Finite Amplitude Steady-State One-Dimensional Waves in Fluidized Beds

Abstract

In this work, we investigate one-dimensional concentration instabilities that occur in fluidized beds. We use the averaged equations of motion for fluidized beds and use closure relations for the stress tensors available in the literature. A linear stability analysis is carried out in order to characterize the frequency, the propagation velocity, and the growth rates of small amplitude disturbances. A fully nonlinear transient numerical solution of the governing equations is also obtained. The linear and nonlinear growth and saturation of concentration waves as they saturate, i.e., as they reach a finite amplitude steady-state, is explored. The one-dimensional governing PDEs are recast into a nonlinear ODE in the frame of reference moving with the velocity of the saturated waves. We propose a numerical method to solve this eigenvalue problem, the result of which leads to the concentration profile, the wavelength, and the propagation velocity of the saturated waves. The results are compared with the predictions of the linear theory, with the fully nonlinear transient numerical simulations, and with the experimental data available. We explore some of the limits of validity of the linear theory and of the closure models.