New Type of Draft Tube Spout-Fluid Bed. Part 2: Modeling and Design of the Acceleration Section of the Riser for the Pneumatic Transport of 1 mm Glass Spheres

A new type of draft tube spout-fluid bed (DTSFB) generically similar to a circulating fluidized bed (CFB) possesses a unique combination of capabilities as a fluid-particle contacting system or multiphase reactor. This paper focuses on the modeling of the acceleration section of the draft tube (a riser) of a DTSFB for the dilute phase pneumatic transport of 1 mm glass spheres in a turbulent air stream. An analytical expression for a dimensionless dynamic pressure gradient distribution in the acceleration section (y(ζ)) is derived from the solution of an isoperimetric problem of the calculus of variations. The gradient is shown to be a function of a single dimensionless parameter, which is the ratio of the dynamic pressure drop in the acceleration section and the dynamic pressure gradient in the nonacceleration section. It is shown that ∫0 1 y(ζ) dζ is maximized in the acceleration section of the riser. As a result, the pressure drop required to overcome the resisting forces in the acceleration section is minimized in any run. The pressure distribution obtained by integrating the pressure gradient is shown to be an excellent fit to experimental data. The modeling of the acceleration section is based on the one-dimensional, two-fluid continuity and momentum equations plus the variational pressure gradient distribution making it possible to calculate the solids fraction, gas and particle phase velocities, and effective drag coefficient distributions in the riser from five independent equations. The particle–wall friction factor in the model is calculated from a theory developed by the authors in a previous paper.