PII S0730-725X(97)00240-3
● Original Contribution
PARAMAGNETIC TRACER CONCENTRATION EVOLUTION BY NMR
RELAXATION TIME MAPPING: APPLICATION TO ARIS-TAYLOR DISPERSION
Y.E. KUTSOVSKY,* V. ALVARADO,* L.E. SCRIVEN,* H.T. DAVIS,* AND B.E. HAMMER†
*Department of Chemical Engineering and Materials Science, and †Department of Radiology, University of Minnesota,
Minneapolis, MN 55455
A procedure to study tracer dispersion was proposed and tested for the case of tracer spreading in tube flow.
Concentration maps of paramagnetic tracers Gd
3
were measured in time through direct measurements of spin
lattice relaxation time T
1
obtained by using a two-point stimulated echo pulse sequence. The procedure was used
to test the linear dependence of Peclet number on inverse velocity in the range of flow rates 0.3–1.2 cc/min.
© 1998 Elsevier Science Inc.
INTRODUCTION
The spreading of particles in fluids under the combined
action of convection and molecular diffusion is called
dispersion. The phenomenon occurs, for example, when
two fully miscible fluids are brought into contact and the
concentration fronts are spread by convection and mo-
lecular diffusion.
1
Such spreading is of great importance
in technological applications such as oil recovery, envi-
ronmental science and chemical engineering.
Dispersion in steady flow is the subject here. It is
commonly studied by using tracers. Once steady-state
flow with mean velocity U is established, a tracer is
released in the fluid at the inlet section of the sample of
interest and the mean tracer concentration is measured as
a function of time or position. A typical dispersion ex-
periment is sketched in Fig. 1. The output profile is a
function of the shape of the input profile and the tracer
spreading by flow and diffusion in the sample.
The evolution of tracers in the typical dispersion
experiment is commonly described by the one-dimen-
sional convection-dispersion equation
C
t
+ U
C
z
= D
2
C
z
2
(1)
This equation looks like the convection-diffusion equa-
tion, but D is not the diffusivity. It is the axial dispersion
coefficient. This analogous quantity is a strong function
of both the flow properties and the molecular diffusivity.
The concentration C is the cup-mixed mean, i.e., the
cross-sectional average not the local concentration value.
Very few data have been published that yield the value of
the dispersion coefficient. One well-known case is tracer
dispersion in Poiseuile flow, which was analyzed theo-
retically by Taylor, 1953
2
and Aris, 1956.
3
This case is
commonly referred to as Taylor dispersion or Taylor-
Aris dispersion. With the contribution of axial molecular
diffusion included, Taylor’s formula for the axial disper-
sion coefficient D in a tube of circular cross-section with
radius R is:
D = D
m
+
U
2
R
2
48D
m
(2)
D
m
is the molecular diffusivity.
Acoustic
4
or electrical
5
detection techniques are often
employed to measure tracer concentration, but their spa-
tial resolution is limited to a few millimeters. To over-
come limitations in spatial resolution, nuclear magnetic
resonance (NMR) has been used to track paramagnetic
agents in liquid.
6,7
Concentration of paramagnetic tracers
can be deduced from signal intensity, which is a function
of spin-lattice relaxation time, T
1
.
8
The ability to determine dispersion properties of a
RECEIVED 7/31/96; ACCEPTED 8/29/97.
Address correspondence to Dr. B. Hammer, University
of Minnesota, Cntr. Interdisciplinary Appl. Mag. Res.,
Department of Radiology, Box 292 Mayo, 420 Delaware
Street, S.E., Minneapolis, MN 55455 USA. E-mail:
hammer@ciamr.umn.edu
Magnetic Resonance Imaging, Vol. 16, No. 1, pp. 63–71, 1998
© 1998 Elsevier Science Inc. All rights reserved.
Printed in the USA.
0730-725X/98 $19.00 + .00
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