Journal of Applied Physiology, Vol 72, Issue 1 312-320, Copyright © 1992 by American Physiological Society
Gas dispersion in volume-cycled tube flow. I. Theory
D. Elad, D. Halpern and J. B. Grotberg
Biomedical Engineering Department, Robert R. McCormick School of Engineering and Applied Science, Northwestern University, Evanston, Illinois 60208.
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).
