from Figure 2 that both water and Br moved from the
t0
Wi = ∫ Fw (i)dt
warmer to cooler parts in the soil columns. The unfro-
(5)
zen part (segments 110) of soil columns lost both wa-
0
ter and Br . It is interesting that a few segments in the
frozen part near the 0C isotherm also lost both water
t0
and Br and that the minimum value of C + appears in
Bi = ∫ FB (i)dt
^
(6)
these frozen segments.
0
where t0 is the duration of the experiment (22 days).
If we assume that Br is transported mainly by the
ANALYSIS OF DATA
movement of unfrozen water, the mass flux FB of Br is
We will consider the balance of water and Br in the
given as
part of soil columns Vi consisting of segments 1i. We
will assume that the initial dry density of each segment
FB (i) = Ci Fw (i)
(7)
is equal to the final dry density of each segment mea-
sured at the end of the experiment. The loss of water Wi
where Ci is the concentration of Br in unfrozen water
from the part Vi during the experiment is calculated as
for segment i. From eq 6 and 7 we obtain
(
)
i
Wi = v∑ ρ j w0 - wj
(3)
t0
Bi = ∫ Ci FB (i)dt.
j=1
(8)
0
where v is the volume of each segment and ρj and wj are
the dry density and the water content of segment j re-
For a special case in which Ci remains constant at the
spectively. Similarly, the loss of Br , Bi, from the part Vi
initial value C0, from eq 5 and 7 we obtain
during the experiment is calculated as
Bi = C0Wi .
(9)
(
)
i
Bi = v∑ ρ j C0 - Cj
^
^
(4)
It follows from eq 9 that both Wi and Bi attain their
j=1
maxima at the same segment. This obviously did not
^
where Cj is the Br content of segment j.
occur in our experiments.
The calculated values of Wi for experiments 14 are
Since Br is excluded from growing ice and is con-
plotted together with those for experiments 58 in Fig-
fined to unfrozen water, the concentration of Br , Ci,
must be a nondecreasing function of i for i ≥ 10 when a
ure 3, where open circles are the data points from ex-
periments 14, while solid circles are the data points
linear temperature field is established at the beginning
from experiments 58. For instance, in Figure 3a the
of experiments. Because of the Br exclusion from ice,
calculated values of Wi for experiment 1 are plotted to-
Ci increases with i in segments where ice is present; as
gether with those for experiment 5. Experiments 1 and
a result, the segment number of the maximum Bi be-
5 are conducted under the same conditions, except that
comes greater than that of the maximum Wi.
Br is absent in experiment 1. Suppose that the soil was
The Br ion may not be completely confined to un-
packed uniformly in each column and that Br does not
frozen water. However, let us assume this complete
affect the movement of unfrozen water. Then it is an-
confinement, as it appears to be a good approximation.
ticipated that the calculated values of Wi for experiment
We will also assume that the phase equilibrium of wa-
1 should be equal to those for experiment 5. Since the
ter holds true in the experiments. Then, the unfrozen
uniform packing of soil is quite difficult, Figure 3 indi-
water content is approximated by the equilibrium un-
cates that the level of Br content used in experiments
frozen water content determined by the nuclear mag-
58 does not significantly affect the mobility of unfro-
netic resonance technique on a separate sample of
zen water.
Morin clay. Under these assumptions the con-
The calculated values of Wi and Bi for experiments
centrations of Br in unfrozen water at both the begin-
58 are presented in Figure 4. It is easy to see from
ning and the end of the experiments were calculated.
these figures that both Wi and Bi attain their maxima
The results of calculations for experiments 58 are
and that the segment number of the maximum Bi is
presented in Figure 5, where nondimensional quanti-
+
greater than that of the maximum Wi. Let Fw(i) and
ties C0 and C+(t) are defined as the concentrations of
FB(i) be mass fluxes of water and Br , respectively,
Br in unfrozen water divided by C0 at the beginning
from segment i to segment i + 1, then Wi and Bi are
and the end of the experiment respectively. For in-
+
given as
stance, the calculated values of C0 and C+(t) for ex-
5
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