We will formulate the basic equations in terms of a convective system with mass flowing through the

stationary upper surface at constant velocity *U*, as shown in Figure 14. The governing equations are

2

α1

-*U * 1 - 1 = 0

0≤*x*≤ *X*

(C1)

(C1a)

(C1b)

2

α2

-*U * 2 - 2 = 0

(C2)

(C2a)

=*G*

(C2b)

(C2c)

The initial temperature at the beginning of freeze is

(C2d)

The energy balance at the phase change interface for the freeze process is

( *X*, *t*) - *k*2 2 ( *X*, *t*) + ρ2l*U *= ρ2l

.

(C3)

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 per-

iod, the heat flow from the thawed zone will be less than the deep geothermal heat flow. The energy bal-

ance at the freezing front can also be written as two equations (Lunardini 1981b)

2

2

*T *( *X*, *t*)

= ρ1lα1

-*k*1 1

+ *k*2 2

(C4)

2

*T *( *X*, *t*)

2

= ρ2lα 2

-*k*1

+ *k*2 2

(C5)

.

Quadratic temperature profiles in regions 1 and 2 that satisfy the boundary conditions are

2

+ (*a*1X - ∆*T*1)

(C6)

* X*

* X*

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