APPENDIX C: HEAT BALANCE INTEGRAL EQUATIONS
FOR SYNGENETIC PERMAFROST GROWTH
Method 1
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
T1
T
T
α1
-U 1 - 1 = 0
0≤x≤ X
(C1)
x2
x
t
T1( X, t) = Tf
(C1a)
T1(0, t) = Ts
(C1b)
2
T2
T
T
α2
X ≤ x ≤ X +δ
-U 2 - 2 = 0
(C2)
x2
x
t
T2 ( X, t) = Tf
(C2a)
T2 ( X + δ, t)
=G
(C2b)
x
T2 ( X + δ, t) = ( X + δ)G + To .
(C2c)
The initial temperature at the beginning of freeze is
Ti = To + Gx.
(C2d)
The energy balance at the phase change interface for the freeze process is
T
dX
T1
( X, t) - k2 2 ( X, t) + ρ2lU = ρ2l
k1
.
(C3)
x
dt
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 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)
T ( X, t) T1( X, t)
T1( X, t)
= ρ1lα1
-k1 1
+ k2 2
(C4)
x
x2
x
x
2
T ( X, t)
2
T1( X, t) T2 ( X, t)
T2 ( X, t)
= ρ2lα 2
-k1
+ k2 2
(C5)
.
x
x2
x
x
Quadratic temperature profiles in regions 1 and 2 that satisfy the boundary conditions are
2
x - X
x - X
T1 = Tf + a1X
+ (a1X - ∆T1)
(C6)
X
X
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