Page 663 - Bird R.B. Transport phenomena

§20.5 'Taylor Dispersion" in Laminar Tube Flow 643

Fig. 20.5-1. Sketch show-
I Parabolic velocity ing the axial spreading of
Pulse I ' z = 0 profile in a circular tube z = L a concentration pulse in
injected ' \ Taylor dispersion in a
here
circular tube.

t = l/<v z >

§20.5 "TAYLOR DISPERSION" IN LAMINAR TUBE FLOW

Here we discuss the transport and spreading of a solute "pulse'' of material A intro-
duced into fluid В in steady laminar flow through a long, straight tube of radius R, as
shown in Fig. 20.5-1. A pulse of mass m A is introduced at the inlet 2 = 0 over a very short
period near time t = 0, and its progress through the tube is to be analyzed in the long-
time limit. Problems of this type arise frequently in process control (see Problem 20C.4),
1
medical diagnostic procedures, and in a variety of environmental applications. 2
A short distance downstream from the inlet, the 0-dependence of the mass fraction
distribution will die out. Then the diffusion equation for a> (r, z, t) in Poiseuille flow with
A
constant \x, p, and ЯЬ takes the form
АВ
(20.5-1)

This equation is to be solved with the boundary conditions

B.C. 1 and 2: at r = 0 and at r = R, -^ = 0 (20.5-2)
oY
which express the radial symmetry of the mass fraction profile and the impermeability of
the tube wall to diffusion. For this long-time analysis it is not necessary to specify the exact
shape of the pulse injected at t = 0. No exact analytical solution is available for the mass
fraction profile u> {r, z, t)—even if an initial condition were clearly formulated—but Tay-
A
lor ' 3 4 gave a useful approximate analysis that we summarize here. This involves getting
from Eq. 20.5-1 a partial differential equation for the cross-sectional average mass fraction
Г27Г (R
o) r dr d6
A
Jp JQ
= — (o r dr (20.5-3)
A
rdrdB
which can then be solved to describe the behavior at long times.

1 J. B. Bassingthwaighte and C. A. Goresky, in Section 2, Volume 3 of Handbook of Physiology, 2nd
edition, American Physiological Society, Bethesda, Md. (1984).
2 P. С Chatwin and C. ML Allen, Ann. Rev. Fluid Mech., 17,119-150 (1985); В. Е. Logan,
Environmental Transport Processes, Wiley-Interscience, New York (1999), Chapters 10 and 11; J. H.
Seinfeld, Advances in Chemical Engineering, Academic Press, New York (1983), pp. 209-299.
3
G. I. Taylor, Proc. Roy. Soc. A219,186-203 (1953).
4
G. I. Taylor, Proc. Roy. Soc, A225, 473^77 (1954).