Fully developed travelling wave solutions and bubble formation in fluidized beds

It is well known that most gas fluidized beds of particles bubble, while most liquid fluidized beds do not. It was shown by Anderson, Sundaresan & Jackson (1995), through direct numerical integration of the volume-averaged equations of motion for the fluid and particles, that this distinction is indeed accounted for by these equations, coupled with simple, physically credible closure relations for the stresses and interphase drag. The aim of the present study is to investigate how the model equations afford this distinction and deduce an approximate criterion for separating bubbling and non-bubbling systems. To this end, we have computed, making use of numerical continuation techniques as well as bifurcation theory, the one- and two-dimensional travelling wave solutions of the volume-averaged equations for a wide range of parameter values, and examined the evolution of these travelling wave solutions through direct numerical integration. It is demonstrated that whether bubbles form or not is dictated by the value of Omega = (rho(s) nu(t)(3)/Ag)(1/2), where rho(s) is the density of particles, nu(t) is the terminal settling velocity of an isolated particle, g is acceleration due to gravity and A is a measure of the particle phase viscosity. When Omega is large (> similar to 30), bubbles develop easily. It is then suggested that a natural scale for A is rho(s) nu(t) d(p), so that Omega(2) is simply a Froude number.