T2 (X, t) = Tf
T2 ( X + δ, t)
= G.
(2b)
x
The initial temperature at the beginning of freeze is
Ti = To + Gx .
(2c)
The maximum depth at any time to which the temperature disturbance will be felt is X + δ. Then
T2 (X + δ, t) = (X + δ)G + To.
(2d)
The energy balance at the phase change interface for the freeze process is
T
T
dX
k1 x1 (X , t) - k2 2 (X , t) = ρ2l
.
(2e)
x
dt
The energy balance at the freezing front can also be written as two equations (Lunardini 1981b)
T (X , t)
T2 (X , t) T1(X , t)
T1(X , t)
2
2
+ k2
-k1 1
= ρ1lα1
(3)
x2
x
x
x
2
T1( X, t) T2 ( X, t)
T ( X, t)
T2 ( X, t)
2
= ρ2lα 2
-k1
+ k2 2
x
(4)
.
x2
x
x
Because of the initial temperature distribution, during freeze the heat flow to the interface from the
thawed region will exceed the geothermal heat flow until equilibrium is established. Likewise, during a
thaw period the heat flow from the thawed zone will be less than the deep geothermal heat flow.
An approximate solution to this problem will be obtained using the heat balance integral technique (see
Lunardini 1991). In this method, the differential equations are solved on average over a finite volume of
material rather than at each point of the region. The integration of the energy equations over the regions
where temperature changes are occurring, 0 ≤ x ≤ X + δ, detailed by Lunardini (1981b) is
d
X+δ
X
∫
∫ T2 ( x, t)dx - ρ1lX + (ρ2c2 - ρ1c1)Tf X
ρ1c1 T1( x, t)dx + ρ2c2
dt
X
0
T (0, t)
- ρ2c2 ( X + δ)To + ( X + δ) = -k1 1
G
+ k2G .
(5)
x
2
Quadratic temperature profiles in regions 1 and 2 that satisfy the boundary conditions are chosen as
2
x - X
x - X
+ (a1X - ∆T1)
T1 = Tf + a1X
(6)
X
X
(x - X) 2
(
)
x-X
[
]
T2 = Tf + G(δ + 2 X ) + 2∆T
- GX + ∆T
(7)
δ
δ2
where
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