a transform (or mapping function) on a distribution Example: static mixer (Chen et al. 1972)
Markov chains have been introduced Can they help for granular mixing
simulation?
Static mixer
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J. Doucet, N. Hudon, F. Bertrand and J. Chaouki
Motivation
Markov chain needs the probabilities of transition between elements of a state space
By measuring these transitions from the flow, we can construct a stochastic process
The challenge: how to construct the process?
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J. Doucet, N. Hudon, F. Bertrand and J. Chaouki
Previous work on granular mixing using Markov chains 1 and 2D systems [Chen et al. (1972), Chou et al. (1977), Rippie
and Chou (1978), Aoun-Habbache et al. (2002), Berthiaux et al. (2005)]
No convergence analysis (is the method consistent?) No connection with current Markov chain properties to
investigate mixing The operator is constructed experimentally
Is it appropriate to speed up DEM mixing simulations?
Previous work
Time line
DEM feed inMarkov chain extrapolationChain training
Learning time Endpoint
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J. Doucet, N. Hudon, F. Bertrand and J. Chaouki
Theory and definitions
We want to determine the probability of moving from one place to another in a state space
It is assumed independent of time (stationarity)
A stochastic process (the evolution of the system)
A state space
A transition matrix
Probability of transition at iteration n from i to j
Probability measure
The particle is in state i at iteration n
The particle was in state j
at iteration n-1
For all n
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J. Doucet, N. Hudon, F. Bertrand and J. Chaouki
Theory and definitions
Example: displacement of tracers over a grid
42 tracers= 4/42
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J. Doucet, N. Hudon, F. Bertrand and J. Chaouki
Theory and definitions
Consider the N-dimensional state vector
By stationarity, the state of the system at time n is given by:
OperatorInitial probability distribution in S
Probability distribution after n iterations of
the map
How do we get this operator from experimental data?
Probability of being in state i at iteration l
i
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J. Doucet, N. Hudon, F. Bertrand and J. Chaouki
Construction of the operator
Probability of going from state i to j at time tn
Time average over NLT iterations
Indicator function (1 if p is in state i at time t)
Number of particles in i at time t
DEM feed inMarkov chain extrapolationChain training
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Application
Benchmark: cylindrical tumbler We investigate the impact of
The number of states The time step of the chain The learning time
As a measure of performance, we compare this DEM-based Markov chain method with:
RSD curves Segregation profiles obtained from the complete
DEM solution
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J. Doucet, N. Hudon, F. Bertrand and J. Chaouki
Effect of the number of states
Finer mesh
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J. Doucet, N. Hudon, F. Bertrand and J. Chaouki
Effect of the number of states
N=253
DEM
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J. Doucet, N. Hudon, F. Bertrand and J. Chaouki
Effect of the number of states
N=1813
DEM
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Effect of the number of states
N=3587
DEM
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Effect of the number of states
N=5595
DEM
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J. Doucet, N. Hudon, F. Bertrand and J. Chaouki
Effect of the time step and learning time
Rule of thumb: time step = time of autocorrelation of the system
Weak effect of the learning time
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Operator properties
Several properties can be extracted from P Invariant distribution (i.e. as , mixed state) Rate at which this distribution is reached (mixing
time) The dynamical properties (KS and topological
entropies)
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Operator properties
Assume that is the invariant distribution Order the eigenvalues of P such that
Denoting the second largest eigenvalue modulus (SLEM) by
We can show that (Diaconis et al. 1991):
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J. Doucet, N. Hudon, F. Bertrand and J. Chaouki
Operator properties
The mixing rate is defined as
The mixing time is defined as
For the drum problem:
Expected (P has spectral radius 1)SLEM
Mixing is mainly limited by axial diffusion in the tumbler
=175 rotations
Time for the distance to the invariant state to decrease by a factor e (2.718)
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Conclusion and future work
Parameters of the Markov process were investigated Time step, size of the state space, learning time
It was applied to monosized cohesionless particles in a 3D cylindrical drum RSD curves Segregation patterns Under certain conditions, the mixing mechanisms can be
described by a linear map Characterization of the operator
Example: mixing rate and mixing time Extrapolation of DEM data under certain condition using
Markov chain is feasible.
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Acknowledgements
Financial support of the Natural Science and Engineering Research Council of Canada
Financial support of the Research and Development center of Ratiopharm Operations
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Operator properties
Is there an invariant probability distribution? What happens to as ?
The answer is obtained by solving the eigenvalue problem
Since the spectral radius is 1, there is at least one eigenvalue = 1.
Left eigenvector Eigenvalue
Invariant distribution
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Operator properties
How fast does the system reaches this invariant distribution? Assume the eigenvalues of P being ordered as
Denoting the second largest eigenvalue modulus (SLEM) by
We can show that (Diaconis et al. 1991):
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Operator properties
Kolmogorov-Sinai entropy (Gaspard, 1998, Lecomte et al. 2005)
Consider the left and right eigenvectors of P:
Construct the matrix and the invariant vector
Kolmogorov-Sinai entropy (LB of sum of Lyapunov Exponents)
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Operator properties
Topological entropy (Balmforth et al. 1994)
Construct the transition matrix
The topological entropy is given by the logarithm of the largest eigenvalue of .
UB of the sum of Lyapunov Exponents
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Theory and definitions
Suppose the following system where P is the desired operator
Can we map the system evolution by a simple linear operator?