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A non-local model for heterogeneous material flow on conveyor belts

Paola Goatin, Simone Göttlich, Fabian Ziegler

TL;DR

The paper addresses a two-dimensional system of non-local conservation laws for heterogeneous particle flow on bounded domains, incorporating boundary effects in the non-local flux through an augmented density $r_\Omega$. It develops a Roe-type finite-volume scheme with dimensional splitting and proves convergence to an entropy weak solution, establishing positivity, $L^1$/$L^\infty$/$BV$ bounds and discrete entropy inequalities, as well as Lipschitz dependence on initial data. A key novelty is handling the flux discontinuity via a smoothed Heaviside activation and embedding boundaries within the non-local terms, extending scalar results to a multi-class, obstacle-bearing setting. Numerical experiments with two particle classes and a diverter demonstrate size-based segregation and mixing that align with microscopic simulations, supporting the method's relevance for conveyor-belt technologies with heterogeneous materials.

Abstract

In this paper, a finite volume approximation scheme is used to solve a non-local macroscopic material flow model in two space dimensions, accounting for the presence of boundaries in the non-local terms. Based on a previous result for the scalar case, we extend the setting to a system of heterogeneous material on bounded domains. We prove the convergence of the approximate solutions constructed using the Roe scheme with dimensiona splitting, where the major challenge lies in the treatment of the discontinuity occurring in the flux function. Numerical tests show a good agreement with microscopic simulations.

A non-local model for heterogeneous material flow on conveyor belts

TL;DR

The paper addresses a two-dimensional system of non-local conservation laws for heterogeneous particle flow on bounded domains, incorporating boundary effects in the non-local flux through an augmented density . It develops a Roe-type finite-volume scheme with dimensional splitting and proves convergence to an entropy weak solution, establishing positivity, // bounds and discrete entropy inequalities, as well as Lipschitz dependence on initial data. A key novelty is handling the flux discontinuity via a smoothed Heaviside activation and embedding boundaries within the non-local terms, extending scalar results to a multi-class, obstacle-bearing setting. Numerical experiments with two particle classes and a diverter demonstrate size-based segregation and mixing that align with microscopic simulations, supporting the method's relevance for conveyor-belt technologies with heterogeneous materials.

Abstract

In this paper, a finite volume approximation scheme is used to solve a non-local macroscopic material flow model in two space dimensions, accounting for the presence of boundaries in the non-local terms. Based on a previous result for the scalar case, we extend the setting to a system of heterogeneous material on bounded domains. We prove the convergence of the approximate solutions constructed using the Roe scheme with dimensiona splitting, where the major challenge lies in the treatment of the discontinuity occurring in the flux function. Numerical tests show a good agreement with microscopic simulations.
Paper Structure (15 sections, 9 theorems, 110 equations, 4 figures)

This paper contains 15 sections, 9 theorems, 110 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Existence
  4. Positivity
  5. L 1 bound
  6. L infinity bound
  7. BV bound
  8. Discrete entropy inequalities
  9. Lipschitz continuous dependence on initial data
  10. Numerical results
  11. Test setting
  12. Discretisation and solution properties
  13. Conclusion
  14. Acknowledgements
  15. Technical Lemma

Key Result

Theorem 2.2

Let $\boldsymbol{\rho}_o \in (\mathbf{L^\infty} \cap \mathbf{BV}) (\Omega; {\mathbb{R}}^N_+)$ for $\Omega\in{\mathbb{R}}^2$. Let assumptions vs, H and eta hold. Then, for all $T>0$, there exists a unique entropy weak solution $\boldsymbol{\rho} \in (\mathbf{L^\infty} \cap \mathbf{BV}) ([0,T] \times where $\mathcal{C}^c_\infty$ is defined in eq:Cinf, $\mathcal{K}^c_1$ is defined in eq:K1defroe, $\

Figures (4)

  • Figure 1: Schematic view of the static field of the conveyor belt.
  • Figure 2: The smaller particles (blue) start first and are pushed to the edge of the obstacle by the coming larger particles (red).
  • Figure 3: The larger particles (red) start first with the following smaller particles (blue) creeping through the mass of larger particles.
  • Figure 4: Outflow diagram for both test cases. By penetrating the mass of large particles, most of the mass of smaller particles leaves the area around the obstacle faster than the larger particles even if the large particles start first.

Theorems & Definitions (11)

  • Definition 2.1
  • Theorem 2.2
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Proposition 3.5
  • Remark 3.6
  • Corollary 3.7
  • Lemma 3.8
  • Proposition 3.9
  • ...and 1 more