1 1
b1 =  +
+ C21φ
3 ST
1
b2 =  C21φ.
3
The derivatives of β and g can be found from the following equations
a +a a
dβ
= β′ = 5 1 4
(C14)
dσ
a3 + a2a4
dg
= g′ = a1  a2β′
(C15)
dσ
where
(σ + φ)β(β + 2)
α 21
a1 =
1 
m
m
α 21(σ + φ)
[2σ(β + 1) + 2φ]
a2 =
m2
1 2ρ21(g  1)
a3 =

 k21σ
g
ST
2ρ
1
a4 = 21 + 2 β
ST
g
a5 = k21(β + 2)
[
]
m = β σ(β + 2) + 2φ .
The problem has now been reduced to a simple numerical quadrature of eq C12 using the auxiliary rela
tions of eq C13C15. A FORTRAN program to carry out the integration is listed as PFTSYNB.FOR in Ap
pendix E.
Phase change model verification
A simplification of this problem can be solved in a closed form. Consider the case of a soil initially
thawed at Tf and with a zero geothermal gradient G. The problem is then one of a single phase only with eq
C1, C1a,b, C3, C4, and C9 governing the freeze process. The temperature is chosen as
2
x  X  c P2 x  X
T = Tf + P
(C16)
X 2l X
where
l
P=
R = 1 + 2ST  1.
R,
c
The location of the freeze interface is given by
K3 K2 X + K3
K1
t=
X 
ln
(C17)
K2
K3
K2
31