(13b) Modeling of the Acceleration Section of a Pneumatic Transport Riser Using the Variational Calculus Pressure Gradient Profile

A second generation (type 2) draft tube spout fluid bed is developed by separating the draft tube (riser) and annulus effluents to provide a means of independently varying the fluid flow rate through and the pressure drop across the riser while maintaining independent conditions in the annulus moving packed bed. The result of the separation of the two effluents is a device that is generically similar to a circulating fluidized bed, however it has more operational flexibility. The type 2 DTSFB allows the solids circulation rate to be determined by two independent conditions in the non-acceleration section of the riser. As a result, the riser is central to the design and operation of a type 2 DTSFB.

Particles undergoing pneumatic transport experience a significant acceleration length (H_ra) in the riser resulting in a changing particle fraction and velocity. Closure of the one-dimensional two fluid model may be obtained by adding a function that describes the dynamic pressure gradient in the acceleration section and specifying three independent variables in the non-acceleration section of the riser.

The theoretically predicted and experimentally verified pressure gradient profile obtained from a solution of an isoperimetric problem of the variational calculus describes the experimentally obtained dynamic pressure gradient profile using only two readily measured physically relevant pressure measurements (P(0)-P(1)  and P’(1)). The acceleration parameter that results from taking the ratio of the dynamic pressure drop across and the pressure gradient at the end of the acceleration section determines the shape of the pressure gradient function.

The variational pressure gradient distribution describes a minimized pressure gradient over the length of the acceleration section. While the variational function does not describe the mechanics of the process, it does demonstrate that the particles and fluid interact to minimize the pressure gradient throughout the acceleration process. The variational pressure gradient distribution then provides a fifth relationship that allows the voidage profile to be determined from specifying three independent variables, such as gas and particle mass flow rates. 

An additional expression obtained from the variational pressure gradient distribution provides a means of predicting the solids circulation rate using a correlation between the normalized inertial term and the acceleration parameter.  This provides a means for determining the operation of a type 2 DTSFB using relatively few process inputs while eliminating the need for specifying the drag coefficient.