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



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



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



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?



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



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




Example: displacement of tracers over a grid

42 tracers= 4/42




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



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



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



Effect of the number of states

Finer mesh




N=253

DEM




N=1813

DEM




N=3587

DEM




N=5595

DEM



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



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)



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):



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)



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.



Acknowledgements

Financial support of the Natural Science and Engineering Research Council of Canada

Financial support of the Research and Development center of Ratiopharm Operations





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



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):



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)



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




Suppose the following system where P is the desired operator

Can we map the system evolution by a simple linear operator?

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Modeling of Granular Mixing using Markov Chains and the Discrete Element Method

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