Gas dispersion in volume-cycled tube flow. I. Theory

A mathematical theory is derived for the dispersion of a contaminant bolus introduced into a fully developed volume-cycled oscillatory pipe flow. The convection-diffusion equation is solved for a tracer gas bolus by expressing the local concentration field as a series expansion of derivatives of the area-averaged concentration. The local, as well as the area-averaged, concentration is determined for a uniform initial slug or Gaussian bolus. The effect of various flow parameters such as Womersley parameter, Schmidt number, and tidal volume is investigated. The overall dispersion is characterized by a time-averaged effective diffusion coefficient, which for long duration coincides with previous dispersion theories based on a constant linear axial concentration profile. The effective diffusion coefficient can be determined from the local time history of concentration, independent of the spatial location or the initial tracer bolus. Furthermore the local peaks of the concentration-time curve follow a decaying curve dictated by the time-averaged effective diffusion coefficient. Thus the theory is directly applicable for dispersion measurements in oscillatory tube flows, a basis for the pulmonary airways application, as shown by Gaver et al. (J. Appl. Physiol. 72: 321-331, 1992).