Mathematical description of the hydrodynamic regimes of an asymptotic model for two-phase flow arising in PFBC boilers.

摘要

Two-phase systems where a dense phase of small particles is fluidized with a gas flow appear in many industrial applications, among which the fluidized bed combustors are probably the most important. A homogenization technique allows us to formulate the mathematical model in form of the compressible Navier-Stokes system type with some particularities: 1) the volumetric fraction of the dense phase (analogous to the density in the Navier-Stokes equations) may vanish, 2) the constitutive viscosity law may depend in a nonlinear form on this density, 3) the source term is nonlinear and coupled with state equations involving drag forces and hydrodynamic pressure, and 4) the state equation for the collision pressure of dense phase blows up for finite values of the density. We develop a rigorous theory for a special kind of solutions we call stationary clouds. Such solutions exist only under restrictions on the geometry of combustor and on the boundary conditions that usually meet in engineering applications. In return, these solutions have a stationary one-dimensional structure very simple and, from them, it is possible to reconstruct much of the dynamics of the whole system, responding to most of the practical issues of interest. Finally, we study the linear stability for the trivial solutions corresponding to uniform fluidized states injecting plane wave perturbations in our equations. Depending on the parameters of the equations of state describing the collisions between solid particles, hydrodynamic pressure, and the values of blowing boundary condition, we can draw detailed abacus separating stable regions of unstable regions where bubbles appear. Then, we use the dispersion relations of this multidimensional linearized model, combined with the stationary phase theorem, to approach the profiles and the evolution of the bubbles appearing in unstable regimes, and verify that the obtained results adjust to the observations.