Mesure et caractérisation du mélange dans les systèmes granulaires denses

This thesis develops fundamental tools for the characterization of mixing in the case of, without being limited to, granular flows.

The first chapter presents a method based on radioactive particle tracking (RPT) and this chapter generalizes this approach to complex systems of more than two (2) dimensions. Details of the theoretical approach and the algorithm are presented. The nature of the signal obtain enables a thorough analysis of the particle trajectories by recursing to chaos theory and Lagrangian turbulence (Ottino, 1989). We also present the details related to the computation of attractor dimensions, entropies and Lyapunov exponents by assuming that ergodicity as a fundamental hypothesis to extrapolate the Lagrangian measurements to the entire class of particles.

Elements from chaos theory however fail in providing mixing curves at the macroscopic scale (for both time and space). Current approaches for the calculation of the intensity of segregation are based on Danckwerts (1952). These indexes have major issues related to the invariance of the measure. This research presents two (2) new definitions for mixing for granular materials and a new measure is developed by using principal component analysis and provides an invariant measure of mixing with strong connection with space and properties of the particles. The derived measure is also found to be an upper bound of the mixing curve.

As a result from the presence of segregation in dense granular flows, conventional particle tracking methods are not very suitable for investigation of mixing systems dealing with large particle size distributions. This is due to the fact that the system is no longer ergodic as per classes of particle size are driven into specific subregions of the domain and remain trapped. This research developed a new approach which relies on a cluster of radioactive particles. These particles have the same particle size distribution as the inert particles in the system, which makes this approach more suitable for dense granular assemblies. As a result, the concentration of radioactive tracer introduced initially across the system can be precisely monitored with an error less than 0.005% in weight.

Second part of the thesis is oriented on mathematical models and simulation methods for mixing of dense granular systems. It discusses aspects of the verification and validation of the models using a conventional benchmark, the discharge of silos and hoppers. An important result shows that there is a selective sensitivity upon the particle-particle and particle-wall friction coefficient. These parameters were found to have the most sensitivity on the model. We also investigate the effect of the time step on the velocity profiles, granular temperature profiles and mixing curves. We conclude that the minimum time step derived from the Hertz contact law yields in wrong simulations results, even at a macroscopic scale, like the circulation time and the rate of decay of the mixing curve.

Finally, this thesis proposes a new model based on Markov chain theory to simulate mixing of granular particles. This approach is similar to that of the mapping functions. However, by using the Markov chain formalism, some important characteristics of mixing can be derived directly from the transition matrix. As a result, we show that the invariant state of the system (or well-mixed state) can be obtain from the eigenvectors of the Hamiltonian. The rate of mixing and the mixing time can be obtained from the second largest eigenvalue. Also, upper and lower bound for the sum of Lyapunov exponents are obtained from the K-S and Ruelle entropies that can be extracted from the operator (Gaspard, 1998).