Non-markovian models of blocking in concurrent and countercurrent flows

Recurrence dynamics of particulate transport with reversible blockage: From a single channel to a bundle of coupled channels

We model a particulate flow of constant velocity through confined geometries, ranging from a single channel to a bundle of N_{c} identical coupled channels, under conditions of reversible blockage. Quantities of interest include the exiting particle flux (or throughput) and the probability that the bundle is open. For a constant entering flux, the bundle evolves through a transient regime to a steady state. We present analytic solutions for the stationary properties of a single channel with capacity N≤3 and for a bundle of channels each of capacity N=1. For larger values of N and N_{c}, the system's steady state behavior is explored by numerical simulation. Depending on the deblocking time, the exiting flux either increases monotonically with intensity or displays a maximum at a finite intensity. For large N we observe an abrupt change from a state with few blockages to one in which the bundle is permanently blocked and the exiting flux is due entirely to the release of blocked particles. We also compare the relative efficiency of coupled and uncoupled bundles. For N=1 the coupled system is always more efficient, but for N>1 the behavior is more complex.

Stochastic models of multi-channel particulate transport with blockage

Particle conveying channels may be bundled together. The limited carrying capacity of the constituent channels may cause the bundle to be subject to blockages. If coupled, the blockage of one channel causes an increase in the flux entering the others, leading to a cascade of failures. Once all the channels are blocked, no additional particles may enter the system. If the blockages are of finite duration, the system reaches a steady state with an exiting flux that is reduced compared to the incoming one. We propose a stochastic model consisting of N _c channels, each with a blocking threshold of N particles. Particles enter the system's open channels according to a Poisson process, with an equally distributed input flux of intensity Λ. In an open channel the leading particle exits at a rate μ and a blocked channel unblocks at a rate [Formula: see text], where [Formula: see text]. We present and explain the methodology of an analytical description of the behavior of bundled channels. This leads to exact expressions for the steady-state output flux, for [Formula: see text], which promises to extend to arbitrary N _c and N. The results are applied to compare the efficiency of conveying a particulate stream of intensity Λ using a single, high capacity (HC) channel with multiple channels of a proportionately reduced low capacity (LC). The HC channel is more efficient at low input intensities, while the multiple LC channels have a higher throughput at high intensities. We also compare [Formula: see text] coupled channels, each of capacity N  =  2 with the corresponding number of independent channels of the same capacity. For [Formula: see text], if [Formula: see text], the coupled channels are always more efficient. Otherwise the independent channels are more efficient for sufficiently large Λ.

Cascading blockages in channel bundles

Flow in channel networks may involve a redistribution of flux following the blockage or failure of an individual link. Here we consider a simplified model consisting of N(c) parallel channels conveying a particulate flux. Particles enter these channels according to a homogeneous Poisson process and an individual channel blocks if more than N particles are simultaneously present. The behavior of the composite system depends strongly on how the flux of entering particles is redistributed following a blockage. We consider two cases. In the first, the intensity on each open channel remains constant while in the second the total intensity is evenly redistributed over the open channels. We obtain exact results for arbitrary N(c) and N for a system of independent channels and for arbitrary N(c) and N=1 for coupled channels. For N>1 we present approximate analytical as well as numerical results. Independent channels block at a decreasing rate due to a simple combinatorial effect, while for coupled channels the interval between successive blockages remains constant for N=1 but decreases for N>1. This accelerating cascade is due to the nonlinear dependence of the mean blocking time of a single channel on the entering particle flux that more than compensates for the decrease in the number of active channels.

Generalized model of blockage in particulate flow limited by channel carrying capacity

We investigate stochastic models of particles entering a channel with a random time distribution. When the number of particles present in the channel exceeds a critical value N, a blockage occurs and the particle flux is definitively interrupted. By introducing an integral representation of the n-particle survival probabilities, we obtain exact expressions for the survival probability, the distribution of the number of particles that pass before failure, the instantaneous flux of exiting particles, and their time correlation. We generalize previous results for N=2 to an arbitrary distribution of entry times and obtain exact solutions for N=3 for a Poisson distribution and partial results for N≥4.