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# A model for a network of conveyor belts with discontinuous speed and capacity

• * Corresponding author: Adriano Festa
This work was partially supported by the Haute-Normandie Regional Council via the M2NUM project and the project GO 1920/7-1 by the German Research Foundation (DFG).
• We introduce a macroscopic model for a network of conveyor belts with various speeds and capacities. In a different way from traffic flow models, the product densities are forced to move with a constant velocity unless they reach a maximal capacity and start to queue. This kind of dynamics is governed by scalar conservation laws consisting of a discontinuous flux function. We define appropriate coupling conditions to get well-posed solutions at intersections and provide a detailed description of the solution. Some numerical simulations are presented to illustrate and confirm the theoretical results for different network configurations.

Mathematics Subject Classification: Primary: 90B30; Secondary: 35L65, 65M25.

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• Figure 1.  A conveyor belt in a brewery. Image courtesy of Sidel Blowing & Services SAS

Figure 2.  Characteristics in the non-congested case

Figure 3.  Trajectories in the congested case

Figure 4.  Solution in the congested case: evolution of three shock waves

Figure 5.  Scheme of the two cases considered of one-to-two junction: passive (left) and active (right)

Figure 6.  Choice of the merging parameter $q$

Figure 7.  Regularized flux function $f_{ \xi, i}$

Figure 8.  Test 1: non-congested case with $a_{{1}} = 1$ and $a_{{2}} = 2$

Figure 9.  Test 2: congested case with $a_{{1}} = 2$ and $a_{{2}} = 1$

Figure 10.  Test 2: space-time diagram for the congested case

Figure 11.  Test 3: "passive" junction with distribution parameter $\mu = 0.5$

Figure 12.  Test 4: "active" junction with distribution parameter $\mu = 0.5$

Figure 13.  Test 5: merging junction with parameter $q = 0.3$

Table 1.  Decreasing step sizes (left), decreasing smoothing parameter $\xi$ (right)

 $\Delta x$ $\Delta t$ error $\xi$ error 0.1 $2 \cdot 10^{-4}$ 0.0842 $5 \cdot 10^{-2}$ 0.0051 0.05 $\phantom{2 \cdot }10^{-4}$ 0.0381 $2 \cdot 10^{-2}$ 0.0042 0.01 $5 \cdot 10^{-5}$ 0.0184 $\phantom{1 \cdot} 10^{-2}$ 0.0039 0.01 $2 \cdot 10^{-5}$ 0.0073 $5 \cdot 10^{-3}$ 0.0037 0.005 $\phantom{2 \cdot }10^{-5}$ 0.0057 $2 \cdot 10^{-3}$ 0.0035
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