9 - Pipe Bends, Secondary Flows, and Fluid Heterogeneities

Abstract

This chapter develops the mathematical ideas needed to model flow in pipelines where the radius of curvature of the axis significantly exceeds duct cross-dimensions. For smaller radii of curvature, secondary viscous flows in the cross-plane appear, in addition to centrifugal effects. Bends in pipelines and annuli are interesting because they are associated with losses; that is, to maintain a prescribed volume flow rate, a greater pressure drop is required in pipes with bends than those without. This is true because the viscous stresses that act along pipe walls are higher. Newtonian calculations similar to those performed for concentric plate Pouseuille flow show that, when pressure gradient is prescribed, volume flow rate again decreases as the radius of curvature tends to zero. The velocity and stress solutions obtained are also useful in determining how and where debris settles within the duct. Numerous factors enter that include particle size, shape and distribution, buoyancy effects, local velocities, and gradients. It is suggested that results will vary for pipelines with noncircular cross-sections, non-Newtonian flow, or both.