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

Abstract

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.

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

“…(30) for λτ > 2 ln(2) and λ t = 2e ν sin(gν) + ge λτ −g − 2 sin(gν)e −ν + e ν (sin(gν) + g cos(gν)) + 1 (31) for λτ < 2 ln(2), where g = |1 − 4e −λτ | and ν = λτ 2 (note that these correct the expressions given in [22]). The two stationary quantities p o (λ) and j(λ) are obtained by inserting this result in Eqs.…”

“…An instantaneous blockage occurs if N particles are simultaneously present, and lasts for time τ b > τ . In the limit τ b → ∞, there is no steady state and the exiting flux falls to zero [22]. Here, we focus instead on reversible blockages, during which, the N particles are retained, and no more may enter the channel.…”

“…In this limit, the mean blockage time can be estimated by noting that a blockage occurs when a batch of particles enters in a finite duration τ , leading to t = τ (N −1)! (λτ ) N [22]. Expanding p o (λ) to first order gives…”

“…and For N = 3, the complete kinetic description of the model is cumbersome so we restrict our attention to the stationary quantities for which analytical expressions have been obtained [22]. In particular, the mean time to blockage starting from an empty channel is given by λ t = 2e ν sinh(gν) + ge λτ −g − 2 sinh(gν)e −ν + e ν (sinh(gν) + g cosh(gν)) +1…”

“…Particles enter at random times according to a Poisson process of intensity λ and exit, if no blockage occurs, after a fixed transit time τ . Subsequently, several generalizations were studied, including a higher blocking threshold (N > 2) [22], an inhomogeneous entering flux [23], and multiple channels [24,25]. When the blockage is reversible, the system is reactivated after a constant waiting time, τ b .…”