where e7 is defined as
e7 (T1) = e6 - eo ν1[1 - (a4 / a3 )eo ] / [1 - (a4 / a3 )eos ]
(208)
^
eos = eo (Ts ).
(209)
It follows from eq 207 that α1g > α1s if e7 is positive. This implies that there exists a point
(α1g, αo) on L- such that W1 vanishes and that W1 may be negative for α1 > α1g. When W1 is
c
˙
negative, then g2 (T1) is positive and the uniqueness of the solution is not warranted.
Hence, if e7 is positive, the solution of Proposition 15 is unique under condition given as
e6(T1) < 0 and e7(T1) > 0 for T1 < Ts and α1 ≤ α1g.
(210)
On the other hand, if e7 ≤ 0, then α1g ≤ α1s. This clearly implies that W1 > 0 on L- and that a
c
solution of Proposition 15 is unique. We will present our finding by the following proposi-
tion.
Proposition 16
The solution of Prop. 15 is unique if either e6 (T1) ≥ 0 or e7 (T1) ≤ 0 for T1 < Ts. When e7 (T1)
> 0 for T1 < Ts , the solution is unique if eq 210 holds true.
APPLICATIONS
We will describe the use of traveling wave solutions obtained above for the empirical
verification of the model M1 below. It is known (Andersland and Anderson 1978) that the
empirically determined function ν (T) under equilibrium conditions takes a form given as:
- A1
ν(T ) = Ao T
(211)
T <0
for
where Ao and A1 are positive constants. Experimental methods were proposed to deter-
mine K1 (Williams and Burt 1974, Horiguchi and Miller 1983) and K2 (Perfect and Williams
1980). Horiguchi and Miller (1983) empirically found that K1 of several frozen porous me-
dia also takes the same form as eq 211. Since ν and K1 are known to be bounded, we will
use forms given as
A ≤T <0
wo
ν(T ) =
(212)
if Vo > 0
w (A / T )b3
A >T
o
A ≤T <0
Ko
K1(T ) =
(213)
K (A / T )b1
A >T
o
where A is a small negative number, b1 and b3 are positive numbers. When Vo = 0, ν (T) is
not needed in the balance equations of mass and heat. However, Ki (i = 1,2) implicitly takes
account of the composition of the frozen fringe.
Recently, Nakano and Takeda (1994) empirically found that K2 (T) of Kanto loam can be
described in the same form as eq 211 for T < Tσ. Using eq 37, we will describe K2 as
K20 = γKo
A ≤T <0
K2 (T ) = γ K1(T )
Tσ ≤ T < A
(214)
K (A / T )b2
T < Tσ .
20
26
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