# Stochastic Models for Transport in a Fluidized Bed

In this paper we study stochastic models for the transport of particles in a fluidized bed reactor and compute the associated residence time distribution (RTD). Our main model is basically a diffusion process in [0,A] with reflecting/absorbing boundary conditions, modified by allowing jumps to the origin as a result of transport of particles in the wake of rising fluidization bubbles. We study discrete time birth-death Markov chains as approximations to our diffusion model. For these we can compute the particle distribution inside the reactor as well as the RTD by simple and fast matrix calculations. It turns out that discretization of the reactor into a moderate number of segments already gives excellent numerical approximations to the continuous model. From the forward equation for the particle distribution in the discrete model we obtain in the diffusion limit a partial differential equation for the particle density p(t,x) $\frac{\partial}{\partial t} p(t,x) =\frac{1}{2} \frac{\partial^2}{\partial x^2}[D(x) p(t,x)] -\frac{\partial}{\partial x}[v(x) p(t,x)] -\lambda(x) p(t,x)$ with boundary conditions $p(t,1)=0$ and $\frac{1}{2}\frac{\partial}{\partial x}[D(x) p(t,x)]_{|x=0}-v(0)p(t,0) -\int_0^1 \lambda(x)p(t,x) dx =0.$ Here v(x) and D(x) are the velocity and the diffusion coefficients and $\lambda(x)$ gives the rate of jumps to the origin.\par We also study a model allowing a discrete probability of jumps to the origin from the distributor plate, thus incorporating the experimentally observed fact that a fixed percentage of particles gets caught in the wake of gas bubbles during their formation at the bottom of the reactor. It turns out that this effect contributes to an extra term to the boundary condition at x=0.

Finally, we model the particle flow in the wakes of rising fluidization bubbles and derive $\lambda(x)$ as well as v(x). We compare our results with experimental data.

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