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

[object Object],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 ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Motivation ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Static mixer

Motivation ,[object Object],[object Object],[object Object]

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Previous work Time line DEM feed in Markov chain extrapolation Chain training Learning time Endpoint

Theory and definitions ,[object Object],[object Object],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

Theory and definitions ,[object Object],42 tracers = 4/42

Theory and definitions ,[object Object],[object Object],Operator Initial 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 t n Time average over N LT iterations Indicator function (1 if p is in state i at time t ) Number of particles in i at time t DEM feed in Markov chain extrapolation Chain training

Application ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Effect of the number of states Finer mesh

Effect of the number of states N=253 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 ,[object Object],[object Object],[object Object],[object Object]

Operator properties ,[object Object],[object Object],[object Object],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 ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Acknowledgements ,[object Object],[object Object]

Operator properties ,[object Object],[object Object],[object Object],[object Object],Left eigenvector Eigenvalue Invariant distribution

Theory and definitions ,[object Object],Can we map the system evolution by a simple linear operator?

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

Recommandé

Recommandé

Contenu connexe

Tendances

Tendances (20)

En vedette

En vedette (20)

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

Similaire à Modeling of Granular Mixing using Markov Chains and the Discrete Element Method (20)

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