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APPENDIX C: HEAT BALANCE INTEGRAL EQUATIONS FOR SYNGENETIC PERMAFROST GROWTHcontinue
Table C1. Comparison of closed solution (G = 0) and numerical quadrature (G = 0.0001)
CR95_08
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1 1
b
1
=  +
+
C
21
φ
3
S
T
1
b
2
= 
C
21
φ.
3
The
derivatives
of β
and
g
can
be
found
from
the
following
equations
a
+
a a
d
β
= β′ =
5 1 4
(C14)
d
σ
a
3
+
a
2
a
4
dg
=
g
′ =
a
1

a
2
β′
(C15)
d
σ
where
(
σ + φ
)
β
(
β + 2
)
α
21
a
1
=
1 
m
m
α
21
(
σ + φ
)
[
2σ
(
β + 1
)
+ 2φ
]
a
2
=
m
2
1 2ρ
21
(
g  1
)
a
3
=


k
21
σ
g
S
T
2ρ
1
a
4
=
21
+
2
β
S
T
g
a
5
=
k
21
(
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
T
f
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 P
2
x

X
T
=
T
f
+
P
(C16)
X
2l
X
where
l
P
=
R
= 1 + 2
S
T
 1.
R
,
c
The
location of
the
freeze
interface
is
given
by
K
3
K
2
X +
K
3
K
1
t
=
X

ln
(C17)
K
2
K
3
K
2
31
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