Using eq 5, we will obtain the balance equation of heat for the mixture (Nakano 1990)
given as
(
)
q=L λ +z
(6)
2
x
where z (x,t) is defined as
T
T
+ (c1 - c2 )T λ 2 - ∑ ρiuici
Lz = -c
(7)
t
x
i
c = c1ρ1 + c2ρ2 + c3ρ3 .
(8)
We will consider a special case in which a frost front x = n0 (t) moves with a constant
speed, namely
d
-
no (t) = -no = Vo ≥ 0.
˙
(9)
dt
Hereafter we will exclude the case of negative Vo where melting occurs. We will introduce
a new independent variable ξ defined as
ξ = x - not - no (0).
˙
(10)
For the sake of convenience we will define new dependent variables f1 (ξ) and f2 (ξ) as
f1 = ρ1(u1 u3)
(11)
f2 = ρ2(u2 u3).
(12)
Therefore, fi (i = 1,2) is the mass flux of either B1 or B2 relative to the mass flux of soil
particles. Using eq 10, 11 and 12, we reduce eq 3 to
(ρ1V)′ = - f1′ - λ 2
(13)
(ρ V )′ = - f ′ + λ
(14)
2
2
2
(ρ V )′ = 0
(15)
3
where primes denote differentiation with respect to ξ and V (ξ) is defined as
V = u3 - no .
˙
(16)
Similarly we will reduce eq 6 and 7 to
(
)
q′ = - (kT ′)′ = L λ 2 + z
(17)
(
)
(
)
Lz = - c1 f1 + c2 f2 + cV T ′ + c1 - c2 λ 2T.
(18)
QUASI-STEADY PROBLEM
A freezing soil may be considered to consist of three parts: the unfrozen part Ro, the
frozen fringe R1 and the frozen part R2, as shown in Figure 1. We will also make seven
assumptions:
4
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